Line Formation Theory for the Multiterm Atom with Hyperfine Structure in a Magnetic Field

LINE FORMATION THEORY FOR THE MULTITERM ATOM WITH HYPERFINESTRUCTURE IN A MAGNETIC FIELD

R. Casini and R. Manso Sainz

High Altitude Observatory, National Center for Atmospheric Research,1 P.O. Box 3000, Boulder, CO 80307-3000

Received 2004 November 19; accepted 2005 January 13

ABSTRACT

Within the framework of the quantum theory of polarized line formation, in the limit of complete frequency re-distribution and of the collisionless regime, we derive explicit formulae describing the statistical equilibrium andthe radiative emission of a multiterm atom with hyperfine structure, in the presence of an external magnetic field.The formulae we obtained for the radiative rates of the statistical equilibrium equations and for the radiative coef-ficients of the transfer equation for polarized radiation can be applied to investigate the formation of spectral linesfor which both fine-structure and hyperfine-structure effects are important (e.g., the D1 and D2 lines of Na i in thesolar atmosphere).

Subject headinggs: atomic processes — line: profiles — polarization — Sun: magnetic fields

1. INTRODUCTION

The study of spectral line shapes and polarization is by and large the most powerful tool to investigate the stability and dynamics ofastrophysical plasmas and their interaction with cosmic magnetic fields. Virtually all thermodynamic properties of a plasma leavetheir signatures in the shape of spectral lines (e.g., thermal and pressure broadening and line asymmetries induced by velocity gra-dients), while polarization of the emitted radiation invariably results from the breaking of rotational symmetry in the atomic systeminteracting with photons (e.g., the Zeeman effect induced by a directional magnetic field, impact polarization by anisotropic collisionsin the plasma, and resonance scattering polarization by anisotropic illumination of the plasma). Understanding the physics of an emit-ting plasma, and in particular the testing of dynamical theories aimed at this goal, inevitably requires the ability to reproduce thespectral features and polarization signatures of the observed radiation from the plasma.

Both spectral and polarization analyses of radiation involve photon-demanding observational techniques, and so the sensitivity wecan attain is intrinsically limited by the intensity of the radiation we receive, as well as by the efficiency of our instrumentation. This isobviously the reason why our Sun is the most studied object via spectropolarimetric techniques. In turn, the analysis of solar radiationoffers an ideal test of our understanding of polarized line formation. A typical example is given by the so-called second solar spectrum(e.g., Stenflo&Keller 1997; Gandorfer 2000), which is the linear polarization signature from resonance scattering that is ubiquitous inthe radiation coming from regions mostly near the solar limb. When observed with high polarimetric sensitivity (better than 10�3 ofthe intensity), this second solar spectrum reveals a plethora of structures that challenge our understanding of the solar conditions underwhich they are formed and also our very ability to reproduce them. The D1 and D2 lines of Na i, respectively at 5895.9 and 5890.08,are probably the best-known examples of these structures because of the complexity of their shape and its variety across the solar disk.There have been several attempts made in the past few years to pin down all the ingredients that are needed to model these lines (e.g.,Landi Degl’Innocenti 1998; Casini et al. 2002; Trujillo Bueno et al. 2002; Fluri et al. 2003). Currently, we know that hyperfine struc-ture plays an essential role, but at the same time we also must account for quantum interferences between the fine-structure levelsfrom which these two lines originate. We know that radiation transport in a realistic atmospheric model is fundamental to reproducingthe complex shape of these lines, whereas very little work has been devoted to understanding the role of magnetic fields (Landolfi &Landi Degl’Innocenti 1985; Casini et al. 2002; Trujillo Bueno et al. 2002). Past work on the theory of partial redistribution (PRD) inresonance lines (e.g., Rees & Saliba 1982; Saliba 1985) suggests that the trilobate shape of the D2 line is likely determined by PRDeffects. This suggestion seems to be confirmed by more recent investigations (Fluri et al. 2003). However, we must note that LandiDegl’Innocenti (1998) was able to reproduce the same line shape assuming instead a radiation anisotropy factor independent of wave-length (thus avoiding self-consistent PRD calculations), while adopting a specific depth dependence for the ground-level polarization.At the present time, we are still unable to correctly quantify the relative effects of PRD and ground-level polarization in the modelingof the D lines of Na i, as a completely self-consistent theory of resonance scattering polarization with PRD in complex atoms is stilla work in progress (Landi Degl’Innocenti et al. 1996; Bommier 1997a, 1997b).

In this paper we try to fill some of the above gaps by presenting a general formalism for the description of resonance scattering po-larization from an atomic system endowed with both fine structure (FS) and hyperfine structure (HFS) and subject to an external mag-netic field. We base our derivation on the quantum theory of polarized line formation in the limit of complete frequency redistribution(CRD), and in the collisionless regime, developed by Landi Degl’Innocenti (1983, 1984, 1985; see also Landi Degl’Innocenti &Landolfi 2004). In the atomic system described by our formalism, we allow for the possibility that besides quantum interferences be-tween distinct F levels within a given J level, quantum interferences between distinct J levels within an atomic term (usually speci-fied by its L and S values, assuming that the approximation of L-S coupling is valid, and by a corresponding atomic term label, � ) also

1 The National Center for Atmospheric Research is sponsored by the National Science Foundation.

1025

The Astrophysical Journal, 624:1025–1037, 2005 May 10

# 2005. The American Astronomical Society. All rights reserved. Printed in U.S.A.

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may be important. In fact, the observational signature in the solar spectrum of these ‘‘superinterferences’’ was first observed andcorrectly interpreted by Stenflo (1980) for the H and K lines of Ca ii, and they also play a fundamental role in determining theresonance scattering polarization of the D1 and D2 lines of Na i.

In order to meet the requirements of CRD, we have to make the assumption that the incident radiation field has no spectral struc-ture (i.e., is spectrally flat) over the frequency interval of the separation between interfering levels. In the case of the Na i linesmentioned above, when formed in the solar atmosphere, this condition is only fulfilled for interferences between distinct F levelswithin the same J level, whereas it generally does not hold in the case of the superinterferences that are created between the upperlevels of D1 and D2. In fact, these lines, which are formed in the atomic transitions 2S1/2–

2P1/2 and2S1/2–

2P3/2, respectively, appear inthe solar spectrum as two deep absorption lines separated by a broad continuum. Therefore, the intensity variations of the emergentradiation from the solar atmosphere around the wavelengths of the D1 and D2 lines occur within a frequency interval that corre-sponds to the separation of the interfering J levels of the 2P term. On the other hand, the radiation spectrum of the two lines issufficiently flat in the line cores, where the interferences between F levels occur. Nonetheless, the restriction of spectrally flat radiationdoes not necessarily prevent the possibility of modeling properly the formation of the Na iD lines in the solar atmosphere (see the endof x 3).

In the CRD approximation, the problem of computing the polarization of the scattered radiation by an atom subject to an externalfield can be separated into two stages. The first stage involves the determination of the excitation state of the atom in the external field,illuminated by the incident radiation. Once the solution to this problem is known, we are able to compute the reemitted (scattered)radiation and its polarization characteristics.We remark that the possibility of describing radiation scattering as a two-stage problem isa direct consequence of the hypothesis of CRD. In fact, since CRD is equivalent to noncoherent scattering of radiation, it becomespossible to describe scattering as the temporal succession of two incoherent, first-order atom-photon processes, one of absorption ofradiation (which determines the excitation state of the atomic gas) and one of reemission (where the excitation state previously de-termined is used to calculate the scattered radiation). PRD requires instead the possibility of describing both coherent and noncoherentscattering, and therefore higher order atom-photon processes must also be taken into account.

Following this scheme, in x 2 we derive the expressions for the radiative rates that enter the equations of the statistical equilibrium(SE) of the multiterm atom with HFS. The solution of the SE problem is given in terms of the density matrix elements of the atomicsystem, which describes both the population of the atomic levels and their state of polarization andmutual interference. Next, in x 3wederive the expressions for the coefficients entering the transfer matrix in the radiative transfer (RT) equation for polarized radiation.This equation governs the production and the transport of polarized radiation through a gas that is generally optically thick, and itallows the computation of the intensity and the polarization state of the scattered radiation, when the density matrix of the atomicsystem is known at every point in the gas by having previously solved the SE problem. In both xx 2 and 3 we adopt the formalism of theirreducible spherical tensors (e.g., Fano 1957; Landi Degl’Innocenti & Landolfi 2004), because it provides a more direct interpre-tation of the various quantities in terms of physical symmetries of the interaction processes of the atomic system with the radiationfield. Similarly, for the description of the polarization state of the incident and scattered radiation we adopt the formalism of the Stokesparameters (e.g., Chandrasekhar 1960; Born &Wolf 1983). The theoretical development in this paper is limited to the description of acollisionless atomic plasma, so the state of excitation of the atomic system is completely determined by the incident radiation only.Within the limit of the impact approximation (see, e.g., Lamb & Ter Haar 1971), the collisional rates can be added to the rates for thecorresponding radiative processes. Collisions can be an important ingredient in the formation of the Na i D lines, but the estimate ofthe collisional cross sections is still a work in progress (see, e.g., Kerkeni & Bommier 2002).

In x 4 we apply our formalism to the investigation of the Na i D lines, under the restrictive hypothesis (for solar applicationpurposes) that the illuminating spectrum is totally flat across the wavelength range of the two lines. The results in that section onlyserve the purpose of demonstrating the profile’s general structure that we can expect to be able to interpret within our formalism, evenunder the approximations of CRD and of optically thin plasmas (however, see x 3 for a scheme that allows one to describe the transportof polarized radiation in a dense plasma, which will be the subject of a future paper).

2. THE EQUATIONS OF STATISTICAL EQUILIBRIUM

In this section we derive the SE equations. These represent the quantum mechanical evolution equation for the statistical operatorof the atomic system, �A, projected on some basis of choice for the Hilbert space of the atom. If we select the basis of the eigenstatesof the atomic Hamiltonian, j�i, the SE equations have the form (see Landi Degl’Innocenti 1983; Landi Degl’Innocenti & Landolfi2004)

d

dt�� 0 ¼ �i!�� 0�� 0 �

X� 0 0� 0 0 0

hRA(�; �

0; � 00; � 000)þ RS(�; �0; � 00; � 000)þ RE(�; �

0; � 00; � 000)i�� 0 0� 0 0 0

þX�l� 0

l

TA(�; �0; �l; �

0l )��l� 0

lþX�u� 0

u

hTS(�; �

0; �u; �0u)þ TE(�; �

0; �u; �0u)i��u� 0

u: ð1Þ

In equation (1), only radiative processes are considered. The meaning of the different ‘‘transfer rates,’’ TA,E,S , and ‘‘relaxationrates,’’ RA,E,S , associated with these processes is illustrated in Figure 1, in terms of absorption and emission (both spontaneous andstimulated) processes.

The atomic Hamiltonian,HA , for the model atom that we consider in this work includes both FS and HFS contributions. Because ofthe assumption of L-S coupling, the general ket of the standard basis of eigenstates ofHA, in the absence of external fields, has the form

j��¼ j�LSJIFM

�ð2Þ

CASINI & MANSO SAINZ1026 Vol. 624

and energy E� ¼ f!�LSIJF , independent of M. The index � enumerates the different L-S terms of the model atom. If, instead, a mag-

netic field B is present and we take the direction of B as the quantization axis, then we must include in HA the contribution of themagnetic Hamiltonian,2

HB ¼ f!B(Jz þ Sz); ð3Þ

where !B is the Larmor frequency for the applied field. The magnetic Hamiltonian mixes states of the form in equation (2) hav-ing different J and F values (i.e., J and F are no longer good quantum numbers), so the eigenstates ofHA in the presence of B have theform

j�LSI ; �M�¼

XJF

CJF� (�LSI ;M )j�LSJIFM

�; ð4Þ

with eigenvalues k�(�LSI ;M ). The index � spans the dimension of the subspace with Fz � Jz þ Iz ¼ M within the atomic config-uration �LSI.

Because of our choice of the quantization axis,HA is real symmetric, so the coefficientsCJF� (�LSI ;M ) can also be chosen to be real

quantities, satisfying the orthogonality properties

X�

CJF� (�LSI ;M )CJ 0F 0

� (�LSI ;M )¼ �JJ 0�FF 0 ; ð5aÞXJ F

C JF� (�LSI ;M )CJF

� 0 (�LSI ;M )¼ �� 0 : ð5bÞ

The atomic Hamiltonian can then be expressed through its spectral representation,

HA ¼X�L

X�M

k�(�LSI ;M )j�LSI ; �M��LSI ; �M j

¼X�L

X�M

XJ F

XJ 0F 0

k�(�LSI ;M )CJF� (�LSI ;M )CJ 0F 0

� (�LSI ;M ) j�LSJIFM��LSJ 0IF 0M j; ð6Þ

from which we derive the important sum ruleX�

k�(�LSI ;M )CJF� (�LSI ;M )CJ 0F 0

� (�LSI ;M ) ¼��LSJIFM jHAj�LSJ 0IF 0M

�: ð7Þ

In order to derive explicit expressions for the SE equations, we introduce the correspondences

� � �LSI ;�M ; �l � �lLlSI ; �lMl; �u � �uLuSI ; �uMu;

� 0 � �LSI ; �0M 0; � 0l � �lLlSI ; �

0lM

0l ; � 0

u � �uLuSI ; �0uM

0u;

� 00 � �LSI ; �00M 00;

� 000 � �LSI ; �000M 000;

ð8Þ

2 Because of its negligible contribution, we do not include the interaction between the magnetic field and the nuclear spin.

Fig. 1.—Grotrian diagram illustrating the different radiative processes in the multiterm atom. Straight arrows indicate spontaneous emission processes, whereaswiggly lines indicate radiation-induced processes (absorption and stimulated emission).

MULTITERM ATOM WITH HYPERFINE STRUCTURE 1027No. 2, 2005

through which the atomic density matrix becomes, with evident notation,

�� 0 � �LSI�(�M ; �0M 0) ��LSI ; �M j�j�LSI ; �0M 0�

¼XJ F

XJ 0F 0


� 0 (�LSI ;M 0)�LSI�(JFM ; J 0F 0M 0); ð9aÞ

and conversely, using equation (5b),

�LSI� (JFM ; J 0F 0M 0) ¼X�� 0


� 0 (�LSI ;M 0)�LSI�(�M ; �0M 0): ð9bÞ

Thus, there are ½(2Lþ 1)(2S þ 1)(2I þ 1)�2 density matrix elements for each atomic term.In order to express the SE equations for the density matrix in the standard representation, �LSI� (JFM ; J 0F 0M 0), we must apply the

double summation of equation (9b), X�� 0


� 0 (�LSI ;M 0) ; ; ð10Þ

to both sides of equation (1). However, we finally want to work with the irreducible spherical-tensor representation of the statisticaloperator. For this, we will also need the transformation formula (see, e.g., Landi Degl’Innocenti 1984),

�LSI� (JFM ; J 0F 0M 0) ¼XKQ

(�1)F�M�K

F F 0 K

M �M 0 �Q

� ��LSI�K

Q (JF; J0F 0); ð11aÞ

and its inverse,

�LSI�KQ (JF; J

0F 0) ¼XMM 0

(�1)F�M�K

F F 0 K

M �M 0 �Q

� ��LSI� (JFM ; J 0F 0M 0); ð11bÞ

having adopted the shorthand notation (Varshalovich et al. 1988, hereafter VMK88)

�ab : : : n ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(2aþ 1)(2bþ 1) : : : (2nþ 1)

p: ð12Þ

Both operations (10) and

XMM 0

(�1)F�M�K

F F 0 K

M �M 0 �Q

� �; ð13Þ

must be applied to both sides of equation (1).

2.1. Evaluation of the Magnetic Kernel

We first concentrate on the imaginary term in equation (1),

!�� 0�� 0 � f�1hk�(�LSI ;M )� k� 0 (�LSI ;M 0)

i�LSI�(�M ; �0M 0):

We make the substitution in equation (9b) and also apply the operations in equations (10) and (13) to this expression. After usingthe identities in equations (5a) and (7), we find

!�� 0�� 0 !(10)-(13) X

J 0 0F 0 0J 0 0 0F 0 0 0

XK 0Q 0

N (�LSI ; JJ 0FF 0KQ; J 00J 000F 00F 000K 0Q0)�LSI�K 0

Q 0 (J00F 00; J 000F 000); ð14Þ

where

N (�LSI ; JJ 0FF 0KQ; J 00J 000F 00F 000K 0Q0) ¼XMM 0

(�1)F0 0�F�KK 0

F F 0 K

M �M 0 �Q

� �F 00 F 000 K 0

M �M 0 �Q0

� �

; f�1 �J 0J 0 0 0�F 0F 0 0 0

��LSJIFM jHAj�LSJ 00IF 00M

�� JJ 0 0�FF 0 0

��LSJ 000IF 000M 0jHAj�LSJ 0IF 0M 0��

ð15Þ


is called the magnetic kernel. In order to explicitly evaluate it, we need the expression for the matrix elements of the atomicHamiltonian in the presence of B. This is given by


�¼ �JJ 0�FF 0 f!�LSI

JF þ �0B(�1)LþSþJþJ 0þI�Mþ1�JJ 0FF 0F F 0 1

�M M 0

� �J J 0 1

F 0 F I

�

; �JJ 0 (�1)LþSþJþ1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJ (J þ 1)

p

�J

þ (�1) J�J 0�S

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS(S þ 1)

p J J 0 1

S S L

� �; ð16Þ

as can be determined by applying the Wigner-Eckart theorem and its corollaries (e.g., Brink & Satchler 1993) to equation (3).The contribution of the FS and HFS to equation (16) determines a diagonal contribution to the magnetic kernel. To evaluate this, we

only need the orthogonality properties of the 3j symbols in order to perform the summation over the (degenerate) magnetic quantumnumbers. We find

N (�LSI ; JJ 0FF 0KQ; J 00J 000F 00F 000K 0Q0)diag ¼ �JJ 0 0�J 0J 0 0 0�FF 0 0�F 0F 0 0 0�KK 0�QQ 0!�LSIJF; J 0F 0 : ð17aÞ

When this term is substituted into equation (1), it contributes a depolarizing term for the coherences (JF, J 0F 0) on the left-hand sideof the equation. Physically, this corresponds to the relaxation of the atomic coherences between two interfering levels due to thepartial lifting of their degeneracy, determined by their energy separation.

The magnetic (nondiagonal) contribution to the magnetic kernel is evaluated with the help of equation (6) of VMK88 (p. 454),

N (�LSI ; JJ 0FF 0KQ; J 00J 000F 00F 000K 0Q0)n:d: ¼ �QQ 0!B(�1)F0 0 þ F 0 0 0�Q�KK 0

K K 0 1

�Q Q 0

� �

;

"�J 0J 0 0 0�F 0F 0 0 0�LSI (JF; J

00F 00)K K 0 1

F 00 F F 0

� þ �JJ 0 0�FF 0 0 (�1)KþK 0

�LSI (J000F 000; J 0F 0)

K K 0 1

F 000 F 0 F

� #; ð17bÞ

where we introduced the Larmor frequency !B ¼ �0B/ f and the function

�LSI (JF; J0F 0)¼ (�1)LþSþJþJ 0þIþF�JJ 0FF 0

J J 0 1

F 0 F I

�

;

"�JJ 0 (�1)LþSþ Jþ1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJ (J þ 1)

p

� J

þ (�1)J�J 0�S

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS(S þ 1)

p J J 0 1

S S L

� #: ð18Þ

This nondiagonal term is responsible for the relaxation of atomic coherences induced by an external magnetic field (‘‘Hanle effect’’)and also for the alignment-to-orientation conversion mechanism (e.g., Kemp et al. 1984; Landi Degl’Innocenti & Landolfi 2004). Wenote that equation (18) allows us to rewrite equation (16) in the more compact form


�¼ �JJ 0�FF 0 f!�LSI

JF þ �0B(�1)1�F�M�LSI (JF; J0F 0)

F F 0 1

�M M 0

� �: ð19Þ

2.2. Evaluation of the Radiative Rates

Because of the assumption that the radiation illuminating the atom does not have spectral structure over the frequency range inwhich quantum interferences within a term exist (in compliance with the limit of CRD, in which the present theory is known to bevalid; see x 1), the radiative rates are independent of the magnetic field strength, and so we can derive their expressions directly in thestandard representation without passing preliminarily through the energy-eigenstate representation. This possibility ultimately de-pends on the fact that the spectral details of the atomic configurations and the energy shifts induced by the external field are missingfrom the frequency dependence of the radiation illuminating the atom. They instead appear explicitly in the representation basis of theeigenstates of the atomic Hamiltonian (see eq. [4]), but they cancel out because of the orthogonality properties in equations (5a) and(5b). Below, we formally prove this property in the case of the absorption transition rate, TA , but the same argument applies directly tothe other transition rates and also to the relaxation rates, with small modifications.

Under this ‘‘flat-spectrum’’ approximation, TA satisfies the transformation property

TA(�LSI ; �M ; �0M 0;�lLlSI ; �lMl; �0lM

0l ) ¼

XJ F

XJ 0F 0

XJlFl

XJ 0lF 0l

C JF� (�LSI ;M )CJ 0F 0

� 0 (�LSI ;M 0)CJlFl

�l(�lLlSI ;Ml)C

J 0lF 0l

� 0l

(�lLlSI ;M0l )

; TA(�LSI ; JFM ; J 0F 0M 0;�lLlSI ; JlFlMl; J0l F

0l M

0l ):


The above expression must be multiplied by ��l� 0l� �lLlSI� (�lMl; �

0lM

0l ), expressed through equation (9a), and summed over the

states �l and � 0l . Use of the orthogonality property in equation (5a) gives

X�l� 0

l

TA(�; �0; �l; �

0l )��l� 0

l�XJ F

XJ 0F 0

C JF� (�LSI ; M )C J 0F 0

� 0 (�LSI ; M 0)

;X�lLl

XJlFlMl

XJ 0lF 0lM 0

l

TA(�LSI ; J FM ; J 0F 0M 0;�lLlSI ; JlFlMl; J0l F

0l M

0l )

�lLlSI� (JlFlMl; J0l F

0l M

0l );

where for convenience we renamed (JFM )! (J FM ) and (J 0F 0M 0) ! (J 0F 0M 0).We then apply the operation in equation (10) to bothsides of equation (1) to pass to the standard representation and use again the orthogonality property in equation (5a), which intro-duces a product ofKronecker � functions between barred and unbarred quantumnumbers. Therefore, we have proved our initial statementthat

X�l� 0

l

TA(�; �0; �l; �

0l )��l� 0

l!(9) X

�lLl

XJlFlMl

XJ 0lF 0lM 0

l

TA(�LSI ; JFM ; J 0F 0M 0;�lLlSI ; JlFlMl; J0l F

0l M

0l )

�lLlSI�(JlFlMl; J0l F

0l M

0l ): ð20Þ

We then consider the correspondences

n � �LSJIFM ; nl � �lLlSJlIFlMl; nu � �uLuSJuIFuMu;

n0 � �LSJ 0IF 0M 0; n0l � �lLlSJ0l IF

0l M

0l ; n0u � �uLuSJ

0uIF

0uM

0u;

n00 � �LSJ 00IF 00M 00;

n000 � �LSJ 000IF 000M 000;

ð21Þ

and for notational convenience, we indicate with !nu nl the (average) transition frequency between the terms (�uLuSI ) and (�l Ll SI ). Limitingthe present derivation to electric-dipole transitions, we then have (see, e.g., Landi Degl’Innocenti 1983; Landi Degl’Innocenti & Landolfi2004)

TA(n; n0; nl; n

0l ) �

16�3

f2c

Xqq 0

(�1)qþq 0J�q�q 0 (!nnl )��LSJIFM jdqj�lLlSJlIFlMl

� ��LSJ 0IF 0M 0jdq 0 j�lLlSJ

0l IF

0l M

0l

��; ð22aÞ

TS(n; n0; nu; n

0u) �

16�3

f2c

Xqq 0

(�1)qþq 0J�q�q 0 (!nun)��uLuSJ

0uIF

0uM

0ujdqj�LSJ

0IF 0M 0��uLuSJuIFuMujdq 0 j�LSJIFM��; ð22bÞ

TE(n; n0; nu; n

0u) �

4

3fc3!3nun

Xq

�� uLuSJ

0u IF

0uM

0ujdqj�LSJ 0IF 0M 0� ��u LuSJu IFuMujdqj�LSJIFM

��; ð22cÞ

RA(n; n0; n00; n000) � 8�3

f2c

Xnu

Xqq 0

(�1)qþq 0J�q�q 0 (!nun)

;��JJ 0 0�FF 0 0�MM 0 0

��u LuSJu IFuMujdqj�LSJ 0IF 0M 0� ��u LuSJu IFuMujdq 0 j�LSJ 000IF 000M 000��

þ �J 0J 0 0 0�F 0F 0 0 0�M 0M 0 0 0��u LuSJu IFuMujdqj�LSJ 00IF 00M 00� ��u LuSJu IFuMujdq 0 j�LSJIFM

�� ; ð22dÞ

RS(n; n0; n00; n000) � 8�3

f2c

Xnl

Xqq 0

(�1)qþq 0J�q�q 0 (!nnl )

;��JJ 0 0�FF 0 0�MM 0 0

��LSJ 000IF 000M 000jdqj�lLlSJlIFlMl

� ��LSJ 0IF 0M 0jdq 0 j�lLlSJlIFlMl

��þ �J 0J 0 0 0�F 0F 0 0 0�M 0M 0 0 0

��LSJIFM jdqj�lLlSJlIFlMl

� ��LSJ 00IF 00M 00jdq 0 j�lLlSJlIFlMl

�� ; ð22eÞ

RE(n; n0; n00; n000) � 2

3fc3

Xnl

Xq

!3nnl

;��JJ 0 0�FF 0 0�MM 0 0

��LSJ 000IF 000M 000jdqj�lLlSJlIFlMl

� ��LSJ 0IF 0M 0jdqj�lLlSJlIFlMl

��þ �J 0J 0 0 0�F 0F 0 0 0�M 0M 0 0 0

��LSJIFM jdqj�lLlSJlIFlMl

� ��LSJ 00IF 00M 00jdqj�lLlSJlIFlMl

�� ; ð22f Þ


where d is the electric-dipole operator, whose general matrix elements can be evaluated through the Wigner-Eckart theorem and itscorollaries (Brink & Satchler 1993):

��LSJIFM jdqj� 0L0SJ 0IF 0M 0� ¼ (�1)LþSþJþJ 0þI�Mþ1�LJJ 0FF 0

F F 0 1

�M M 0 q

� �J J 0 1

F 0 F I

� L L0 1

J 0 J S

� ��Ljjdjj� 0L0�: ð23Þ

The polarization tensors of the radiation field, Jqq 0 (!ul), must be expressed in terms of their irreducible spherical components, as

Jqq 0 (!ul) ¼XKQ

(�1)qþ1 �Kffiffiffi3

p1 1 K

q �q0 �Q

� �JKQ (!ul): ð24Þ

The definition of the irreducible spherical tensors JKQ (!) is

JKQ (!) ¼X3i¼0

Idk

4�TKQ (i; k) Si (!; k); ð25Þ

where (S0; S1; S2; S3) � (I ;Q;U ;V ) is the Stokes vector of the incident radiation (Chandrasekhar 1960; Born & Wolf 1983) andTKQ (i; k) are irreducible spherical tensors defining the propagation direction k and the reference direction of linear polarization in the

particular reference frame of the magnetic field previously introduced (see Fig. 2), in which the solution density matrix is calculated.These tensors are given in Table 1 (see also Landi Degl’Innocenti & Landolfi 2004).

Equation (23) shows that all radiative rates can ultimately be expressed in terms of the Einstein coefficients of the transition betweenthe terms (�uLuSI ) and (�lLlSI ):

A(�uLuSI ! �lLlSI )¼4

3fc3!3nunl

��uLujjdjj�lLl

�� 2; ð26aÞ

B(�uLuSI ! �lLlSI )¼4�3c2

f

1

!3nunl

A(�uLuSI ! �lLlSI ) ¼16�3

3f2c

��uLujjdjj�lLl

�� 2; ð26bÞ

(2Ll þ 1)B(�lLlSI ! �uLuSI )¼ (2Lu þ 1)B(�uLuSI ! �lLlSI ): ð26cÞ

We illustrate in some detail the steps to calculate the expressions of the radiative rates in the irreducible spherical-tensor represen-tation. Although we explicitly refer to the case of TA, an analogous procedure also applies to the other transition rates and to the re-laxation rates, with only minor modifications.

We apply equation (13) to the right-hand side of equation (20), with �lLlSI� (JlFlMl; J0l F

0l M

0l ) expressed through equation (11a) and TA

given by equation (22a). We then use equation (23) to express the product of dipole matrix elements and equation (24) for the radiation-field tensors. We thus obtain a product of five 3j symbols summed on the sextuplet (M ;M 0;Ml;M

0l ; q; q

0), which is evaluated throughequation (14) of VMK88 (p. 456). The procedure forTS is identical, whereas forTEweobtain a contraction of only four 3j symbols, whichis evaluated through equation (8) of VMK88 (p. 454). In the evaluation of RA and RS, we also obtain contractions of five 3j symbols, but

Fig. 2.—Geometry defining the propagation vector, k, and the reference direction of linear polarization, e1(k), in the frame of reference adopted for thediagonalization of the atomic Hamiltonian. The plane of polarization, generated by the basis vectors for linear polarization, e1(k) and e2(k), is normal to thepropagation vector k. The position angle � corresponds to the position of the acceptance axis of a linear polarizer through which 1

2(I þ Q) is measured.


these are instead evaluated through equation (13) of VMK88 (p. 456). Finally, in the case of RE, the contraction of four 3j symbols isdirectly evaluated through the orthogonality relations. In the case of the relaxation rates, it is also possible to sum explicitly over thequantum numbers J and F associated with the summations over terms that appear in equations (22d)–(22f ). This is accomplished viaequation (18) of VMK88 (p. 466) in the case of RA and RS, and through the orthogonality properties of 6j symbols in the case of RE.

Following this calculation procedure, the SE equations in the formalism of the irreducible spherical tensors can finally be written inthe form

d

dt�LSI �K

Q (JF; J0F 0) ¼�

XJ 0 0F 0 0

XJ 0 0 0F 0 0 0

XK 0Q 0

hiN (�LSI ; JJ 0FF 0KQ; J 00J 000F 00F 000K 0Q0)

þ RA(�LSI ; JJ0FF 0KQ; J 00J 000F 00F 000K 0Q0)þ RS(�LSI ; JJ

0FF 0KQ; J 00J 000F 00F 000K 0Q0)

þ RE(�LSI ; JJ0FF 0KQ; J 00J 000F 00F 000K 0Q0)

i�LSI�K 0

Q 0 (J00F 00; J 000F 000)

þX�lLl

XLlJl

XL 0lJ 0l

XKlQl

TA(�LSI ; JFJ0F 0KQ;�lLlSI ; JlFlJ

0l F

0l KlQl)

�lLlSI�Kl

Ql(JlFl; J

0l F

0l )

þX�uLu

XLuJu

XL 0uJ

0u

XKuQu

hTS(�LSI ; JFJ

0F 0KQ;�lLuSI ; JuFuJ0uF

0uKuQu)

þ TE(�LSI ; JFJ0F 0KQ;�lLuSI ; JuFuJ

0uF

0uKuQu)

i�u Lu SI�Ku

Qu(JuFu; J

0uF

0u); ð27Þ

with the expressions for the various radiative rates given by

TA(�LSI ; JFJ0F 0KQ;�lLlSI ; JlFlJ

0l F

0l KlQl) ¼ (2Ll þ1)B(�lLlSI ! �LSI )(�1)J

0�JþJ 0l�JlþF 0

l�FlþKlþQl�JJ 0JlJ

0lFF 0FlF

0l

;L Ll 1

Jl J S

� L Ll 1

J 0l J 0 S

� J Jl 1

Fl F I

� J 0 J 0

l 1

F 0l F 0 I

� XKrQr

ffiffiffi3

p�KKlKr

K Kl Kr

�Q Ql �Qr

� � K Kl Kr

F Fl 1

F 0 F 0l 1

8><>:

9>=>;JKr

Qr(!nnl );

ð28aÞTS(�LSI ; JFJ

0F 0KQ;�uLuSI ; JuFu J0uF

0uKuQu) ¼ (2Lu þ 1)B(�uLuSI ! �LSI )(�1) J

0�JþJ 0u�JuþF 0

u�FuþKuþKrþQu�JJ 0JuJ 0uFF

0FuF 0u

;L Lu 1

Ju J S

� L Lu 1

J 0u J 0 S

� J Ju 1

Fu F I

� J 0 J 0

u 1

F 0u F 0 I

� XKrQr

ffiffiffi3

p�KKuKr

K Ku Kr

�Q Qu �Qr

� � K Ku Kr

F Fu 1

F 0 F 0u 1

8><>:

9>=>;JKr

Qr(!nun);

ð28bÞTE(�LSI ; JFJ

0F 0KQ;�lLuSI ; JuFuJ0u F

0uKuQu) ¼ �KKu

�QQu(2Lu þ 1)A(�uLuSI ! �LSI )(�1) J

0�JþJ 0u�JuþF 0þF 0

uþKþ1�JJ 0Ju J 0uFF

0FuF 0u

;L Lu 1

Ju J S

� L Lu 1

J 0u J 0 S

� J Ju 1

Fu F I

� J 0 J 0

u 1

F 0u F 0 I

� F F 0 K

F 0u Fu 1

� ; ð28cÞ

TABLE 1

Polarization Tensors TKQ (i; k)

i ¼ 0 i ¼ 1

T 00 (0; k) ¼ 1 T 0

0 (1; k) ¼ 0

T10 (0; k) ¼ 0 T1

0 (1; k) ¼ 0

T11 (0; k) ¼ 0 T1

1 (1; k) ¼ 0

T 20 (0; k) ¼ 1½ / 2

ffiffiffi2

p� ��(3 cos2#� 1) T 2

0 (1; k) ¼ � 3½ / 2ffiffiffi2

p� �� cos 2� sin2#

T 21 (0; k) ¼ �

ffiffiffi3

p�/2Þ sin # cos # e i’ T 2

1 (1; k) ¼ �ffiffiffi3

p�/2Þ( cos 2� cos #þ i sin 2�) sin # ei’

T 22 (0; k) ¼

ffiffiffi3

p�/4Þ sin2# ei2’ T 2


p�/4Þ cos 2� 1þ cos2#ð Þ þ i 2 sin 2� cos #½ �ei2’

i ¼ 2 i ¼ 3

T 00 (2; k) ¼ 0 T 0

0 (3; k) ¼ 0

T10 (2; k) ¼ 0 T1

0 (3; k) ¼ffiffiffiffiffiffiffi3=2

pcos #

T11 (2; k) ¼ 0 T1


p�/2Þ sin # ei’

T 20 (2; k) ¼ 3½ / 2

ffiffiffi2

p� �� sin 2� sin2# T 2

0 (3; k) ¼ 0

T 21 (2; k) ¼

ffiffiffi3

p�/2Þ( sin 2� cos #� i cos 2�) sin # ei’ T 2

1 (3; k) ¼ 0

T 22 (2; k) ¼

ffiffiffi3

p�/4Þ sin 2� 1þ cos2#ð Þ � i 2 cos 2� cos #½ �ei2’ T 2

2 (3; k) ¼ 0

Notes.—This table lists the components of the irreducible spherical tensors TKQ (i; k) for Q � 0. The components with negative Q are obtained

from the conjugation property, TK�Q(i; k) ¼ (�1)Q TK

Q (i; k)�. The angular parameters #, ’, and � for a given propagation direction k and ref-

erence direction of linear polarization are defined in Fig. 2.


RA(�LSI ; JJ0FF 0KQ; J 00J 000F 00F 000K 0Q0) ¼

X�uLu

(2Lþ 1)B(�LSI ! �uLuSI )(�1)LuþS�IþFþQ 0þ1

;XKrQr

ffiffiffi3

p�KK 0Kr

K K 0 Kr

Q �Q0 Qr

� �L L Kr

1 1 Lu

� JKr

Qr(!nun)

;1

2

"�JJ 0 0�FF 0 0 (�1) J

0�J 0 0 0þKr�J 0J 0 0 0F 0F 0 0 0K K 0 Kr

F 000 F 0 F

� J 0 J 000 Kr

F 000 F 0 I

� J 0 J 000 Kr

L L S

�

þ �J 0J 0 0 0�F 0F 0 0 0 (�1) J�J 0 0þF 0 0�F 0þKþK 0�JJ 0 0FF 0 0

K K 0 Kr

F 00 F F 0

� J J 00 Kr

F 00 F I

� J J 00 Kr

L L S

� #; ð28dÞ

RS(�LSI ; JJ0FF 0KQ; J 00J 000F 00F 000K 0Q0) ¼

X�lLl

(2Lþ1)B(�LSI ! �lLlSI )(�1) LlþS�IþFþQ 0þ1

;XKrQr

ffiffiffi3

p�KK 0Kr

K K 0 Kr

Q �Q0 Qr

� �L L Kr

1 1 Ll

� JKr

Qr(!nnl )

;1

2

"�JJ 0 0�FF 0 0 (�1) J

0�J 0 0 0�J 0J 0 0 0F 0F 0 0 0

K K 0 Kr

F 000 F 0 F

� J 0 J 000 Kr

F 000 F 0 I

� J 0 J 000 Kr

L L S

�

þ �J 0J 0 0 0�F 0F 0 0 0 (�1) J�J 0 0þF 0 0�F 0þKþK 0þKr�JJ 0 0FF 0 0K K 0 Kr

F 00 F F 0

� J J 00 Kr

F 00 F I

� J J 00 Kr

L L S

� #;

ð28eÞ

RE(�LSI ; JJ0FF 0KQ; J 00J 000F 00F 000K 0Q0) ¼ �KK 0�QQ 0�JJ 0 0�J 0J 0 0 0�FF 0 0�F 0F 0 0 0

X�lLl

A(�LSI ! �lLlSI ): ð28f Þ

These equations can be specialized to treat the case of a multiterm atom devoid of HFS (such as, e.g., the He i atom) by simply lettingI ¼ 0, and F � J accordingly. In fact, it is easy to verify that equation (27), with equations (17a)–(17b) and equations (28a)–(28f ),reduces to the appropriate SE equations for this particular case (see Landi Degl’Innocenti & Landolfi 2004).

3. THE RADIATIVE TRANSFER EQUATION

In x 2 we presented the equations that govern the SE of an atom with fine and hyperfine structure subject to an external mag-netic field and illuminated by a radiation field that can be both anisotropic and polarized. The solution of this problem, which isachieved by solving equation (27), is represented by the irreducible spherical components of the density matrix operator for the atomicsystem, �LSI�K

Q (JF; J0F 0), which specifies not only the populations of the atomic levels, but also the polarization of those levels, as

well as their mutual interference due to the effect of the atomic fine structure and the applied fields.When the atomic density matrix is known, it is possible to compute the radiation that is emitted by an ensemble of atoms through

both spontaneous and stimulated processes. In an optically thick plasma, the production and transport of polarized radiation of an-gular frequency !, propagating in the direction k, are governed by the RT vector equation (Landi Degl’Innocenti 1983; LandiDegl’Innocenti & Landolfi 2004)

d

dsI(!; k) ¼ �

hK(a)(!; k)�K(s)(!; k)

iI(!; k)þ J(!; k); ð29Þ

where I � (I ;Q;U ;V ) is the Stokes vector of the polarized radiation, K(a))and K(s))are the absorption and stimulated-emissionmatrices,

K(a;s) �

�(a;s)I �

(a;s)Q �

(a;s)U �

(a;s)V

�(a;s)Q �

(a;s)I �

(a;s)V ��

(a;s)U

�(a;s)U ��

(a;s)V �

(a;s)I �

(a;s)Q

�(a;s)V �

(a;s)U ��

(a;s)Q �

(a;s)I

0BBBBBB@

1CCCCCCA; ð30Þ

and finally J � ("I ; "Q; "U ; "V ) is the emission vector. If the gas is optically thin, then it is easy to show that the Stokes vector of theradiation received by the observer is simply proportional to the emissivity vector J.


The expressions for the various radiative coefficients, �i, �i, and "i (i ¼ 0, 1, 2, and 3), for a transition between the lower term l andthe upper term u are (see, e.g., Landi Degl’Innocenti 1983; Landi Degl’Innocenti & Landolfi 2004)

�(a)i (!; k)¼ N

4�2!

fc

X�u�l�

0l

XMuMlM

0l

Xqq 0

(�1)qþq 0RenT�q�q 0 (i; k)

�lLlSI� (�lMl; �0lM

0l )�(!

�uLuSI ;�lLlSI�uMu;�lMl

� !)

;��uLuSI ; �uMujdqj�lLlSI ; �lMl

� ��uLuSI ; �uMujdq 0 j�lLlSI ; �

0lM

0l

��o; ð31aÞ

�(s)i (!; k)¼ N

4�2!

fc

X�l�u�

0u

XMlMuM 0

u

Xqq 0

(�1)qþq 0Re

nT�q�q 0 (i; k)�uLuSI� (�0

uM0u; �uMu)�(!


�!)

;��uLuSI ; �uMujdqj�lLlSI ; �lMl

��uLuSI ; �

0uM

0ujdq 0 j�lLlSI ; �lMl

��o; ð31bÞ

�(a; s)i (!; k) ¼ �(a; s)

i (!; k)fRe ! Img; (i ¼ 1; 2; 3) ð32aÞ

"i(!; k) ¼f

4�3c2!3nunl

�(s)i (!; k) ¼ A(�uLuSI ! �lLlSI )

B(�uLuSI ! �lLlSI )�(s)i (!; k): ð32bÞ

The spherical tensors Tqq 0 (i; k) define the geometry of the observer (line-of-sight direction k and reference direction for linearpolarization measurements) in the reference frame of the magnetic field, in which the solution density matrix is calculated (see Fig. 2).They are related to the corresponding irreducible spherical tensors, TK

Q (i; k), given in Table 1, through an expression formally identicalto equation (24).

Using the same substitutions as in the derivation of the radiative rates of the SE equations, we get, after some straightforwardalgebra,

�(a)i (!; k) ¼ N

f!

4�(2Ll þ 1)B(�lLlSI ! �uLuSI )

XKQ

XKlQl

ffiffiffi3

p�KKl

;X�l�u

XJlFlJ

0lF 0lJ 0 0lF 0 0l

XJuFuJ 0

uF0u

XMlM

0lMu

Xqq 0

(�1) J0u�JuþJ 0

l�JlþF 0 0

l�Mlþq 0þ1�JuJ 0

uJlJ0lFuF 0

uFlF0l

;CJlFl

�l(�lLlSI ;Ml)C

J 0 0lF 0 0l

�l(�lLlSI ;Ml)C

JuFu

�u(�uLuSI ;Mu)C

J 0uF

0u

�u(�uLuSI ;Mu)

;Fu Fl 1

�Mu Ml �q

� �F 0u F 0

l 1

�Mu M 0l �q0

� �1 1 K

q �q0 �Q

� �F 00l F 0

l Kl

Ml �M 0l �Ql

� �

;Ju Jl 1

Fl Fu I

� J 0u J 0

l 1

F 0l F 0

u I

� Lu Ll 1

Jl Ju S

� Lu Ll 1

J 0l J 0

u S

�

; RenTKQ (i; k)

�lLlSI�Kl

Ql(J 00

l F00l ; J

0l F

0l )�(!

�uLuSI ; �lLlSI�uMu; �lMl

� !)o; ð33aÞ

�(s)i (!; k) ¼ N

f!

4�(2Lu þ 1)B(�uLuSI ! �lLlSI )

XKQ

XKuQu

ffiffiffi3

p�KKu

;X�u�l

XJuFuJ 0

uF0uJ

0 0u F 0 0

u

XJlFlJ

0lF 0l

XMuM 0

uMl

Xqq 0

(�1) J0u�JuþJ 0

l�JlþF 0

u�Muþq 0þ1�JuJ 0uJlJ

0lFuF 0

uFlF0l

;CJuFu

�u(�uLuSI ;Mu)C

J 0 0u F 0 0

u�u

(�uLuSI ;Mu)CJlFl

�l(�lLlSI ;Ml)C

J 0lF 0l

�l(�lLlSI ;Ml)

;Fu Fl 1

�Mu Ml �q

� �F 0u F 0

l 1

�M 0u Ml �q0

� �1 1 K

q �q0 �Q

� �F 0u F 00

u Ku

M 0u �Mu �Qu

� �

;Ju Jl 1

Fl Fu I

� J 0u J 0

l 1

F 0l F 0

u I

� Lu Ll 1

Jl Ju S

� Lu Ll 1

J 0l J 0

u S

�

; RenTKQ (i; k)

�uLuSI�Ku

Qu(J 0

uF0u; J

00u F

00u )�(!


� !)o: ð33bÞ

We verified that for I ¼ 0, expressions (33a) and (33b) reduce to the ones valid for the multiterm atom devoid of HFS (see LandiDegl’Innocenti & Landolfi 2004).

Because of the explicit dependence of the (complex) line profile �(!) on the energy of the atomic eigenstates, it is not possible toachieve for the radiative coefficients the same kind of simplification that we were able to exploit in the case of the radiative rates. Onlyif the spectral details of the line profile induced by the external fields can be ignored will equations (33a) and (33b) reduce to a formthat no longer depends on the energy-eigenstate representation. This is the case, for instance, if the Zeeman splitting induced by the


applied magnetic field is negligible compared to the typical broadening of the line profile, or more simply if one is only interested inthe polarization signal integrated across the line spectral range.

If the Zeeman splitting can be neglected in the frequency dependence of the line profile, then all dependences on the energy-eigenvector expansion coefficients disappear, thanks to the orthogonality relation in equation (5a). It then becomes possible to con-tract the product of four 3j symbols (using eq. [8] of VMK88, p. 454) and also to perform the summations over the J and F numbersof the final level of the transition (using eq. [18] of VMK88, p. 466). We finally obtain

�(a)i (k)¼ N

f!

4�(2Ll þ1)B(�lLlSI ! �uLuSI )

XKQ

XJlFlJ

0lF 0l

(�1)LuþSþJ 0l�Jl�I�F 0

lþKþ1

ffiffiffi3

p�JlJ

0lFlF

0l

;Ll Ll K

1 1 Lu

� Ll Ll K

Jl J 0l S

� Jl J 0

l K

Fl F 0l I

� TKQ (i; k)

�lLlSI�KQ (JlFl; J

0l F

0l )Re

n� !�uLuSI ;�lLlSI �!� �o

: ð34aÞ

�(s)i (k)¼ N

f!

4�(2Lu þ 1)B(�uLuSI ! �lLlSI )

XKQ

XJuFu J 0

uF0u

(�1)LlþSþJu�J 0u�I�Fuþ1

ffiffiffi3

p�JuJ 0

uFuF 0u

;Lu Lu K

1 1 Ll

� Lu Lu K

Ju J 0u S

� Ju J 0

u K

Fu F 0u I

� TKQ (i; k)

�uLuSI�KQ (J

0uF

0u; JuFu)Re

n� !�uLuSI ;�lLlSI � !� �o

: ð34bÞ

If, instead, the Zeeman splitting cannot be neglected in equations (33a) and (33b), then the needed expansion coefficients,CJF� (�LSI ;M ), and eigenvalues, k�(�LSI ;M ), must be determined by diagonalization of the atomic Hamiltonian, equation (19),

according to the first-order theory of stationary perturbations (e.g., Messiah 1961).Although the derivation of equations (33a) and (33b) is in compliance with the hypothesis of CRD, it is clear that after the first

scattering event, the radiation field is no longer spectrally flat. This poses a consistency problem when applying the present theory tothe transfer of radiation in a medium that is not optically thin.

One way to overcome this difficulty is to interpret each JKQ component of the radiation field tensors entering the SE equations as anaverage over the spectral range of interest (e.g., an intensity-weighted average over the multiplet or a single-frequency value). Averysimple strategy can then be developed from this assumption. One first relies on the fact that in a stellar atmosphere, to a very goodapproximation, the atomic polarization of the levels is basically set by the radiation field anisotropy and not by the (low) degree oflight polarization. Therefore, the radiation field is calculated under the usual (unpolarized) non-LTE hypothesis, and from it one thenevaluates the frequency-averaged JKQ tensor components at all heights in the atmosphere (we assume here the simplest case of a one-dimensional atmosphere). After solving the corresponding SE problem, one can calculate the radiative transfer coefficients "i and �i,which allow one to evaluate the emergent polarized radiation after one single formal solution of the RT equation.

4. ILLUSTRATIVE RESULTS

We created a numerical code incorporating the formalism presented in the previous sections. Because of the complexity ofthe atomic systems that need to be described through this formalism, it is not possible to conduct many exhaustive numericalchecks of the code. Even for the simplest system, that of H i Ly, the density matrix �LSI�K

Q (JF; J0F 0) includes 160 elements.

We checked that the program reproduces the results for the multiterm atom devoid of HFS (e.g., He i) if we explicitly specify I ¼ 0in the atomic data file and also verified that forcing the diagonality of the SE equations over the J quantum numbers would repro-duce the results for the multilevel atom with HFS (e.g., Na i neglecting the superinterferences). In the more general case, we checkedthat the solution density matrix satisfied the proper conjugation properties and the normalization condition of the total atomicpopulation.

The ultimate check is obviously to try reproducing observations of line profiles that are known to form under the conditions ofour formalism. The main difficulty here is the lack of such observations, mostly because of the restrictive assumption of CRD and ofthe single-scattering approximation. Whereas the latter can be eventually overcome by implementing our program into a radiativetransfer code, the CRD limitation is built into the formalism. The natural choice of the solar-limb observations of the Na i lines as a testof our code is likely to be affected by both these restrictions. Nonetheless, even with our simplifying assumptions, our code is able toreproduce the global pattern of the superinterferences between D1 and D2 and represents a starting point for future investigations ofthe formation processes of these lines.

In Figure 3, we show a plot of the fractional linear polarization Q /I of the D1 and D2 lines for a particular configuration of themagnetic field (see caption). This plot corresponds to an off-limb observation with the scattering center located at a height of 0.01 R�.Whereas the typical trilobate shape of D2 cannot be reproduced by our simple modeling for an optically thin plasma, the shape of D1,and in particular the presence of a central peak in its core, closely resembles observations of this line published in recent work (e.g.,Sheeley &Keller 2003). Departures of the line shapes in Figure 3 from typical observations of the D1 and D2 lines are likely to be dueto radiative transfer effects and possibly PRD effects. We are currently working on the implementation of our formalism into aradiative transfer code to verify this hypothesis.

In Figure 4, we present more calculations of the fractional linear polarization of D1 for several magnetic field strengths (0–50 G)and inclinations (0

�–90

�). In addition to the expected sensitivity of the line core’s shape and amplitude to the magnetic field strength

and inclination, we also notice a variation of the zero-crossing wavelength of the superinterferences between D1 and D2. This mightprove to be a new interesting diagnostic tool for the investigation of solar magnetism.


Fig. 4.—Synthetic profiles of Q/I of the Na i D1 line observed off-limb, for the same scattering geometry, illumination conditions, and plasma temperature ofFig. 3. The magnetic field ranges from 0 to 50 G, with different inclinations from the local vertical (0�, 30�, 60�, and 90�), and it is assumed to be uniformly dis-tributed in azimuth. The reference wavelength in these plots is that of the peak intensity of D1.

Fig. 3.—Synthetic profiles of the fractional linear polarization Q/I of the D1 and D2 lines of Na i observed off-limb (90� scattering) at a height of 0.01 R� , in thepresence of a field of 45 G, inclined 90� from the local vertical and uniformly distributed in azimuth. The reference wavelength in this plot is that of the peakintensity of D–>1, at 5896 8 (a normalized plot of the intensity profiles is superposed for reference: dotted line). We implemented the flat-spectrum approximationusing a Planckian illumination at 6000 K.

5. CONCLUSIONS

In this paper we presented a general formalism that allows us to model the resonance scattering polarization of spectral lines fromatoms with both fine structure and hyperfine structure in the presence of arbitrary magnetic fields. In addition to the D1 and D2 lines ofNa i considered in this paper, the formation of other resonance lines of great diagnostic interest can be investigated with thisformalism, such as the D lines of other elements in the groups of alkaline metals and ionized alkaline earth metals. For instance, the Li iand Be ii doublets (respectively at 6707.9 and 6707.88 and at 3131.1 and 3130.48), like the Na iD lines, are characterized by upperlevels 2P1/2 and

2P3/2 that are close enough in energy to require the treatment of both fine- and hyperfine-structure coherences. How-ever, we should note that to model the observed pattern of the Li i doublet in the second solar spectrum (Stenflo et al. 2000), weprobably also need to consider isotopic shifts, whereas observations of the Be ii doublet (Stenflo 2003) indicate that continuum po-larization should be accounted for. Laboratory observations of the K i doublet (at 7699.0 and 7664.9 8) offer the possibility of study-ing a case similar to the Na i D lines, but in the case in which fine-structure coherences are negligible. The same is true for the D1 andD2 lines of Ba ii in the second solar spectrum at 4934.1 and 4554.08, respectively. In particular, the D1 line of Ba ii, like the D1 line ofNa i, has a well-documented observed polarization profile (Stenflo et al. 2000), and among the solar lines it represents the most prom-inent example of the simpler scenario in which fine-structure coherences can be neglected. The modeling of the D2 line of Ba ii is alsoof great interest for solar diagnostics because of the significant amplitude of the observed Q /I signal (of the order of 1%; Stenflo& Keller 1997; Gandorfer 2002). We point out that Stenflo (1997) was able to explain the triplet shape of this line in terms of relativeabundances of the barium isotopes and their hyperfine-structure separations.

Our formalism is limited by the assumption of complete redistribution of the incident radiation and also lacks a formulation of theeffect of collisions on the scattered polarization signal. Depolarizing (elastic) collisions are easily implemented, when their crosssections are known. In theory, inelastic collisions can also be added to our formalism, under the simplifying hypothesis of the impactapproximation, although one needs first to develop a description of the transition-inducing collisional cross sections based on theformalism of the irreducible spherical tensors, in a way similar to the one presented here for the radiative cross sections.

Lifting the constraint of complete redistribution is a daunting goal for atomic systems of high complexity. A self-consistent theoryof partial redistribution effects in scattering polarization for complex atoms is a work in progress. Even if one can possibly envision inthe near future the construction of a general formalism that enables a complete description of radiation scattering from an atom ofarbitrary complexity (such as the ones that necessitate the formalism presented in this paper), the numerical problem that wouldoriginate even from the simplest cases might prove to be extremely difficult to solve. The inclusion of radiative transfer to simulaterealistic line formation conditions brings in even more challenges to this already intricate problem. It is possible that in the end,one will have to resort to simpler, heuristic formulations of the scattering problem for complex atoms and other clever approximationsthat still succeed in reproducing the main physical aspects of resonance line polarization (see, e.g., the formulation of LandiDegl’Innocenti et al. [1996] using the concept of ‘‘metalevels’’). This is also true for the description of the radiative transfer problemin a realistic atmosphere. For instance, the simple radiative transfer scheme described at the end of x 3 will allow us to account forpotentially interesting effects not yet considered in the formation of the polarization signature of the Na i D lines.

The authors wish to thank E. Landi Degl’Innocenti (University of Florence, Italy) and J. Trujillo Bueno (Instituto de Astrof ısica deCanarias, Spain) for insightful discussions on the formation of the Na i D lines. We also are grateful to the anonymous referee forcritical comments that helped us to improve the presentation of this work.

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Line Formation Theory for the Multiterm Atom with Hyperfine Structure in a Magnetic Field

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