# The polarization of the solar Ba ii D1 line with partial frequency
redistribution
and its magnetic sensitivity

Ernest Alsina Ballester Instituto de Astrofísica de Canarias, E-38205 La
Laguna, Tenerife, Spain  Departamento de Astrofísica, Universidad de La
Laguna, E-38206 La Laguna, Tenerife, Spain  Tanausú del Pino Alemán Instituto
de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain  Departamento
de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain
Javier Trujillo Bueno Instituto de Astrofísica de Canarias, E-38205 La Laguna,
Tenerife, Spain  Departamento de Astrofísica, Universidad de La Laguna,
E-38206 La Laguna, Tenerife, Spain  Consejo Superior de Investigaciones
Científicas, Spain

###### Abstract

We investigate the main physical mechanisms that shape the intensity and
polarization of the Ba ii D1 line at $4934$ Å via radiative transfer numerical
experiments. We focus especially on the scattering linear polarization arising
from the spectral structure of the anisotropic radiation in the wavelength
interval spanned by the line’s hyperfine structure (HFS) components in the odd
isotopes of barium. After verifying that the presence of the low-energy
metastable levels only impacts the amplitude, but not the shape, of the D1
linear polarization, we relied on a two-term atomic model that neglects such
metastable levels but includes HFS. The D1 fractional linear polarization
shows a very small variation with the choice of atmospheric model, enhancing
its suitability for solar magnetic field diagnostics. Tangled magnetic fields
with strengths of tens of gauss reduce the linear polarization and saturation
is reached at roughly $300$ G. Deterministic inclined magnetic fields produce
a $U/I$ profile and, if they have a significant longitudinal component, a
$V/I$ profile, whose modeling requires accounting for HFS and the Paschen-Back
effect. Because of the overlap between HFS components, the magnetograph
formula cannot be applied to infer the longitudinal magnetic field. Accurately
modeling the D1 intensity and polarization requires an atomic system that
includes the metastable levels and the HFS, the detailed spectral structure of
the radiation field, the incomplete Paschen-Back regime for magnetic fields,
and an accurate treatment of collisions.

Radiative transfer – Scattering – Polarization – Sun: atmosphere

##  1 Introduction

High-precision spectropolarimetric observations in quiet regions close to the
solar limb reveal a wealth of linearly polarized features in spectral lines,
known as the second solar spectrum (e.g., Ivanov, 1991; Stenflo & Keller,
1997). Such polarization signals arise from the scattering of anisotropic
radiation (i.e., scattering polarization) within the solar atmosphere. Through
measurements of the scattering polarization, valuable information on the
properties of the solar atmosphere can be obtained. Indeed, the scattering
polarization in spectral lines is generally sensitive to the magnetic field
via the Hanle effect (e.g., Stenflo, 1994; Landi Degl’Innocenti & Landolfi,
2004), which enables practical diagnostics of solar magnetic fields,
especially in the upper chromosphere and transition region as well as in solar
prominences or spicules (e.g., the review by Trujillo Bueno & del Pino Alemán,
2022), or at sub-resolution scales (e.g., Trujillo Bueno et al., 2004) which
cannot easily be accessed with more widespread techniques such as those based
on the Zeeman effect.

The D lines of Ba ii encode valuable information on the atmospheric properties
of the lower solar chromosphere (see Appendix B for the formation height of
D1). Over the last two decades, a considerable volume of spectropolarimetric
observations of the Ba ii D2 lines has been acquired (e.g., Faurobert et al.,
2009; López Ariste et al., 2009; Ramelli et al., 2009) and several theoretical
investigations on its large scattering polarization signal and its sensitivity
to the Hanle effect have been carried out (e.g., Belluzzi et al., 2007b;
Faurobert et al., 2009; Smitha et al., 2013). The first observations of the
linear polarization of the solar Ba ii D1 line revealed two positive peaks,
with the blue (red) one above (below) the continuum level (see Stenflo &
Keller 1997; also Stenflo et al. 2000). The fact that, in quiet regions, this
line and the D1 line of Na i did not simply present a depolarized feature was
regarded as surprising, because the upper and lower fine-structure (FS) levels
of these lines have angular momentum $J=1/2$ (i.e., they are intrinsically
unpolarizable). A compelling explanation for these features was eventually put
forward by Belluzzi & Trujillo Bueno (2013), which relied on the HFS present
in all sodium isotopes and in the odd barium isotopes (135Ba and 137Ba, which
represent $18\%$ of the total). Their modeling took into account the frequency
correlations between the incident and scattered radiation, that is, including
partial frequency redistribution (PRD) effects. Thus, they could account for
the spectral structure of the anisotropic radiation field over the wavelength
intervals spanned by these lines’ HFS components and showed that this gives
rise to linear polarization signals comparable to the observed ones.
Subsequently, Alsina Ballester et al. (2021) modeled the Na i D lines
accounting for this spectral structure and additionally considering the
frequency redistribution effects of elastic collisions and magnetic fields.
Their calculations showed that scattering polarization signals of substantial
amplitude can be produced in the intrinsically unpolarizable D1 lines, even in
the presence of gauss-strength magnetic fields typical of the quiet Sun.
Moreover, the D1 line was also shown to be sensitive to such magnetic fields,
adding to its diagnostic interest.

In this work, we carry out an analogous investigation for the Ba ii D1 line in
which, for the first time, we jointly account for scattering polarization with
PRD effects, the HFSof the atom, quantum interference between atomic states
belonging to the same term, and magnetic fields of arbitrary strength. The D
lines of both Ba ii and Na i originate from resonance transitions between a
lower $s$ term and an upper $p$ term. The upper FS levels of the D1 and D2
lines have $J=1/2$ and $3/2$, respectively, and the ground term has a single
FS level with $J=1/2$. Nevertheless, there are important differences between
the atomic structure of the two atomic species. The separation between the
upper levels of the Ba ii D1 line at $4934$ Å and of the D2 line at $4554$ Å
is much larger than the corresponding separation for the case of the Na i
atom. Moreover, for the isotopes with nuclear spin, the HFS splitting of the
upper and lower levels of Ba ii is roughly one order of magnitude larger than
that of the corresponding levels of Na i. It is also noteworthy that the Ba ii
system has a metastable term $5d\,^{2}\mathrm{D}$, whose two FS levels have
significantly lower energies than the upper term of the D lines. To our
knowledge, the impact of the metastable levels on the D1 intensity and
polarization has not been studied to date.

This paper is organized as follows. In Section 2, we introduce the basic
assumptions and the most general atomic model considered in this work. In
Section 3, we theoretically study the impact on the D1 line of the metastable
levels. Building on these results, Section 4 is focused on a series of
numerical experiments on the Ba ii D1 line, in which the metastable levels are
neglected. We study how the intensity and polarization patterns of the line
are impacted by the HFS splitting and the quantum interference between HFS and
FS levels, as well as the sensitivity of the line to different atmospheric
models and to magnetic fields, both isotropically distributed and
deterministic. Conclusions are outlined in Section 5. Information about the
atomic quantities used in this work and additional figures, including those
pertaining to the Ba ii D2 line, can be found in the appendices.

##  2 Formulation of the problem

Our theoretical investigation aims at highlighting the impact of various
physical mechanisms on the intensity and polarization of the Ba ii D1 line.
Such investigations are based on a series of spectral syntheses, obtained
through the numerical solution of the radiative transfer (RT) problem out of
local thermodynamical equilibrium (LTE) conditions. We account for PRD
effects, in order to suitably account for the spectral structure of the
incident radiation field, which can introduce a linear polarization signal in
the D1 line as explained in Belluzzi & Trujillo Bueno (2013). For the sake of
reducing computational cost, we decouple the angular and frequency dependence
introduced by the Doppler effect in scattering processes by making the angle-
averaged (AA) approximation (Rees & Saliba, 1982), except where otherwise
noted. We considered one-dimensional (1D) semiempirical atmospheric models,
namely those introduced in Fontenla et al. (1993) — hereafter FAL models — and
the M${}_{\mbox{\scriptsize{CO}}}$ model of Avrett (1995) — hereafter FAL-X.
In particular, we considered the FAL-C model except where otherwise noted. The
lines of sight (LOS) for which we show the synthetic profiles are given by
$\mu=\cos\theta$, where $\theta$ is the heliocentric angle. In order to
consider scattering polarization profiles of substantial amplitude, we take
$\mu=0.1$ except where otherwise noted. In all cases, we take the reference
direction for positive $Q$ to be parallel to the nearest limb.

Figure 1: Grotrian diagram for the most general Ba ii model considered in this
work, which includes three terms: $6s\,^{2}\mathrm{S}$ (ground term),
$5d\,^{2}\mathrm{D}$ (metastable term), and $6p\,^{2}\mathrm{P}^{\mathrm{o}}$
(upper term for the D lines). The latter two consist of two FS levels each.
The figure also displays the HFS for the case of the odd isotopes (energies
not to scale). The solid lines show the permitted radiative transitions that
couple the atomic states of the system. The red and blue lines indicate the
transitions pertaining to the D1 and D2 lines, respectively.

Except where otherwise noted, we considered the seven stable isotopes of
barium throughout this work. Of these, only the two odd isotopes (135Ba and
137Ba) have a nonzero nuclear spin with $I=3/2$ and thus HFS. Relevant atomic
quantities, including the isotopic abundances and shifts and the HFS
coefficients can be found in Appendix A. Outside sunspots, Ba i is a minority
species and we thus consider all barium atoms to be in the Ba ii and Ba iii
stages. In the most general case, we consider five FS levels of Ba ii, namely
$6s\,^{2}\mathrm{S}_{1/2}$ (the ground level), $5d\,^{2}\mathrm{D}_{3/2}$ and
$5d\,^{2}\mathrm{D}_{5/2}$ (the metastable levels), and
$6p\,^{2}\mathrm{P}^{\mathrm{o}}_{1/2}$ and
$6p\,^{2}\mathrm{P}^{\mathrm{o}}_{3/2}$ (the upper levels of D1 and D2,
respectively). The other levels in this ionization stage, which have at least
twice the energy of the $6p$ term, are not considered in this work. The
Grotrian diagram for this atomic model, including the HFS, can be found in
Figure 1. Each HFS level is indicated by its corresponding quantum number $F$.
We observe that the energies of the HFS levels of the
$5d\,^{2}\mathrm{D}_{5/2}$ metastable level decrease with $F$, as a
consequence of its negative magnetic dipole HFS coefficient $\mathcal{A}$ (see
Appendix A). The solid lines connecting the various $F$ levels indicate the
permitted radiative transitions between them, with red and blue lines
pertaining to the D1 and D2 lines, respectively. We only account for the
ground level of Ba iii, which we consider suitable for determining the
ionization balance.

The results of the RT calculations and the corresponding analysis are
presented in the following two sections. In Section 3, we study the impact of
the metastable levels on the D1 linear polarization in the absence of magnetic
fields, using the HanleRT (del Pino Alemán et al., 2016, 2020) synthesis code.
After establishing that the metastable levels modify the amplitude but not the
shape of the D1 scattering polarization, in Section 4 we study the impact of
the HFS, the atmospheric model, and the magnetic fields on the polarization
patterns of the D1 line, neglecting the metastable levels. Such numerical
investigations were carried out using the RT code for a two-term model
introduced in Alsina Ballester et al. (2022).

##  3 The impact of the 5$d$ metastable levels in the zero-field case

At present, no RT code exists that can simultaneously account for PRD effects,
the five above-mentioned atomic levels of Ba ii, the quantum interference
between levels belonging to the same term, the HFS, and magnetic fields in the
incomplete Paschen-Back (IPB) effect regime. However, we can still gain
valuable insights into the physics that shape the intensity and polarization
patterns of the Ba ii D lines by employing different numerical approaches that
can each account for most of the aforementioned phenomena.

In the present subsection, we made use of the HanleRT numerical code for the
synthesis of the intensity and polarization of the D lines. The HanleRT code
accounts for scattering processes with PRD effects following the formalism
introduced by Casini et al. (2014, 2017a, 2017b) and, in its present version,
can consider multi-term 111In contrast to a multi-level atomic model, a multi-
term model accounts for the quantum interference between different FS levels
of the same term (see, e.g., Section 7.5 of Landi Degl’Innocenti & Landolfi,
2004). atomic systems without HFS. A multi-level modeling that includes the
HFS and the quantum interference between the $F$ levels pertaining to a given
$J$ level can be achieved with HanleRT by making the formal substitutions
$S\rightarrow I$, $J\rightarrow F$, and $L\rightarrow J$, considering the HFS
splitting introduced in Appendix A. This treatment neglects quantum
interference between FS levels which, as confirmed in 4.2, is a reasonable
assumption. This approach is otherwise correct in the absence of magnetic
fields. However, in the presence of magnetic fields, the same substitution
leads to an incorrect expression of the magnetic Hamiltonian (e.g., Janett et
al., 2023). Thus, the calculations with HanleRT presented in this work are
restricted to the nonmagnetic case.

For the calculations carried out with HanleRT, we considered only the five
most abundant isotopes; we neglected the contribution from the two least
abundant stable isotopes because data on the isotopic shifts of their
corresponding metastable levels is, as far as we are aware, not presently
available. The abundance of the remaining five isotopes was adjusted
accordingly. We expect the error incurred to be negligible, because of the low
abundance of the omitted isotopes which, moreover, have no nuclear spin.

The population of the ground term, $\mathcal{N}_{\ell}$, was kept fixed during
the iterative solution of the non-LTE RT problem, while letting the overall
population of the Ba ii levels evolve freely. The reason for this is two-fold.
First, the population and ionization balance is calculated considering only
the 138Ba isotope. This population is then distributed among the isotopes
according to their abundances and, for those with HFS, the populations in a
given FS level are distributed among the $F$ levels according to their
statistical weight. Secondly, the metastable levels are critical for the
population balance of the atom. If we were to completely fix the populations,
we would be prescribing the populations in the $F$ levels whereas, if we were
to leave them completely free (thus ensuring mass conservation), we would not
be able to analyze the actual impact on the linear polarization, because the
population balance will significantly change the intensity profile. We
consider this a reasonable approximation because the population of the lower
level is much larger than that of the other levels of the system.

Figure 2: Stokes $I$, normalized to the continuum intensity $I_{\mathrm{c}}$
(upper panel), and fractional linear polarization $Q/I$ profiles (lower panel)
of the D1 line as a function of wavelength. The synthetic profiles are
obtained from calculations using the HanleRT code, accounting for partial
frequency redistribution (PRD) effects in the angle-averaged (AA) case and
accounting for hyperfine structure (HFS) as discussed in the text. The black
and red curves correspond to calculations including and neglecting the
metastable levels, respectively. For all the figures presented in the main
text, the spectral range is $1.2$ Å wide and is centered on the D1 line. A
line of sight (LOS) with $\mu=0.1$ is taken and the reference direction for
positive Stokes $Q$ is parallel to the nearest limb.

In Figure 2, we compare the D1 profiles obtained when including and excluding
the $5d\,^{2}\mathrm{D}$ metastable levels in the atomic model. The figure
shows the intensity normalized to the continuum intensity at $2$ Å to the red
of the line center, $I_{c}$, and the fractional linear polarization pattern
$Q/I$. We find an absorption profile in $I/I_{c}$, which is clearly broadened
due to the HFS. The inclusion of the metastable levels does not appear to have
any impact on the intensity profile (as long as the fixed lower term
population is calculated considering the full atomic model); see also Appendix
C, where the same behavior is found in the corresponding profile for D2.

The $Q/I$ pattern presents a positive blue peak and a negative red one, whose
amplitudes decrease when accounting for the metastable levels. This may be
attributed to the transfer of population imbalances and quantum interference
between magnetic sublevels (i.e., atomic polarization) from the
$6p\,{}^{2}\mathrm{P}^{\mathrm{o}}_{1/2}$ upper level to the
$5d\,^{2}\mathrm{D}_{3/2}$ metastable level. Indeed, we note that the
$6p\,{}^{2}\mathrm{P}^{\mathrm{o}}_{1/2}$ level only presents atomic
polarization due to the spectral structure of the incident radiation field.
The overall shape of the profile obtained when including or neglecting the
metastable levels is very similar, with the blue (red) peak remaining positive
(negative). This similarity suggests that, if the aim is to qualitatively
study the sensitivity of these profiles to specific physical mechanisms such
as those driven by the magnetic field, one can reasonably model the Stokes
profiles of the D1 line with a two-term atomic model that neglects the
metastable levels (but accounts for the atomic HFS). Indeed, this is the
atomic model that is considered in the following sections of this work.
Regardless, one must be aware that suitably reproducing spectropolarimetric
observations of the Ba ii lines will require the inclusion of such metastable
levels. We note that the shape of the D2 scattering polarization profile is
modified by the metastable levels to a far greater degree than that of D1. The
discussion of the D2 line can be found in Appendix C.

HanleRT can also solve the non-LTE RT problem for polarized radiation
accounting for PRD effects while relaxing the AA approximation (i.e., fully
accounting for the frequency-angular coupling due to the Doppler effect).
Figure 3 shows the comparison between the fractional linear polarization $Q/I$
profiles resulting from calculations with and without the AA approximation.
For such calculations, we considered a three-level atomic system (i.e.,
without the metastable levels) with HFS. We find a good agreement between the
two calculations, which highlights the suitability of the AA approximation for
modeling the linear polarization pattern of the Ba ii D1 line, at least in the
absence of magnetic fields. This contrasts with the results of the analogous
investigation for the Na i D1 line reported in Janett et al. (2023), in which
such approximation was found to have a clear impact on the shape of the $Q/I$
profile. Although it is not shown here, were able to reproduce such findings
in the nonmagnetic case using HanleRT. Such differences may be attributed to
the fact that the HFS splittings of the upper and lower levels of the Ba ii D1
line (for the isotopes with nonzero nuclear spin) are more than one order of
magnitude larger than those of the corresponding levels of Na i. The
separation between the HFS components of the Ba ii D1 line is proportionally
larger, reducing potential spectral overlaps between them due to the Doppler
effect. Making the full angle-dependent treatment of scattering processes
likewise has no impact on the intensity profile of the Ba ii D1 line. The
synthetic profiles presented in the rest of this work were obtained under the
AA approximation.

Figure 3: Fractional linear polarization $Q/I$ profiles for the Ba ii line as
a function of wavelength. The synthetic profiles were computed using HanleRT,
accounting for PRD effects both under the AA approximation (black curve) and
considering the fully angle-dependent case (red curve). In the atomic model,
the metastable levels were neglected but the HFS was taken into account.

##  4 The impact of the HFS, atmospheric model, and magnetic fields

In this section we continue investigating the formation of the intensity and
polarization profiles of the Ba ii D1 line in optically thick atmospheres. The
numerical approach considered here assumes a two-term atomic model and does
not allow for the inclusion of the metastable levels, whose impact on the
amplitude of the D1 scattering polarization is not negligible. On the other
hand, it does allow for investigations accounting for the quantum interference
between states within the same FS and/or HFS levels while in the presence of
magnetic fields of arbitrary strength and orientation.

###  4.1 Numerical approach

The synthetic Stokes profiles presented in this section were obtained through
the following two-step approach.

Step 1: Compute a number of quantities to be used as input for the second
step, including the collisional rates, continuum quantities, and the
population of the ground term $\mathcal{N}_{\ell}$. Such calculations are
carried out by solving the non-LTE problem without polarization using the RH
code of Uitenbroek (2001). The considered atomic system includes the ground
level of Ba iii and the five levels of Ba ii discussed in Section 2, but not
the HFS. Because the metastable levels are included in the calculations in
this step, they yield a more accurate value for $\mathcal{N}_{\ell}$ than when
considering a two-term atomic system. More details on such calculations can be
found in Appendix A.

Step 2: Obtain the synthetic Stokes profiles for the D lines by solving the
non-LTE RT problem in the polarized case via the numerical code described in
Alsina Ballester et al. (2022). It is suitable for a two-term atomic system
and thus does not account for the metastable levels of Ba ii, but it can
include the HFS of the odd isotopes. Unless otherwise noted, all seven stable
isotopes are considered.222Each coefficient of the RT equation can be taken as
a linear combination of the contribution from each single isotope, weighted by
its relative abundance (e.g., Alsina Ballester, 2022). The code can account
for scattering polarization with both PRD effects under the AA approximation
and magnetic fields in the incomplete Paschen-Back effect regime. The ground
term is assumed not to have atomic polarization, because elastic collisions
with neutral hydrogen are expected to suppress the ground level atomic
polarization, as they do for the metastable levels (see Derouich, 2008). Thus,
each of the HFS levels of the ground term is populated according to the total
$\mathcal{N}_{\ell}$ and its corresponding statistical weight.
$\mathcal{N}_{\ell}$ is kept fixed throughout the iterative RT calculation for
this step and, because all the RT coefficients are proportional to this value,
the problem is linear.333For the problem to be linear, stimulated emission
must also be neglected, which is a very good approximation for the wavelengths
of interest. Indeed, this assumption is made in the derivations of the PRD
formalisms on which HanleRT and the code described in Alsina Ballester et al.
(2022) are based. The thermal line emissivity is computed as explained in
Alsina Ballester et al. (2022). The potential impact of the collisional
transfer of atomic polarization between different FS or HFS levels is beyond
the scope of this work, and was not taken into account in the calculations for
the profiles presented below.

Figure 4: Stokes $I$, normalized to the continuum intensity $I_{\mathrm{c}}$
(upper panel), and $Q/I$ (lower panel) profiles of the D1 line as a function
of wavelength. In all the figures presented in this section, the profiles were
calculated following the approach described at the beginning of this section.
The colored curves indicate the results of calculations with different
treatments of the hyperfine structure (HFS): fully accounting for it (black),
accounting only for that of the upper (red) or lower (green) term, or
neglecting it entirely (blue). Overlapping curves are dashed for the sake of
visibility.

Because we are considering 1D atmospheric models without bulk velocities, the
problem is axially symmetric along the local vertical (except in the presence
of inclined magnetic fields; see Section 4.4.2). Under such symmetry
conditions, and taking the reference direction for positive Stokes $Q$
parallel to the nearest limb, no Stokes $U$ or $V$ are produced and thus the
corresponding figures are not shown.

###  4.2 The impact of HFS

The black curves in Figure 4 represent the D1 intensity profile normalized to
$I_{c}$ (top panel) and the $Q/I$ profile (bottom panel), obtained as
described above. Such calculations were carried out in the absence of magnetic
fields, considering the FAL-C model and accounting for the HFS of the odd
isotopes, as well as for the quantum interference between all states of the
upper term. Like in the case of the $Q/I$ profile obtained with HanleRT when
neglecting the metastable levels (see Section 3), we find a positive $Q/I$
blue peak and a negative red one. The amplitude of the blue peak is roughly
$0.6\%$ (slightly larger than that found with HanleRT) and the amplitude of
the red one is just above $0.15\%$ (slightly smaller).

Figure 4 also highlights the impact of the HFS of barium on the intensity and
scattering polarization patterns of the D1 line, by presenting a comparison
between the profiles discussed in the previous paragraph, in which the HFS
splitting was fully taken into account, and the profiles obtained by
neglecting it in the $6s\,^{2}\mathrm{S}$ ground term (red curve), the
$6p\,^{2}\mathrm{P}^{\mathrm{o}}$ upper term (blue curve), or both (green
curve). Such splittings were neglected by setting to zero the corresponding
$\mathcal{A}$ and $\mathcal{B}$ HFS coefficients (see Appendix A).

Only $18\%$ of barium atoms have HFS, but this already leads to a substantial
broadening of the D1 intensity profile (in agreement with Belluzzi & Trujillo
Bueno, 2013). This broadening is mostly due to the HFS splitting of the ground
term, which is more than one order of magnitude larger than that for the upper
level of D1. The two-peak scattering polarization pattern can only be
reproduced by accounting for the HFS and, specifically, that of the ground
term. If the latter splitting is neglected, the spectral window spanned by the
various HFS components of the D1 line is very small and the radiation field is
effectively flat within this range. As a result, the key mechanism pointed out
by Belluzzi & Trujillo Bueno (2013), through which scattering polarization is
produced in this intrinsically unpolarizable line, is inhibited. On the other
hand, accounting for the HFS of the ground term but neglecting that of the
upper level of D1 leads to an enhancement of the amplitude of the polarization
peaks. This enhancement occurs because the quantum interference between the
various $F$ levels is maximum if there is no energy separation between them.

We also carried out calculations in which we fully accounted for the HFS but
neglected the quantum interference between states pertaining to different $J$
levels of the upper term (i.e., $J$-state interference) and to different $F$
levels of the same $J$ level of the upper term (i.e., $F$-state interference),
following Appendix C.7 of Alsina Ballester et al. (2022). Neither $J$\- nor
$F$-state interference have an appreciable impact on the scattering
polarization of the D1 line and the corresponding profiles are thus not shown.
Such results were expected, because the separation between the upper FS levels
of the D1 and D2 lines is extremely large and even the HFS splitting in the
upper level of D1 is considerably more than one order of magnitude larger than
the natural width of the line.

###  4.3 The sensitivity to the atmospheric model

The semiempirical 1D atmospheric models considered in this work are
representative of spatial averages of specific regions of the solar atmosphere
and thus cannot account for the full three-dimensional (3D) complexity of the
real solar atmosphere. Moreover, bulk velocities are not taken into account in
this work, despite the dynamic nature of the Sun. Nevertheless, our modeling
can provide valuable insights into the physics that shape the intensity and
polarization patterns of non-LTE lines (see, e.g., Faurobert et al., 2009;
Smitha et al., 2013, in which the Stokes profiles of the Ba ii D2 line were
synthesized considering various semiempirical models and compared them with
observations). Here, we present the synthetic intensity and $Q/I$ profiles of
the Ba ii D1 line obtained with several FAL models other than FAL-C, which was
used in the calculations presented above and is representative of an average
region of the quiet solar atmosphere. The other considered semiempirical
models are FAL-A, which represents relatively faint internetwork regions of
the quiet Sun; FAL-F, representative of particularly bright network regions of
the quiet Sun; and FAL-P, which corresponds to a typical plage region. We also
considered FAL-X, which is representative of an average region of the quiet
solar atmosphere, but with considerably lower temperatures than FAL-C in the
photosphere and up to the middle chromosphere. Throughout the entire
wavelength range taken for the problem (which includes the D1 and D2 lines and
their nearby continuum), the intensity is highest for model P, then F, then C
and X (having very similar values for both models), and is lowest for model A.
However, the D1 intensity profiles, when normalized to $I_{c}$, present a
remarkably similar shape for all considered models. The most appreciable
differences concern the width of the wings, but even these are minor and are
thus not shown here.

Figure 5: Synthetic $Q/I$ profiles of the D1 line as a function of wavelength,
obtained considering different 1D emiempirical atmospheric models. The colored
curves correspond to calculations considering the atmospheric models indicated
in the legend.

The differences between the D1 fractional scattering polarization profiles for
the various considered models are also quite modest, as can be seen in Figure
5. Indeed, the largest differences are found between the $Q/I$ profile
obtained considering the FAL-X model (with a maximum amplitude of
$\sim\\!0.50\%$ in the blue peak) and the other models (whose blue peaks reach
amplitudes between $\sim\\!0.54$ and $0.56\%$).

In order to replicate the spectral smearing due to large-scale velocities
typical of the lower chromosphere and the finite resolution of a typical
instrument, we convolved the synthetic profiles with a Gaussian function with
a FWHM of $70$ mÅ. After such smearing, the $Q/I$ profiles for the various
models are indistinguishable at the plot level (figure not shown), and
differences are only appreciable in the continuum polarization. We emphasize
that even the smeared D1 profiles still present a positive blue peak and a
negative red one, in contrast to the observations reported in Figure 3 of
Stenflo et al. (2000). None of the physical ingredients considered in this
paper (including metastable levels, angle-dependent PRD, and the magnetic
fields, discussed below) can produce a positive red $Q/I$ peak in the D1 line.
For further progress in this respect, we need high-precision
spectropolarimetric observations of this line and to include in our
theoretical modeling the non-coherent continuum scattering investigated by del
Pino Alemán et al. (2014a, b).

Our results indicate that the D1 line is largely insensitive to the
thermodynamical structure of the solar atmosphere and, thus, observable
variations in its scattering polarization should be attributed to other
factors, such as the presence of a magnetic field. In the future, it will be
of interest to investigate the possible sensitivity of the D1 intensity and
polarization signals to changes in atmospheric models that are 3D rather than
1D and dynamic rather than static.

###  4.4 Magnetic fields

The presence of a magnetic field modifies the energy of the magnetic states
$f$ of the Ba ii atom (as illustrated in Figure 2 of Belluzzi et al. 2007b for
the 137Ba isotope; see also Figure 2 of Belluzzi et al. 2007a). This impacts
the polarization of the spectral lines by producing a shift in the $\pi$ and
$\sigma$ components of the line444Throughout this work we refer to such
spectral line polarization as due to the Zeeman effect, even when the shifts
in the $\sigma$ and $\pi$ components do not depend linearly on the magnetic
field because of mixing between states with different $J$ or $F$. and by
modifying the quantum interference between $f$ states. A main point of
interest in this work is to evaluate the magnetic sensitivity of the
polarization patterns of the Ba ii D1 line, thus providing valuable insights
into the potential of this spectral line for diagnostics of chromospheric
magnetic fields. Here we present a series of numerical experiments,
considering magnetic fields of increasing strength that are either
isotropically distributed (see Sect. 4.4.1) or deterministic (see Sect.
4.4.2).

####  4.4.1 Tangled magnetic fields

We analyze the sensitivity of the D1 scattering polarization to magnetic
fields whose orientation changes at scales smaller than the mean free path of
the photons of the line (following Appendix C.6 of Alsina Ballester et al.,
2022, where such fields are called micro-structured), with no preferred
direction. In particular, we consider such fields with an isotropic
distribution of orientations and a fixed strength, which we hereafter refer to
as tangled magnetic fields. Such fields do not break the axial symmetry of the
problem, and thus they do not give rise to any Stokes $U$ or $V$ signal.

Figure 6: Synthetic $Q/I$ profiles of the D1 line as a function of wavelength
obtained in the presence of tangled magnetic fields. The various colored
curves represent the calculations carried out considering the field strengths
indicated in the legend.

We carried out calculations for tangled magnetic fields up to $500$ G,
although in Figure 6 we only show the $Q/I$ profiles obtained for field
strengths up to $300$ G. The intensity profile does not change appreciably
within the considered range of field strengths, for which the magnetic
splitting is much smaller than the Doppler width, and thus the corresponding
figure is not shown. The amplitude of the D1 scattering polarization begins to
decrease appreciably in the presence of fields with strengths of about $15$ G.
As the field strength increases further, the polarization amplitude decreases
monotonically, but this trend begins to halt at about $200$ G. Although it is
not shown in the figure, we also verified that further increases in the field
strength beyond $300$ G barely modify the linear polarization amplitude (i.e.,
saturation is reached). At saturation, the $Q/I$ amplitude of the red peak is
$\sim\\!0.12\%$, which is approximately $1/5$ of the one obtained in the
absence of magnetic fields (roughly $0.60\%$), as expected for the saturation
value for isotropic micro-structured fields in a two-level atom (e.g.,
Trujillo Bueno & Manso Sainz, 1999). Recalling that the D1 scattering
polarization pattern is produced only by the $\sim\\!18\%$ of barium isotopes
that have nonzero nuclear spin (135Ba and 137Ba), we focus the discussion on
such isotopes and their HFS. We verified numerically that neglecting the
magnetic splitting of the ground level does not change the D1 scattering
polarization; its magnetic sensitivity can be mainly attributed to the
splitting of the upper level. Indeed, the Larmor frequency at $15$ G is close
to $1/5$ of the natural width of the line’s upper level; at that point the
splitting between the magnetic states $f$ of any given HFS level of
$6p\,^{2}\mathrm{P}^{\mathrm{o}}_{1/2}$ becomes large enough that their
interference appreciably decreases, reducing their scattering polarization
(the Hanle effect). As the magnetic field increases, so does the splitting
between the $f$ states and thus the interference between them becomes weaker.
At saturation field strengths, the separation between $f$ states of the same
HFSlevel is large enough that the interference between them is negligible.

####  4.4.2 Deterministic magnetic fields

We also investigate the case of deterministic magnetic fields (i.e., those
with a fixed direction rather than an isotropic distribution of orientations),
considering first the specific case of horizontal magnetic fields contained in
the plane defined by the local vertical and the LOS. For an LOS with
$\mu=0.1$, such fields are almost longitudinal. In this case, the problem is
no longer axially symmetric and nonzero $U$ and $V$ signals can arise.

Figure 7: Synthetic $Q/I$ (upper panel) and $U/I$ (lower panel) profiles of
the D1 line as a function of wavelength, obtained in the presence of
deterministic horizontal magnetic fields contained within the plane given by
the local vertical and the LOS. The various colored curves represent the
calculations carried out in the presence of fields with the strengths
indicated in the legend.

Figure 7 shows a series of $Q/I$ and $U/I$ profiles obtained in the presence
of horizontal magnetic fields with a positive projection onto the LOS and the
same strengths considered in Section 4.4.1. Such magnetic fields reduce the
amplitude of the $Q/I$ signal to a greater degree than tangled fields of the
same strengths. For magnetic fields close to saturation (of $200$ G or
stronger), a depolarization pattern is found in $Q/I$. For this geometry, the
Hanle effect also gives rise to a $U/I$ signal, whose amplitude increases with
magnetic field strength until about $35$ G. For stronger fields, the $U/I$
amplitude instead decreases as the magnetic field reduces the interference
between $f$ states.

Figure 8: Synthetic $V/I$ profiles of the D1 line as a function of wavelength,
obtained in the presence of the same horizontal magnetic fields considered in
the previous figure.

The considered horizontal fields also give rise to a $V/I$ pattern with two
positive peaks to the blue of the line center and a negative peak to the red,
as illustrated in Figure 8. In the presence of a 15 G magnetic field, the
amplitudes of these peaks reach roughly $1\%$ and they increase linearly with
field strength within the range considered in this work. This is the behavior
typically associated with the Zeeman effect. The double-peak feature found to
the blue of the line center is due to the large HFS splitting of the upper and
lower levels of the D1 line. Indeed, we verified that a $V/I$ pattern with a
single blue peak is produced instead when the HFS is neglected. This contrasts
with the $V/I$ signals found for the K i D1 line, which arises from a
transition between levels with the same $J$ and $F$ quantum numbers, but which
could be suitably modeled without accounting for the HFS (see Alsina
Ballester, 2022) because its splitting is much smaller than for the analogous
levels of Ba ii. We also verified that one cannot suitably apply the
magnetograph formula (e.g., Section 9.6 of Landi Degl’Innocenti & Landolfi,
2004) to the Ba ii D1 line, calculating the Landé factors according to $L$-$S$
coupling (neglecting HFS).

Finally, we evaluated the suitability of the so-called linear Zeeman
approximation, that is, neglecting the off-diagonal elements of the magnetic
Hamiltonian, which are responsible for the mixing between states with
different $J$ or $F$ eigenstates. We verified numerically that, although the
mixing between $J$ states can be safely neglected, neglecting the mixing
between $F$ states substantially underestimates the amplitude of the circular
polarization patterns. The unsuitability of the linear Zeeman approximation
had also been reported for spectral lines such as H i Lyman-$\alpha$ (see
Alsina Ballester et al., 2019, Appendix A), the Mn i resonance multiplet
around $2800$ Å (del Pino Alemán et al., 2022), or the K i D lines (Alsina
Ballester, 2022), for which the $J$ or $F$ mixings due to the IPB effect are
significant.

Figure 9: Synthetic $Q/I$ (upper panel) and $U/I$ (lower panel) profiles of
the D1 line as a function of wavelength, obtained in the presence of
deterministic vertical magnetic fields. The various colored curves represent
the calculations carried in the presence of fields with the strengths as
indicated in the legend.

We also considered the case of a vertical deterministic magnetic field, which
begins to appreciably impact the linear polarization patterns in the presence
of magnetic fields of about $200$ G, as can be seen in Figure 9. Unlike the
aforementioned horizontal fields, such vertical fields do not impact the
interference between the $f$ states that are degenerate in the nonmagnetic
case (the Hanle effect does not operate). Moreover, the energies of the two
upper HFS levels of the D1 line are too far apart for the interference between
them to play a meaningful role. Instead, the linear polarization signals can
be attributed to the Zeeman effect due to the transverse component of the
magnetic field.

Figure 10: Synthetic $V/I$ profiles of the D1 line as a function of
wavelength, obtained in the presence of the same vertical magnetic fields
considered in the previous figure. In order to consider a larger longitudinal
component of the magnetic field, for this figure we took an LOS with $\mu=1$.

When considering LOS s close to the disk center, vertical magnetic fields are
close to longitudinal and thus give rise to a $V/I$ pattern whose amplitude is
proportional to the longitudinal component, as shown in Figure 10. Although
the shape of the $V/I$ profile is noticeably different from the one shown in
Figure 8 – presenting far wider wing lobes, for instance – there is also a
clear double-peak feature due to the HFS. Of course, the magnetograph formula
and the linear Zeeman approximation are not suitable for this geometry,
either.

##  5 Conclusions

In this work, we carried out a series of numerical experiments to identify the
main physical mechanisms that shape the intensity and polarization patterns of
the D1 line. We obtained the Stokes profiles of these lines through non-LTE RT
calculations, considering semiempirical 1D models of the solar atmosphere and
atomic models that account for both the D1 and D2 lines. Our modeling included
both PRD effects and the HFS of the barium isotopes with nuclear spin ($18$ %
of the total by abundance). This allowed us to study the scattering
polarization arising from the spectral structure of the anisotropic radiation
field over the wavelength interval spanned by the various HFS components of
D1. In order to consider relatively large scattering polarization signals, we
displayed the resulting Stokes profiles at an LOS with $\mu=0.1$.

Here we evaluated the impact of the metastable levels on the D1 line in the
nonmagnetic case using the HanleRT non-LTE code (del Pino Alemán et al., 2016,
2020). This RT code is designed for a multi-term atomic system without HFS,
but through some formal substitutions it can incorporate a multi-level atom
with HFS, which neglects the quantum interference between FS levels but is
otherwise suitable in the absence of magnetic fields. Although the inclusion
of the metastable levels appreciably decreases the amplitude of the $Q/I$
pattern of D1, they have little impact on its shape. We also verified with
HanleRT that the D1 scattering polarization signal can be suitably modeled
making the AA approximation.

For the rest of our investigation, we considered synthetic Stokes profiles
calculated using the non-LTE RT code described in Alsina Ballester et al.
(2022). Although this code can account for PRD effects only under the AA
approximation and cannot account for the metastable levels, it can jointly
include the HFS of the odd isotopes of Ba ii and magnetic fields of arbitrary
strength and orientation. We find that the very large HFS of the ground term
substantially broadens the D1 intensity profile and is responsible for its
two-peak scattering polarization pattern. We also verified that the quantum
interference between FS or HFS levels has no significant impact on the linear
polarization.

Interestingly, our numerical experiments considering the various FAL
semiempirical atmospheric models (which present different stratifications for
parameters such as temperature or density) reveal that neither the D1
continuum-normalized intensity nor the $Q/I$ fractional linear polarization
pattern are strongly sensitive to the different models. Thus, most of the
variation of its linear polarization can instead be attributed to the magnetic
field, enhancing its value for diagnostics of chromospheric magnetic fields.
In the future, the variation of the intensity and polarization of the D1 line
should be investigated when considering dynamic models of the solar atmosphere
that fully account for its 3D complexity.

In this work we considered tangled and deterministic magnetic fields with
strengths of up to $500$ G, although we did not show the Stokes profiles
obtained for fields stronger than $300$ G. The considered fields have no
appreciable impact on the intensity profile. On the other hand, the linear
scattering polarization is clearly sensitive to tangled or horizontal magnetic
fields of roughly $15$ G or stronger via the Hanle effect. Tangled magnetic
fields of increasing strength progressively depolarize the $Q/I$ signal until
reaching saturation at about $300$ G; the $Q/I$ saturation amplitude is
approximately $1/5$ of the amplitude in the nonmagnetic case. Deterministic
horizontal magnetic fields have a stronger depolarizing effect than tangled
fields of the same strength and, at saturation, present an almost completely
depolarized signal. In the presence of such fields, the problem is no longer
axially symmetric, and thus they give rise to a $U/I$ signal. If the magnetic
fields have a substantial longitudinal component, a $V/I$ pattern is produced
through the Zeeman effect, with amplitudes that increase linearly with field
strength and that reach roughly $1\%$ in the presence of longitudinal fields
of $15$ G. Suitably modeling such circular polarization signals requires
accounting for the Paschen-Back effect for HFS. The magnetograph formula,
assuming $L$-$S$ coupling for the Landé factors, does not yield a reliable
estimate of the longitudinal magnetic field from $V/I$. Magnetic fields with
transverse components close to $200$ G or larger also produce appreciable
linear polarization signals due to the Zeeman effect.

These findings highlight the diagnostic value of spectropolarimetric
observations of the Ba ii D1. However, none of the RT codes currently at our
disposal meet all the requirements for a quantitative modeling of the D1
Stokes profiles. In particular, it is necessary to account for scattering
polarization with PRD effects, for the metastable levels and the HFS of the
atomic system, for magnetic fields in the IPB effect regime, and for the
collisional transfer of population and atomic polarization between all levels
of the atomic system. Fortunately, we expect such a code to be developed in
the near future. This would also make it possible to model the D1 together
with D2, whose large scattering polarization is sensitive to considerably
weaker magnetic fields, thus offering complementary information about the
magnetism in the lower solar chromosphere.

We thank the anonymous reviewer for constructive comments that contributed to
improve the manuscript. We gratefully acknowledge the financial support from
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation program (ERC Advanced grant agreement No. 742265). We
also acknowledge support from the Agencia Estatal de Investigación del
Ministerio de Ciencia, Innovación y Universidades (MCIU/AEI) under grant
“Polarimetric Inference of Magnetic Fields” and the European Regional
Development Fund (ERDF) with reference
PID2022-136563NB-I00/10.13039/501100011033. T.P.A.’s participation in the
publication is part of the Project RYC2021-034006-I, funded by
MICIN/AEI/10.13039/501100011033, and the European Union
“NextGenerationEU”/RTRP.

##  Appendix A Atomic quantities

The synthetic Stokes profiles presented in Section 4 were obtained through a
two-step calculation in a similar manner as was done in Alsina Ballester et
al. (2022). The purpose of the first (or preliminary) step is to provide a set
of quantities that are required as input in the second step, in which a two-
term atomic model is considered. Such quantities include the rates of elastic
and inelastic collisions, continuum quantities (the thermal emissivity,
absorption coefficient, and scattering cross-section), and the population of
the lower term. They are computed while solving the non-LTE RT problem
considering an atomic model with more levels than in the two-term system. For
the sake of reducing the computational cost, the problem was solved in the
case of unpolarized radiation, using the RH code of Uitenbroek (2001). The
considered multilevel atomic system consists of five levels for Ba ii – namely
the ground level, the two metastable levels discussed in the main text, and
the upper levels of the D1 and D2 lines – and of the ground level of Ba iii.
It thus includes $5$ continuum transitions and $5$ line transitions. PRD
effects are taken into account for the D1 and D2 lines but not for the other
transitions. We consider this approximation to be suitable for accurately
computing the aforementioned quantities. The inelastic collisions (those that
induce transitions between different terms) were computed taking into account
only the contribution from free electrons, following Seaton (1962). We stored
the rate of collisions that couples the upper level of the D2 line and the
ground level, and set it equal to the broadening rate due to inelastic
collisions $\Gamma_{I}$, to be used in the second step. Regarding the rate of
elastic collisions (those that induce transitions between states that belong
to the same term), the quadratic Stark effect contribution from free electrons
and singly charged ions was computed following Traving (1960) and the van der
Waals contribution from neutral hydrogen and neutral helium was computed
according to Unsöld (1955). The resulting broadening rate for the upper level
of the D2 line was stored as the $\Gamma_{E}$ broadening rate, to be used in
the second step.

The second step of the calculation yields the synthetic Stokes profiles of the
D1 and D2 lines shown in Section 4. Such profiles are obtained by solving the
non-LTE RT problem in the polarized case using the code described in Alsina
Ballester et al. (2022), which considers a two-term system with HFS but cannot
account for the metastable levels. The lower term is the $6s\,^{2}\mathrm{S}$
ground term, which has a single FS level whose energy we take to be zero. The
upper term $6p\,^{2}\mathrm{P}^{\mathrm{o}}$ consists of two FS levels: the
upper level of the D1 line, with $J=1/2$ and an energy of $20261.561$ cm-1,
and the upper level of the D2 line, with $J=3/2$ and an energy of $21952.404$
cm-1. These energies were taken from the NIST database (Kramida et al., 2021).
In this framework, all transitions are assumed to have the same line
broadening555We chose values for the $\Gamma_{E}$ and $\Gamma_{I}$ broadenings
that correspond to the D2 line rather than to D1 or to a weighted mean of the
two. Nevertheless, we verified that this choice has no appreciable impact on
the D1 profile and only a very modest one in the core of the D2 linear
polarization profile, both when considering the FAL-C and the FAL-P models.;
in addition to the collisional contributions discussed above, this broadening
has a radiative contribution $\Gamma_{R}$, which corresponds to the Einstein
coefficient for spontaneous emission of the term. Because the D lines share
the same lower level, the Einstein coefficient of the two lines are identical
if one assumes $L$-$S$ coupling (e.g., Section 7.5 of Landi Degl’Innocenti &
Landolfi, 2004). In reality, their experimental values differ substantially
(see, e.g., Kramida et al., 2021). We take $\Gamma_{R}=1.03\times 10^{8}$ s-1,
which is the average of the Einstein coefficients for the two lines accounting
for the statistical weights of the upper level of each line. In the second
step, we take the damping parameter $a$ that enters the RT coefficients to be
$a={(\Gamma_{R}+\Gamma_{E}+\Gamma_{I})}/{(4\pi\Delta\nu_{D})}$, where
$\Delta\nu_{D}$ is the Doppler width in frequency units.

Table 1:  Isotopic abundances and energy shifts |  |  | Isotope shift (cm-1)
---|---|---|---
| Abundance ($\%$) | $I$ | $6p\,^{2}\mathrm{P}^{\mathrm{o}}_{1/2}$ | $6p\,^{2}\mathrm{P}^{\mathrm{o}}_{3/2}$
130 Ba | $0.106$ | $0$ | $1.185\\!\times\\!10^{-2}$ | $1.242\\!\times\\!10^{-2}$
132 Ba | $0.101$ | $0$ | $9.303\\!\times\\!10^{-3}$ | $9.837\\!\times\\!10^{-3}$
134 Ba | $2.417$ | $0$ | $7.425\\!\times\\!10^{-3}$ | $7.825\\!\times\\!10^{-3}$
135 Ba | $6.592$ | $3/2$ | $1.163\\!\times\\!10^{-2}$ | $1.203\\!\times\\!10^{-2}$
136 Ba | $7.854$ | $0$ | $5.984\\!\times\\!10^{-3}$ | $6.234\\!\times\\!10^{-3}$
137 Ba | $11.232$ | $3/2$ | $9.043\\!\times\\!10^{-3}$ | $9.036\\!\times\\!10^{-3}$
138 Ba | $71.698$ | $0$ | $0.000$ | $0.000$

The HFS of the atomic system is also included in the second step, in which we
account for the seven stable isotopes of barium. Their relative abundance,
nuclear spin, and their corresponding isotopic shifts for the upper levels of
the D1 and D2 lines are displayed in Table 1. The isotopic shifts are given
relative to the 138Ba. The quantities were taken from Table 1 of Belluzzi et
al. (2007b), who themselves took the shifts from Wendt et al. (1984), except
for those for the 134Ba isotope, which were taken from Wendt et al. (1988).

Table 2:  Hyperfine structure coefficients |  135Ba |  137Ba
---|---|---
$\mathcal{A}_{6s\,^{2}\mathrm{S}_{1/2}}$ (cm-1) | $1.198\\!\times\\!10^{-1}$ | $1.341\\!\times\\!10^{-1}$
$\mathcal{A}_{6p\,^{2}\mathrm{P}^{\mathrm{o}}_{1/2}}$ (cm-1) | $2.217\\!\times\\!10^{-2}$ | $2.481\\!\times\\!10^{-2}$
$\mathcal{A}_{6p\,^{2}\mathrm{P}^{\mathrm{o}}_{3/2}}$ (cm-1) | $3.769\\!\times\\!10^{-3}$ | $4.243\\!\times\\!10^{-3}$
$\mathcal{B}_{6p\,^{2}\mathrm{P}^{\mathrm{o}}_{3/2}}$ (cm-1) | $1.968\\!\times\\!10^{-3}$ | $3.085\\!\times\\!10^{-3}$
$\mathcal{A}_{5d\,^{2}\mathrm{D}_{3/2}}$ (cm-1) | $5.567\\!\times\\!10^{-3}$ | $6.329\\!\times\\!10^{-3}$
$\mathcal{B}_{5d\,^{2}\mathrm{D}_{3/2}}$ (cm-1) | $9.658\\!\times\\!10^{-4}$ | $1.486\\!\times\\!10^{-3}$
$\mathcal{A}_{5d\,^{2}\mathrm{D}_{5/2}}$ (cm-1) | $-3.581\\!\times\\!10^{-4}$ | $-4.012\\!\times\\!10^{-4}$
$\mathcal{B}_{5d\,^{2}\mathrm{D}_{5/2}}$ (cm-1) | $1.291\\!\times\\!10^{-3}$ | $1.986\\!\times\\!10^{-3}$

For the isotopes with nonzero nuclear spin, the energies of the various atomic
states depend on the $J$, $F$ and $I$ quantum numbers through the magnetic
dipole ($\mathcal{A}$) and electric quadrupole ($\mathcal{B}$) HFS
coefficients that enter the Hamiltonian for HFS. The nonzero values of such
coefficients are displayed in the four first rows of Table 2, again taken from
Table 1 of Belluzzi et al. (2007b), who themselves took the $\mathcal{A}$
coefficients for the ground level from Becker et al. (1981) and the other
$\mathcal{A}$ and $\mathcal{B}$ coefficients from Villemoes et al. (1993). The
only nonzero $\mathcal{B}$ coefficients correspond to the upper level of the
D2 line. Such coefficients were defined according to the American convention
(the expressions of the elements of the HFS Hamiltonian for such convention
can be found, for instance, in Appendix B of Alsina Ballester et al., 2022).

Figure 11: Height at which the optical depth is unity in the FAL-C
semiempirical atmospheric model as a function of wavelength, in a $1.2$ Å
spectral range centered on the D1 (upper panel) and D2 (bottom panel) lines,
for the LOS s indicated by the colored curves (see the legend). The black
curves represent the broadening due to radiative processes $\Gamma_{R}$ (solid
curves), inelastic collisions $\Gamma_{I}$ (dashed curves), and elastic
collisions $\Gamma_{E}$ (dashed-dotted curves), as a function of atmospheric
height.

The profiles presented in Section 3 were carried out using the HanleRT code,
and accounted for the metastable levels $5d\,^{2}\mathrm{D}_{3/2}$ and
$5d\,^{2}\mathrm{D}_{5/2}$. Their HFS coefficients, taken from Silverans et
al. (1986), are shown in the four bottom rows of Table 2. The isotopic shifts
for the five most abundant isotopes, which were the ones considered in such
same calculations, were obtained from Villemoes et al. (1993).

##  Appendix B Formation Height

Figure 11 shows the height in the FAL-C model at which the optical depth
$\tau_{\nu}$ is equal to unity. It is shown in two $1.2$ Å-wide spectral
ranges, centered on the D1 (discussed in the main text) and the D2 lines
(discussed in Appendix C). This height is a proxy for the formation height of
the line and it is shown for several LOS s. In a 1D atmospheric model, the
optical depth is given by
$\mathrm{d}\tau_{\nu}=-\eta_{I}\,\mathrm{d}z/\\!\mu$, where $z$ is the
atmospheric height. $\eta_{I}$ is the absorption coefficient, which was
computed following Alsina Ballester et al. (2022), taking $\mathcal{N}_{\ell}$
as obtained in step $1$ of the approach described in Section 4.1. We note that
the line core of the D1 forms above the temperature minumum; the heights at
which $\tau=1$ at line center are just below $600$ km in the FAL-C model for
an LOS with $\mu=1$ and above $750$ km for $\mu=0.1$. The D2 forms at slightly
higher regions; the height at which its optical depth is equal to unity in the
FAL-C model is above $600$ km for $\mu=1$ and above $800$ km for $\mu=0.1$.
The $\Gamma_{R}$, $\Gamma_{E}$, and $\Gamma_{I}$ broadenings, shown in both
panels as a function of height for reference, were obtained as explained in
Appendix A.

##  Appendix C Analysis of the D2 Stokes profiles

In the main text, we discussed how various features of the atomic system and
properties of the solar atmosphere impact the intensity and linear
polarization pattern of the Ba ii D1 line, illustrated by the synthetic
profiles displayed in Sections 3 and 4. The profiles presented therein were
calculated considering atomic models that considered not only the D1 line
transitions at $4934$ Å, but also the D2 line transitions at $4554$ Å. In this
appendix, we show the synthetic profiles in the spectral range around the D2
line instead of D1. Many such profiles were obtained through the same
calculations that yielded the profiles shown in the main text. The FAL-C
atmospheric model was considered for all such calculation. Except where
otherwise noted, the profiles are shown for an LOS with $\mu=0.1$, taking the
reference direction for positive Stokes $Q$ parallel to the nearest limb.

Figure 12: Stokes $I$ (upper panel), normalized to the continuum intensity
$I_{\mathrm{c}}$, and $Q/I$ profiles (lower panel) of the D2 line as a
function of wavelength (i.e., in a $1.2$ Å-wide range centered on the D2
line). The synthetic profiles are obtained from calculations using the HanleRT
code, accounting for PRD effects and taking a line of sight (LOS) with
$\mu=0.1$. The black curves correspond to the calculations considering a five-
level atomic model, including the levels belonging to the $6s\,^{2}\mathrm{S}$
ground term, the $6p\,{}^{2}\mathrm{P}$ term that contains the upper levels of
the D lines, and the $5d\,^{2}\mathrm{D}$ metastable term. The red curves
correspond to the calculations in which the metastable term is neglected, for
which a three-level atomic model is thus considered. In both cases, the HFS of
the atomic system is taken into account. The population of the ground level,
$\mathcal{N}_{\ell}$, is kept fixed during the iterative calculation.

The D2 Stokes profiles show in Figure 12 were obtained simultaneously with
those presented in Figure 2. Thus, the figure shows the profiles obtained with
HanleRT using the atomic models discussed in Section 3, taking into account
the HFS of the odd isotopes for all considered levels, both accounting for the
metastable levels and neglecting them. Although the $5d\,^{2}\mathrm{D}$
levels do not substantially change the D2 intensity profile, they have a
crucial impact on its linear polarization, decreasing its line-core amplitude
by roughly $60\%$. The depolarization due to the metastable levels is far
greater than the one reported for the D1 line, and also leads to a far more
apparent change in the shape of its $Q/I$ pattern. For this reason, we deem
the calculations presented in Section 4, for which the metastable levels were
not included, to be less reliable for the D2 line than for D1. Despite this,
they may still provide some insights into the sensitivity of this line to the
HFS or to the magnetic field. We also verified that the dips found in the line
center of the $Q/I$ profile are a consequence of making the AA approximation,
implying that a strictly correct modeling of the D2 line should be carried out
through a fully angle-dependent calculation, in contrast to the case of the D1
line.

Figure 13: Stokes $I$ (upper panel), normalized to $I_{\mathrm{c}}$, and $Q/I$
profiles (lower panel) of the D2 line as a function of wavelength. The
profiles calculated considering different isotopes of barium are shown with
different colored curves. The black curve corresponds to the case in which all
seven stable isotopes are considered with their corresponding abundances,
whereas the red and blue curves correspond to the cases in which only the
138Ba and only the 137Ba isotopes were considered, respectively.

In the rest of this appendix, we present profiles using the numerical code
discussed in Alsina Ballester et al. (2022), most of them being analogous to
those presented in Section 4 for the D1 line. First, we study the impact of
the HFS of the odd isotopes but, unlike in Section 4.2, we do not compare the
intensity and linear polarization profiles obtained by accounting for the HFS
splitting of different FS levels and neglecting it. Instead, as shown in
Figure 13, we compare the profiles obtained considering all seven stable
isotopes with their corresponding abundances (see Appendix A), considering
only the 137Ba isotope, which has nuclear spin $I=3/2$ and HFS, and only
138Ba, for which $I=0$ and thus has no HFS. The inclusion of isotopes with HFS
leads to a broadening of the absorption profile in intensity, much like what
was reported in the D1 line. A comparison between the linear polarization
profiles obtained considering isotopes with $I=0$ and $3/2$ reveals that the
very large HFS of barium depolarizes its line-core $Q/I$ by almost a factor
$4$. This is consistent with the theoretical depolarization when the quantum
interference between different $F$-levels of the upper term is negligible
(e.g., Section 10.22 of Landi Degl’Innocenti & Landolfi, 2004). The full
scattering polarization amplitude is thus mainly produced by the even isotopes
(which have no HFS). This clearly contrasts with the linear polarization
pattern of the D1 line, which is a consequence of the wavelength separation
between the HFS components of the odd isotopes.

Figure 14: Synthetic $Q/I$ profiles of the D2 line as a function of
wavelength, obtained in the presence of tangled magnetic fields of the various
strengths indicated in the legend.

In this Appendix, we also show the D2 profiles obtained in the presence of
both tangled and deterministic magnetic fields, as in Section 4.4, but
considering different field strengths in the range between $0$ and $300$ G.
The intensity profiles are not appreciably affected by fields of such
strengths and thus they are not shown here. The sensitivity of the D2 linear
polarization to tangled magnetic fields (see Section 4.4.1) is illustrated in
Figure 14. Such fields preserve the axial symmetry of the problem and thus no
$U/I$ or $V/I$ signal is produced. For this line, a depolarization is clearly
appreciable in $Q/I$ for fields as weak as $2$ G – considerably lower than
those required to modify the D1 signal. We note that, in contrast to the D1
line, most of the contribution to the D2 scattering polarization comes from
the roughly $82\%$ of isotopes without HFS. Neglecting HFS, the magnetic field
at which the Zeeman splitting of the upper level of D2 is equal its the
natural width (i.e., the Hanle critical field; see e.g., Stenflo, 1994) is
approximately $9$ G. This is fully consistent with the behavior displayed in
Figure 14. For magnetic fields stronger than $100$ G, the linear polarization
amplitude slightly increases, due to the HFS of the odd isotopes, until
reaching saturation (for a further discussion on this enhancement, see Section
10.22 of Landi Degl’Innocenti & Landolfi, 2004).

Figure 15: Synthetic $Q/I$ (top panel) and $U/I$ profiles (bottom panel) of
the D2 line as a function of wavelength, obtained in the presence of
deterministic horizontal fields contained in the plane defined by the local
vertical and the LOS. The various colored curves represent the calculations
carried in the presence of magnetic fields of the strengths indicated in the
legend.

The synthetic $Q/I$ and $U/I$ profiles of the D2 line, obtained in the
presence of deterministic horizontal magnetic fields, contained in the plane
defined by the local vertical and the LOS, are shown in Figure 15. All the
considered fields have a positive projection onto the LOS and the various
curves indicate the same field strengths as in Figure 14. In the case of
horizontal magnetic fields, we observe a stronger depolarization in the line
core for a given field strength and a near-zero $Q/I$ value for fields larger
than $100$ G, in contrast to the substantial saturation value found for
tangled fields. Moreover, horizontal fields induce a rotation of the plane of
linear polarization as the quantum interference between nearby $f$-states is
modified (i.e., Hanle rotation). A maximum in the $U/I$ amplitude is reached
for field strengths close to $5$ G; as the field becomes stronger, the
interference between $f$ states becomes smaller and the linear polarization
fraction decreases. We see no indication of the loop-like behavior in the
polarization diagram (i.e., increases in the $U/I$ amplitude with field
strength after having reached a first maximum) that is found in D2 lines of K
i (see Alsina Ballester, 2022) and of Na i (see Section 10.22 of Landi
Degl’Innocenti & Landolfi, 2004). We attribute the absence of such loops to
the fact that the HFS splitting of the Ba ii isotopes with nuclear spin
(required for the loop-like behavior) is large enough that no level crossings
are reached until field strengths of roughly $50$ G are considered, at which
point the line is almost depolarized.

Figure 16: Synthetic $V/I$ profiles as a function of wavelength, in the
spectral range centered on D2, obtained in the presence of the same horizontal
fields as in the previous figure.

The considered horizontal fields present a substantial longitudinal component
for an LOS with $\mu=0.1$ and, thus, a clear $V/I$ pattern is produced, as
shown in Figure 16. The amplitude of the signal increases linearly with the
field strength, and reaches roughly $1\%$ for $10$ G fields. Like in the case
of the D1 line, reproducing the shape of the $V/I$ pattern requires accounting
for the HFS of the barium atoms. We do not find two distinct blue peaks in the
D2 line, but the HFS splitting does contribute to broadening it appreciably.

Figure 17: Synthetic $Q/I$ (top panel) and $U/I$ (bottom panel) profiles of
the D2 line as a function of wavelength, in the presence of a vertical
magnetic field. The colored curves represent the results of calculations
carried in the presence of magnetic fields of the strengths indicated in the
legend. Figure 18: Synthetic $V/I$ profiles as a function of wavelength, in
the spectral range centered on D2, obtained in the presence of a vertical
magnetic field, for an LOS with $\mu=1$. The colored curves represent the
$V/I$ profiles obtained for the field strengths indicated in the legend.

We also considered the case of a vertical magnetic field, and the resulting
linear polarization patterns for an LOS with $\mu=0.1$ are shown in Figure 17.
Vertical magnetic fields only modify the quantum interference between states
with the same quantum number $f$, and thus its impact on the scattering
polarization is much more modest than that of tangled or horizontal magnetic
fields. Magnetic fields of about $100$ G or stronger have a large transverse
component and introduce a further linear polarization signal due to the Zeeman
effect, whose clearest signature in the figure are the negative peaks at
either side of the line core, with amplitudes that are noticeably larger than
the signals found for the D1 line for the same geometry and field strengths
(see Figure 9). Moreover, vertical magnetic fields also give rise to a
circular polarization pattern in the D2 line due to the Zeeman effect, as is
shown in Figure 18. An LOS with $\mu=1$ was selected for this figure in order
to consider a completely longitudinal magnetic field. For this geometry, we do
find a two-peak structure to the blue of the line center. In contrast to the
D1 $V/I$ profile, the bluemost peak of the D2 presents a larger amplitude than
the one closer to line center. The peak to the red of the line center is much
sharper for the D2 line than for D1, but it also presents a secondary lobe. We
also verified that the amplitude of the D2 $V/I$ is also substantially
underestimated when neglecting the elements of the magnetic Hamiltonian that
are off-diagonal in $F$ (making the linear Zeeman approximation, see Section
4.4.2). Likewise, we also find the magnetograph formula to be unsuitable for
reproducing the D2 $V/I$ pattern while considering the $L$-$S$ coupling scheme
for the effective Landé factor, although the error incurred is not as large as
for the D1 line.

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