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

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3. Modelling the observed polarization

3.1. Effective polarizability

The observed linear polarization depends on a number of factors: (i) The atomic polarizability coefficient [FORMULA]. (ii) The fraction of the emission processes that represent actual scattering transitions. (iii) The "geometric depolarization" factor, determined by the degree of anisotropy of the incident radiation field. (iv) The collisional depolarization factor, i.e., the fraction of the scattering processes that are undisturbed by depolarizing collisions. (v) The Hanle depolarization factor, which expresses how the polarization amplitudes are reduced in the presence of magnetic fields.

There are however many contributors to the solar photons that we observe, including the continuous spectrum and atomic transitions of various other elements. The continuum is weakly polarized due to Thomson scattering at free electrons and Rayleigh scattering at neutral hydrogen. Radiative scattering does not play a significant role in all atomic transitions. Non-scattering lines may depolarize the continuum by removing polarized photons and diluting the continuum radiation with unpolarized photons.

To obtain a representation of the polarization that we can expect to observe from a given atomic multiplet we need to add the contributions from the continuum and from the considered line multiplet, weighted according to the relative number of photons they deliver, i.e., according to the photon emission probability functions. For a given multiplet the photon emission probability is proportional to [FORMULA] of Eq. (16) after it has been convolved by a Gaussian due to thermal and turbulent Doppler broadening. Let us denote this convolved and area-normalized [FORMULA] function by [FORMULA]. We note that [FORMULA] is in general, as shown by Eqs. (16), (12), and (10), a composite of several different Voigt profiles with different weights and central frequencies.

The corresponding function for the continuum is spectrally flat and can be represented by a constant a, the value of which depends on the magnitude of the continuum opacity relative to the line opacity of the considered multiplet. If we ignore the contributions from other spectral lines we can define an "effective" value of [FORMULA] that represents the weighted sum of the line and continuum contributions (Stenflo 1980 ):

[EQUATION]

The additional free parameter b represents the relative continuum polarization, which scales with the fraction of continuum emission processes that represent scattering transitions.

To obtain a fit of the observed polarization curves we have to multiply [FORMULA] with a scale factor that contains the various factors (ii)-(v) mentioned above. In general the factors (iii)-(v) are wavelength dependent, in particular the geometric depolarization factor, and radiative transfer is needed to determine how these factors vary. However, it turns out to be a good approximation for a qualitative discussion of the physics involved to bypass radiative transfer and simply assume that the combined geometric, collisional, and Hanle depolarization factors as well as the incident radiation are spectrally flat over the selected wavelength range. In this case we may obtain the synthetic polarization profile as a function of wavelength directly from [FORMULA] after applying a global, wavelength-independent scale factor.

With a full radiative transfer treatment there would be no free parameters, since their values would be uniquely determined by the model atmosphere used. Our parametrization is introduced to allow us to make useful interpretations while bypassing radiative transfer. It is possible to extend the parametrized model and require that it should simultaneously also fit the wings of the Stokes I profile. This additional constraint would fix the value of the parameter a and remove it from its status as a free parameter (Faurobert-Scholl, private communication). As however such an extension of the model is non-trivial and would involve the use of the Eddington-Barbier relation, it is outside the scope of the present paper.

To summarize, our idealized model for the observed polarization that bypasses radiative transfer thus has three free parameters: the relative continuum opacity a, the continuum polarization parameter b, and the global scale factor.

3.2. Modelling quantum interferences in sodium

Fig. 3 shows an application of this model to explain the observed polarization profile across the Na I D1 and D2 lines that make up multiplet No. 1 of Na I. While the thin solid curve is the observed one (5 arcsec inside the limb at the north pole of the Sun), the thick solid and dashed curves have been obtained from the corresponding solid and dashed [FORMULA] curves in the upper left panel of Fig. 2, using Eq. (18) with [FORMULA], [FORMULA], and a scale factor of 1.01% . Here a is given in units of the value that [FORMULA] has at the wavelength halfways between the two resonant frequencies. While the dashed curve without the quantum interferences is unable to reproduce the sign reversals of the polarization curve around the D1 line for any combination of the free model parameters, the solid curve provides a surprisingly good representation of the gross features of the observed curve, apart from the narrow spectral features in the two line cores.

[FIGURE] Fig. 3. Modelling the observed quantum interference pattern in multiplet No. 1 of sodium. While the upper panel gives the intensity spectrum I, normalized to the intensity [FORMULA] of the local continuum, the lower panel gives the degree of linear polarization, [FORMULA]. Positive values mean that the electric vector of the scattered radiation is preferentially oriented parallel to the nearest solar limb, while negative values represent the orthogonal direction. The thin solid curve was recorded in April 1995 with the ZIMPOL I polarimetric system at the National Solar Observatory (Kitt Peak) with the spectrograph slit 5 arcsec inside the limb at the north pole of the Sun. The thick solid and dashed curves have been obtained by theoretical modelling, the solid with and the dashed without taking the quantum interferences into account.

A sign reversal means that the electric vector of the scattered radiation, which for positive polarization is oriented parallel to the nearest solar limb, changes its orientation by [FORMULA]. This sign reversal is a typical quantum interference effect, in the present case due to coherent superposition of the states with total angular momentum quantum numbers [FORMULA] and [FORMULA]. The same phenomenon has been observed and modeled for the Ca II H and K lines at 3965 and 3933 Å (Stenflo 1980 ), which have the same quantum numbers as the D1 and D2 lines.

For Rayleigh scattering the polarizability can never be negative unless there is quantum interference. If the resonant frequencies of the D1 and D2 lines are [FORMULA] and [FORMULA], the polarizability of the Na I transition as determined by Eq. (8) is

[EQUATION]

To simplify the expression we have omitted the damping constant [FORMULA], which is allowed since it is unimportant when the fine-structure splitting is so much larger than the damping width. Expression (19) shows explicitly that the source of the negative sign is exclusively in the interference term in the nominator.

The modelling in Fig. 3 ignores the hyperfine structure in sodium, so it is natural to expect that the narrow polarization features in the line cores that are not reproduced by our model could be due to hyperfine structure effects. However, we have performed corresponding model calculations for the hyperfine structure multiplets of the D2 and D1 transitions without being able to come close to fitting the core polarization peaks, in spite of the great success of the same theory when applied to the hyperfine structure multiplet of the Ba II 4554 Å line, as we will see below. Thus it appears unlikely that hyperfine structure can have much to do with the narrow polarization peaks of sodium.

The general shape of the D2 profile, with a narrow core peak surrounded by minima followed by wing maxima, is the same as previously found for the Ca I 4227 Å and Ca II K 3933 Å lines ( Stenflo et al.  1980 , 1983a ,b), which have no hyperfine structure (calcium has zero nuclear spin). It has been possible to explain these types of polarization profiles in terms of partial redistribution effects in polarized radiative transfer (Rees & Saliba 1982 ; Saliba 1985 ; Frisch 1996 ). Therefore it appears likely that the D2 line can be explained in this way as well, but solutions of the radiative-transfer problem would be needed to prove this point.

For the D1 line, on the other hand, frequency redistribution within this line is unlikely to provide an explanation of the narrow polarization peak, since the polarizability [FORMULA] near the D1 resonance is close to zero, and the hyperfine splitting does not seem capable of changing the polarizability sufficiently. Then the frequency redistribution process does not seem to have the building blocks out of which a polarization peak could be constructed.

Interestingly, the two 1 Ba II lines at 4554 and 4934 Å have the same quantum numbers and hyperfine structure patterns (although the magnitudes of the splittings are different) as the 1 Na I D2 and D1 lines. The "D1 -type" line Ba II 4934 Å is also observed to have a sharp, pronounced polarization peak, which supports the reality of the Na I D1 peak and indicates that they have a common, yet unidentified, physical origin. For the Ba II line pair the fine structure splitting is so large (380 Å) that quantum interference from the D2 transition at 4554 Å cannot play a significant role. Thus new physics not covered by the present theoretical framework may be needed to explain the narrow D1 polarization peaks of Na I and Ba II.

3.3. Signatures of fluorescent scattering

Let us now turn to multiplet No. 318 of Fe I, which according to Fig. 1 has a complex fine structure splitting pattern with a rich variety of possible fluorescent scattering combinations within the multiplet, which as seen in the lower right panel of Fig. 2 results in a complex structure for the polarizability [FORMULA]. The various combinations of energy levels lead to several doublet lines. Two such pairs are shown in the left and right panels of Fig. 4. Both have a complex polarization structure. Here we will focus the discussion on the 4957 Å line pair, since its polarization feature has remained an unexplained riddle since it was first observed with a Fourier transform spectrometer in 1978 (Stenflo et al.  1983b ). Only now it can be demonstrated that it is a signature of fluorescence within the multiplet, as will be shown below.

[FIGURE] Fig. 4. Two portions of the spectrum that each contain a doublet line pair from multiplet No. 318 of Fe I. The spectra have been pieced together from separate, partially overlapping recordings (obtained in April 1995 with ZIMPOL I at NSO/Kitt Peak, 5 arcsec inside the north polar limb of the Sun), and demonstrate the high degree of reproducibility of the spectral features in the linearly polarized ([FORMULA]) spectrum. Some of the prominent polarization features are due to Ti I, Cr I, C2, and Nd II.

Fig. 4 also gives an example of the structural richness of the "second solar spectrum" ([FORMULA]). Some of the prominent polarization features in the figure are due to various multiplets of Ti I, Cr I, C2, and Nd II, all of which are unexpected surprises. Another strongly polarizing weak line of ionized neodynium (Nd II) has been found at 5249 Å (Stenflo & Keller 1996b ).

Since each recording with the imaging CCD polarimeter (ZIMPOL) only covers about 4 Å , the more extended ranges shown in Figs. 3 and 4 have been pieced together from a series of partially overlapping recordings. In our illustrations these recordings are plotted on top of each other to allow us to judge the reproducibility of the spectral features. As can be seen, all polarization features are reproduced in detail (apart from the minor high-frequency noise ripple - no Fourier filter has been applied to the data). The reproducibility is less good in the intensity spectrum ([FORMULA]) due to inaccurate flat-fielding of the CCD sensor (gain table effects), with efficiency gradients across the field of view. Such gain table effects divide out entirely when forming the fractional polarization ([FORMULA]) image, which explains the better reproducibility of [FORMULA] as compared with I alone.

The observed polarization in the 318 Fe I 4957 Å doublet has remained an enigma since it has seemed to contradict the rules of quantum mechanics. If one assumes that the two lines are formed by resonant scattering, then the 4957.302 Å line, as a [FORMULA] transition, should according to Eqs. (13) and (14) have a polarizability [FORMULA], while the adjacent 4957.603 Å line, as a [FORMULA] transition, should have [FORMULA]. Accordingly one would expect the observed polarization amplitude of the left line of the pair to be [FORMULA] times larger than that of the right line. The observations, both those from 1978 (Stenflo et al.  1983b ) and the present ones of Fig. 4, show the entirely opposite behavior: it is the right line instead of the left one that exhibits a pronounced polarization peak, although it should have a polarizability close to zero according to its quantum numbers for resonant scattering.

The resolution of this apparent mystery comes from the fluorescent contributions, which completely change the resulting polarizabilities. Thus the emission transition [FORMULA] can have three different scattering contributions: [FORMULA], [FORMULA], and [FORMULA]. Similarly, the emission transition [FORMULA] can have the three contributions [FORMULA], [FORMULA], and [FORMULA]. These different possibilities are drawn in Fig. 1 as the dotted and dashed lines in the diagram for 318 Fe I.

With our present theory that can account for all the possible scattering contributions within a multiplet we have done some calculations to illustrate the role of fluorescence and how it can qualitatively completely change the relative distribution of polarizabilities between the various spectral lines of a multiplet. Thus the thick solid line in the upper panel of Fig. 5 has been obtained from the model of Eq. (18) when all the possible fluorescent contributions are accounted for, and we assume equal populations of all the initial states and a spectrally flat incident spectrum. The three free model parameters a, b, and the global scale factor have been adjusted to obtain a reasonable fit to the observed spectrum (represented by the thin, solid curve, taken from the lower right panel of Fig. 4).

[FIGURE] Fig. 5. Portion of the linearly polarized ([FORMULA]) spectrum around the 318 Fe I 4957 Å doublet. The thin curves represent the observations, taken from the lower right panel of Fig. 4, while the thick curves show the results of two theoretical models. The figure serves to illustrate that the non-resonant, fluorescent contributions within a multiplet can qualitatively change the relative distribution of polarizability between the different multiplet lines.

Although the fit obtained this way is far from ideal, the model serves to illustrate what happens if we remove some of the scattering contributions. While the upper panel of Fig. 5 refers to the case when all the contributing initial states with [FORMULA], 4, 5, and 6 are given equal weight, the theoretical curve in the lower panel is obtained if we give zero weight to the initial states with [FORMULA] and 5, so that only the [FORMULA] and 6 states, which are the same as the final [FORMULA] states for this doublet, contribute. The relative continuum opacity and the global scaling parameter have the same values in both diagrams.

In the upper panel it is the right spectral line that dominates the polarization, in agreement with the observations, while in the lower panel it is the left line that stands out, as expected from the values of [FORMULA] for resonant scattering. Although the lower diagram does not only have resonant contributions (the fluorescent transition [FORMULA] also contributes), it is dominated by the resonant contributions.

The polarization feature of the Fe I 4957 Å doublet can thus be regarded as a signature of fluorescence effects within an atomic multiplet.

3.4. Hyperfine structure

Barium occurs in nature with a mixture of isotopes. The even isotopes, dominated by nucleon number 138, contribute 82% of the total abundance. They have zero nuclear spin and thus no hyperfine structure. The odd isotopes, dominated by isotopes 137 and 135, contribute the remaining 18% . They have nuclear spin [FORMULA], like sodium nuclei. The hyperfine splitting pattern for the odd barium isotopes was shown in Fig. 1, where also the relative positions of the unsplit levels of the even isotopes were marked by the thicker, horizontal lines (neglecting the smaller isotope shifts - these shifts are however taken into account in the model calculations when weighting the contributions from the various isotopes). The atomic data used for the Ba II hyperfine structure have been taken from Rutten (1976 ).

Since multiplet No. 1 of Na I (with the D1 and D2 lines) has the same quantum numbers as multiplet No. 1 of Ba II, it also has the same hyperfine structure pattern, so the splitting diagram in Fig. 1 for the odd Ba isotopes also represents the pattern for the Na I D2 line. The corresponding D1 line of barium has the wavelength 4934 Å . The magnitude of the hyperfine splitting is larger in barium by about a factor of five as compared with sodium. It is the hyperfine splitting of the lower state that dominates; the splitting of the upper state is an order of magnitude smaller (cf. the different scale factors used for the lower and upper levels in the plot of Fig. 1).

For sodium the hyperfine splitting of the upper state is comparable in magnitude to the natural, radiative width (inverse life time) of the excited state. Still this minute splitting cannot be neglected since it affects the quantum interferences between the substates that determine the polarizability.

Our theory for the polarizability of multiplets can be directly applied to the case of hyperfine structure multiplets if we only substitute the quantum numbers J, L, and S with the corresponding quantum numbers F (total angular momentum), J (electronic angular momentum), and I (nuclear spin). Here we will discuss the successful application of this theory to explain the observed polarization pattern of the Ba II 4554 Å line, the "D2 line" of multiplet No. 1 that represents a scattering transition with [FORMULA].

Fig. 6 shows the CCD images of the observed intensity (top panel) and fractional linear polarization (bottom panel) around the Ba II 4554 Å line, recorded in April 1995 with the spectrograph slit 5 arcsec inside the Sun's north polar limb. The polarization in the Ba line exhibits a triplet structure. As will be shown below the partially resolved polarization components in the wings of the line are due to the added contributions from the hyperfine structure components of the odd isotopes, while the central component is due to the unsplit even isotopes.

[FIGURE] Fig. 6. Signature of hyperfine structure in barium. The images of the intensity (I) and fractional linear polarization ([FORMULA]) were recorded in April 1995 with ZIMPOL I at NSO/Kitt Peak, with the spectrograph slit 5 arcsec inside the north polar limb of the Sun. While the central peak of the polarization triplet is due to the even isotopes, the satellite peaks in the blue and red line wings are due to the shifted hyperfine structure components of the odd isotopes.

Fig. 7 has been obtained from Fig. 6 by averaging along the spectrograph slit. Superposed on the [FORMULA] diagram are two theoretical curves representing models of the polarization profile, with (thick solid curve) and without (dashed curve) quantum interferences between the split hyperfine structure levels of the excited atomic state.

[FIGURE] Fig. 7. 1-D version of the recording of Fig. 6, obtained by spatially averaging along the spectrograph slit. While the thin solid lines represent the observations, the thick solid and dashed curves in the [FORMULA] diagram have been obtained with our model for the effective polarizability, [FORMULA]. The thick solid line accounts for the quantum interferences between the split hyperfine structure components of the excited state, while the dashed curve ignores the interference terms.

The model results of Fig. 7 have been obtained, as in the Na I and 318 Fe I cases, by calculating and globally rescaling [FORMULA] of Eq. (18) (thus for instance implicitly assuming a spectrally flat incident radiation field). When forming [FORMULA] from Eq. (17), [FORMULA] and [FORMULA] have first been convolved with a Gaussian with a Doppler width of [FORMULA] in velocity units, to crudely account for broadening by line of sight velocities on the Sun. For the comparison with the observations we have in addition smeared the computed [FORMULA] profiles by convolving them with a Gaussian that has a total half width of 47 mÅ , to simulate instrumental broadening and macroturbulence. The diagram to the lower left in Fig. 2 has been obtained this way. Then a global scale factor (accounting for the geometric depolarization) has been applied to make the maxima of the observed [FORMULA] and the scaled [FORMULA] profiles identical.

With the so broadened and normalized [FORMULA] function the two free parameters a and b of the model of Eq. (18) that best fit the observations have been determined. Parameter b, which represents the continuum polarization, thereby plays a subordinate role, since it mainly fixes the background polarization level that is approached at distances far from the line center. It is parameter a (the relative continuum opacity) that dominates the fitting procedure.

Our analysis shows that the partially resolved wing components of the observed polarized profiles are due to the odd barium isotopes, which appear at these wing positions due to the hyperfine structure splitting. The even isotopes, which are not subject to hyperfine structure, are responsible for the central polarization peak, which is larger than the wing peaks since the relative abundance of the even isotopes is larger (82% of the total abundance). Fig. 7 also shows that quantum interferences between the split hyperfine structure components of the excited level play a significant role, although the magnitude of the splitting of the upper state is an order of magnitude smaller than that of the lower state.

Strong lines like the Ba II 4554 Å line are broadened by saturation effects when formed in an optically thick atmosphere. This broadening wipes out subtle effects in the Stokes I profiles. Such saturation does not occur for the polarization profiles. They are therefore narrower, which enhances the visibility of the hyperfine structure splitting.

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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