## 3. Modelling the observed polarization## 3.1. Effective polarizabilityThe observed linear polarization depends on a number of factors: (i) The atomic polarizability coefficient . (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 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 function by . We note that 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 The additional free parameter To obtain a fit of the observed polarization curves we have to multiply 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 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 To summarize, our idealized model for the observed polarization
that bypasses radiative transfer thus has three free parameters: the
relative continuum opacity ## 3.2. Modelling quantum interferences in sodiumFig. 3 shows an application of this model to explain the
observed polarization profile across the Na I
D
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
. 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
and . 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 D For Rayleigh scattering the polarizability can never be negative
unless there is quantum interference. If the resonant frequencies of
the D To simplify the expression we have omitted the damping constant , 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 D The general shape of the D For the D 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 D ## 3.3. Signatures of fluorescent scatteringLet 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 . 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.
Fig. 4 also gives an example of the structural richness of the
"second solar spectrum" (). Some of the
prominent polarization features in the figure are due to various
multiplets of Ti I, Cr I, C 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
() 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 () image,
which explains the better reproducibility of as
compared with 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 transition, should according to Eqs. (13) and (14) have a polarizability , while the adjacent 4957.603 Å line, as a transition, should have . Accordingly one would expect the observed polarization amplitude of the left line of the pair to be 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 can have three different scattering contributions: , , and . Similarly, the emission transition can have the three contributions , , and . 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
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 , 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 and 5, so that only the and 6 states, which are the same as the final 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 for resonant scattering. Although the lower diagram does not only have resonant contributions (the fluorescent transition 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 structureBarium 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 , 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
D 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 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.
Fig. 7 has been obtained from Fig. 6 by averaging along the spectrograph slit. Superposed on the 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.
The model results of Fig. 7 have been obtained, as in the Na I and 318 Fe I cases, by calculating and globally rescaling of Eq. (18) (thus for instance implicitly assuming a spectrally flat incident radiation field). When forming from Eq. (17), and have first been convolved with a Gaussian with a Doppler width of 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 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 and the scaled profiles identical. With the so broadened and normalized
function the two free parameters 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
© European Southern Observatory (ESO) 1997 Online publication: May 26, 1998 |