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Astron. Astrophys. 324, 344-356 (1997)

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2. Quantum theory of line polarizability

2.1. Introduction

For allowed electric dipole transitions the quantum-mechanical amplitude for scattering from initial substate a to final substate f via the intermediate, excited substates b is given by the Kramers-Heisenberg formula

[EQUATION]

[FORMULA] are the polarization vectors of the radiation field, [FORMULA] the frequency of the scattered radiation, [FORMULA] the energy difference between the upper and final levels, and [FORMULA] the damping constant (which is the sum of the radiative and collisional contributions). The summation is done over all the intermediate states b. Since energy conservation requires that [FORMULA], we could also express Eq. (1) in terms of the frequencies [FORMULA] of the incident radiation.

The radiation coherency matrix [FORMULA] that describes the transformation from the polarization state of the incident to that of the scattered radiation is given by

[EQUATION]

(Stenflo 1994 ), where the symbols [FORMULA] and [FORMULA] stand for tensor product and complex conjugation, respectively. For simplicity we have here assumed (as we will do in the rest of the paper) that there is no atomic polarization in the initial state a when summing over all the initial and final magnetic substates that are represented by their magnetic quantum numbers [FORMULA]. This allows us to express the polarizing characteristics of the scattering process in terms of a phase matrix (see below) that is independent of the scattering medium (the model atmosphere used) and only depends on atomic physics. The neglect of the polarization of state a is physically well justified because the life time of the initial state with respect to radiative absorption processes is longer than the radiative life time of the excited state by about two orders of magnitude (cf. Sect.  4). The initial state therefore has plenty of time to be depolarized by collisions and weak magnetic fields.

Note that the initial, intermediate, and final states may have arbitrary Zeeman splittings, and that [FORMULA] in Eq. (1) contains the Zeeman displacements of the magnetic sublevels b and f. The formulation therefore allows for magnetic fields of arbitrary strengths.

To obtain the scattering Mueller matrix [FORMULA] that describes how a Stokes 4-vector is transformed by the scattering process, we form

[EQUATION]

where [FORMULA] is a purely mathematical transformation matrix without physical contents, given explicitly in Stenflo (1994 ). The constant of proportionality is determined by the normalization condition for [FORMULA].

We may expand [FORMULA] in terms of its multipolar components (Stenflo 1996 ):

[EQUATION]

where index K represents the [FORMULA] -multipole. Let us by [FORMULA] define a [FORMULA] matrix that has its ij component equal to unity, while all the other components are zero. Then, in the limit of weak magnetic fields, defined by the requirement that the Zeeman splitting should be much smaller than the Doppler broadening, [FORMULA], while [FORMULA]. This means that [FORMULA] represents isotropic, unpolarized scattering, while [FORMULA] only scatters the circular polarization, which in this limit is decoupled from the linear polarization. Thus the scattered radiation contains circular polarization only if the incident radiation also contains circular polarization. In contrast, unpolarized incident light is the main source of linear polarization in the scattered radiation. Therefore the study of scattering physics focuses on the linear polarization, which is generated by the [FORMULA] matrix.

In the weak-field limit the effect of the Zeeman splitting on the shapes of the absorption line profiles can be disregarded. This means that all the matrix components of [FORMULA] can be assumed to have the same frequency profile, which therefore can be factorized out from [FORMULA] as a common scalar. The remaining, frequency-independent matrix that describes the polarization properties of the scattering process is in its normalized form the so-called phase matrix [FORMULA]. It can be expanded as

[EQUATION]

where, in the case of zero magnetic fields, [FORMULA] is the classical, Rayleigh scattering phase matrix. By definition and normalization [FORMULA] and [FORMULA]. While [FORMULA] for classical scattering the values of these coefficients generally depend on the total angular momentum quantum numbers of the initial, intermediate, and final states a, b, and f. All the quantum-mechanical effects on the scattering process are thus hidden in the [FORMULA] coefficients.

When the magnetic field is weak but not zero it influences the scattering process via the Hanle effect. Explicit expressions for the [FORMULA] matrices in the presence of the weak-field Hanle effect (when the Zeeman splitting is much smaller than the Doppler width) were first given in Stenflo (1978 ) (cf. also Stenflo 1994 ).

If we disregard the here unimportant circular polarization, the phase matrix that results from Eq. (1) can in the case of zero magnetic field be written as

[EQUATION]

where [FORMULA] is the Rayleigh phase matrix that is valid for the case of classical dipole scattering. [FORMULA] thus represents the fraction of scattering processes that occurs as classical dipole scattering, while the remaining fraction, [FORMULA], represents unpolarized, isotropic scattering.

2.2. Raman scattering with quantum interferences

In the present paper the terms fluorescent scattering and Raman scattering will both be used to describe cases when the initial and final states are different. The term fluorescent scattering is used when the excitation occurs near an actual resonance, while Raman scattering can occur at frequencies that are arbitrarily far from the resonant frequencies. Raman scattering is therefore the more general concept and approaches continuously the case of fluorescent scattering when we move closer in frequency towards a resonance. Similarly Rayleigh and resonant scattering both refer to the case when the initial and final states are the same, but while resonant scattering occurs near a resonance, Rayleigh scattering does not have this restriction. In this sense all the three terms fluorescent, Rayleigh, and resonant scattering can be considered to be special cases of Raman scattering (if we consider the case when the initial and final states are identical as a special case of Raman scattering).

The scattering theory developed in the present paper is valid for general Raman scattering. The other terms will be used when addressing special cases.

Let us now in what follows for convenience of notation let a, b, and f refer not as before just to the individual magnetic substates, but instead to the states that are characterized by their total angular momentum quantum numbers (and thus contain magnetic substates, which may in principle be Zeeman shifted). For a given af combination of initial and final states in the limit of weak magnetic fields we may express the Raman scattering Mueller matrix [FORMULA] as

[EQUATION]

The coefficients [FORMULA], which are frequency dependent (see below), contain not only the squared terms but also the interference terms between the various intermediate states of different total angular momenta. Eq. (7) allows us to define a phase matrix of the form (5) if we let

[EQUATION]

If more than one intermediate state contributes to the transition from state a to state f, [FORMULA] will in general be frequency dependent.

The coefficients [FORMULA] can more explicitly be given as

[EQUATION]

(cf. Stenflo 1994 ), where m and n label the intermediate states. The coefficients [FORMULA] depend exclusively on the total angular momentum quantum numbers of levels [FORMULA]. The complex profile function [FORMULA] is given by

[EQUATION]

while

[EQUATION]

Here the exponent r is 0 or 1 (thus determining the sign of the expression) depending not only on the total angular momentum quantum numbers J of all the four levels a, m, n, f involved, but also on their respective orbital angular momentum quantum numbers L (in the case of hyperfine structure splitting [FORMULA] are replaced by [FORMULA]). The [FORMULA] etc. in Eq. (11) are the respective absorption oscillator strengths. Their relative values within a multiplet have been given as algebraic expressions of the L, S, and J quantum numbers in Condon & Shortley (1970 ). Algebraic expressions for the coefficients [FORMULA] and the exponent r have been derived and given in Stenflo (1994 ) and are reproduced in the Appendix in a form that is adapted to the somewhat different notations that we have used here. Chapter 9 of Stenflo (1994) explains how these expressions can be derived from sums of products of 3-j symbols for any [FORMULA] (although the less important [FORMULA] case has not been dealt with explicitly yet).

The off-diagonal terms in Eq. (9) represent quantum mechanical interferences between states with different values of [FORMULA] and [FORMULA]. For [FORMULA] such terms cancel out, so that we get

[EQUATION]

Quantum interferences between excited states of different total angular momenta can only be neglected when the line opacity becomes much smaller than the continuum opacity at distances from the resonant frequencies that are comparable to the fine structure splitting. In many cases this condition is not at all satisfied, e.g. in the case of the Ca II H and K lines at 3965 and 3933 Å , the Na I D1 and D2 lines at 5895.94 and 5889.97 Å , or for the hyperfine structure splitting in lines like Ba II 4554 Å , as we will see below. If only a single excited state b with [FORMULA] needs to be considered, e.g. when we are at a resonance frequency,

[EQUATION]

The diagonal coefficients are given by

[EQUATION]

where the brackets denote 6-j symbols (cf. Landi Degl'Innocenti 1984 ; Stenflo 1994 ). Simple, explicit algebraic expressions have been given by Chandrasekhar (1950 ) for the resonant case [FORMULA] and by Stenflo (1994 ) for the non-resonant (fluorescent or Raman) case [FORMULA].

2.3. Polarizability for an entire multiplet

To obtain the polarizability [FORMULA] for an entire multiplet one has to extend the sums in Eq. (2) over all the possible initial and final states of the multiplet, i.e., one has to add together all the possible fluorescent or Raman scattering contributions within the multiplet. Thereby one needs to account for the multiplicity ([FORMULA]) of these states, which represents the number of magnetic substates for a given J state (in the case of fine-structure splitting). The emission probability from a given excited state scales with [FORMULA], while the absorption probability scales with the relative population [FORMULA] of the initial states. Effectively it means that the oscillator strengths f in Eq. (11) get replaced by their gf values (g being the statistical weight). Thus the scattering matrix for the whole multiplet becomes

[EQUATION]

where [FORMULA] is given by Eqs. (1)-(3) in the case of arbitrary Zeeman splitting, and by Eq. (7) in the case of weak magnetic fields. The above as well as the following expressions are valid for multiplets produced by fine-structure splitting. For a hyperfine structure multiplet the expressions remain the same if we replace J with the total angular momentum quantum number F.

If we introduce a phase matrix as in Eq. (5) with the same normalization as before but now representing the whole multiplet, and define

[EQUATION]

then the polarizability coefficients [FORMULA] become

[EQUATION]

In the general case when the incident radiation is not spectrally flat one has to apply the intensity as an additional weight to the nominator and denominator of Eq. (17). Also [FORMULA] in Eq. (16) needs to be convolved with a Gaussian to account for the thermal and turbulent Doppler broadening before it is inserted in the nominator and denominator of Eq. (17) to form [FORMULA].

Fig. 1 shows the energy level diagrams for two multiplets that we will be dealing with, fine structure multiplet No. 318 of neutral iron, and the hyperfine multiplet of the Ba II 4554 Å line. The solid, dashed and dotted lines show some of the different possible combinations of fluorescent scattering transitions in these multiplets.

[FIGURE] Fig. 1. Energy level diagrams to illustrate scattering transitions in multiplet 318 of Fe I and in the hyperfine multiplet of the Ba II 4554 Å transition. While the relative level shifts within a group of initial (index a), intermediate (index b), or final (index f) states are in the right proportions, we have applied different magnifications to clearly see the levels in the plot. As the hyperfine splitting is so tiny, the energy differences within the [FORMULA] group have been magnified by the factor 42,370 with respect to the energy scale for the [FORMULA] group, while the corresponding magnifications for the [FORMULA] and [FORMULA] groups are 3,390 and 2.5, respectively. Thus the upper level hyperfine splitting in Ba II has been magnified by the factor 12.5 with respect to the lower level splitting. The three thick, unlabeled horizontal lines in the Ba II diagram mark the level positions in the absence of hyperfine splitting. Examples of some allowed fluorescent scattering transitions are drawn and are commented on in the text. Note that for 318 Fe I the energy levels decrease with increasing value of J.

If we sum up all the allowed fluorescent combinations with Eq. (16), then Eq. (17) gives us a polarizability [FORMULA] that can have a very complex wavelength dependence, as shown by Fig. 2, in which the two multiplets of Fig. 1 are represented by the two lower panels. The upper panels represent respectively scattering in the Na I D1 -D2 multiplet and Raman scattering from the ultraviolet Ca II H and K multiplet No. 1 (at 3965 and 3933 Å) into the infrared multiplet No. 2 of Ca II. While the solid curves give the full solutions, the dashed curves show what happens when the interference terms are left out and only the diagonal contributions to [FORMULA] in Eq. (9) are used. The Ba II diagram has been constructed as a sum of the different isotopic contributions, and Doppler and instrumental broadening have been applied, in preparation for the later comparison with the observations in Fig. 7 (cf. Sect.  3.4).

[FIGURE] Fig. 2. The polarizability [FORMULA] when accounting for all the allowed Raman scattering transitions within the multiplets 1 Na I and 318 Fe I, between the two multiplets 1 Ca II and 2 Ca II, and within the hyperfine structure multiplet of the 1 Ba II 4554 Å line. The solid curves represent the full solutions, while the dashed curves show what happens if we ignore the quantum interferences between states of different total angular momenta. Only when the interference terms are taken into account we get the asymptotic behavior that is demanded by the principle of spectroscopic stability.

2.4. Interference and spectroscopic stability

The principle of spectroscopic stability provides us with a powerful tool to check the correctness of our algorithms to compute the polarizability. In the present context we use this principle to determine what must happen in the limit of vanishing fine (or hyperfine) structure splitting. The fine structure is physically due to the electron spin, so the polarizability [FORMULA] in the limit of vanishing splitting is simply obtained by letting the spin quantum number S be zero, which means that we use [FORMULA] in Eq. (14) for [FORMULA]. This limiting value of [FORMULA] must be reached asymptotically when moving away in the spectrum to distances from the resonant wavelengths that are much larger than the wavelength separations between the fine structure components, since at such distances the splitting loses its importance.

Let us use this principle to check the results of Fig. 2. The 1 Na I multiplet represents a transition for which [FORMULA]. With [FORMULA] in Eq. (14) we find that [FORMULA]. The solid curve indeed approaches unity asymptotically when moving either to smaller or larger wavelengths. A [FORMULA] transition is the quantum-mechanical analog to classical dipole-type scattering, for which [FORMULA] of course always has its classical value, unity.

The 1 Ca II [FORMULA] 2 Ca II transition corresponds to [FORMULA]. This gives the asymptotic value [FORMULA], in agreement with the solid curve. Similarly, as the 318 Fe I multiplet is an 7 [FORMULA] D [FORMULA] [FORMULA] transition, we have [FORMULA], which gives [FORMULA], in agreement with the figure.

In the hyperfine case of the Ba II 4554 Å line we have to let the nuclear spin I, which for the odd Ba isotopes is 1.5, go to zero, so that F becomes equal to J. As [FORMULA], for which [FORMULA], this is the asymptotic value, in agreement with the figure.

The dashed curves in Fig. 2 show the solutions for [FORMULA] that are obtained if all the quantum interference (off-diagonal) terms in Eq. (9) are omitted. The asymptotic behavior is then entirely different and violates the principle of spectroscopic stability.

The oscillator strengths f and the sign factor [FORMULA] in Eq. (11) are complicated functions of all the quantum numbers J, L, and S (or F, J, and I in the hyperfine case). If any one of these algebraic expressions or those for the coefficients [FORMULA] would contain an error or a wrong sign, this would immediately show up as a violation of the principle of spectroscopic stability for some combinations of the quantum numbers. We have let our computer program scan through all possible combinations of quantum numbers to verify that the principle of spectroscopic stability is always obeyed.

The principle of spectroscopic stability can be shown to be obeyed also for scattering from each separate initial [FORMULA] state, before summing in Eq. (16) over [FORMULA] (but after summing over the final [FORMULA] states). This is an even more restrictive condition that the algebraic expressions have to satisfy.

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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