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Atomic and Molecular Depolarizing Collision Rates
Abstract
This paper is divided in three parts: after having recalled the different types of collisions with the different types of perturbers and having provided rough orders of magnitude of the collision rates, three cases are discussed. Although the most frequent type of depolarizing collision is the one of the collisions with the surrounding Hydrogen atoms, we discuss in the first part a particular case where the depolarizing collision effect is due to collisions with electrons and protons. This is the case of the Hydrogen lines observed in solar prominences. We recall how the interpretation of polarization observations in two lines has led to the joint determination of the magnetic field vector and the electron and proton density, and we show that this density determination gives results in agreement with the densities determined by interpretation of the Stark effect, provided that this last effect be evaluated in the impact approximation scheme which is indeed more valid than the quasistatic approach at these densities. In the second part, we describe a method that has been recently developed for the computation of the depolarizing rates in the case of collisions with the neutral Hydrogen atom. The case of molecular lines is less favourable, because, even if depolarizing collision rates computation may be soon expected and begin to be done inside the ground level of the molecule, calculations inside the excited states are far from the present ability. In the third part, we present an example where the excited state depolarizing rates were evaluated together with the magnetic field through the differential Hanle effect interpretation, based on the fact that the molecule provides a series of lines of different sensitivities that can be compared. This led to an experimental/observational determination of these rates, waiting for future theoretical computations for comparison.
The spectral line polarization of the radiation emerging from a magnetized
astrophysical plasma depends on the state of the atoms within the medium, whose
determination requires considering the interactions between the atoms and the
magnetic field, between the atoms and photons (radiative transitions), and
between the atoms and other material particles (collisional transitions). In
applications within the framework of the multiterm model atom (which accounts
for quantum interference between magnetic sublevels pertaining either to the
same J-level or to different J-levels within the same term) collisional
processes are generally neglected when solving the master equation for the
atomic density matrix. This is partly due to the lack of experimental data
and/or of approximate theoretical expressions for calculating the collisional
transfer and relaxation rates (in particular the rates for interference between
sublevels pertaining to different J-levels, and the depolarizing rates due to
elastic collisions). In this paper we formally define and investigate the
transfer and relaxation rates due to isotropic inelastic and superelastic
collisions that enter the statistical equilibrium equations of a multiterm
atom. Under the hypothesis that the atom-collider interaction can be described
by a dipolar operator, we provide expressions that relate the collisional rates
for interference between different J-levels to the usual collisional rates for
J-level populations. Finally, we apply the general equations to the case of a
two-term atom with unpolarized lower term, illustrating the impact of inelastic
and superelastic collisions on scattering polarization through radiative
transfer calculations in a slab of stellar atmospheric plasma anisotropically
illuminated by the photospheric radiation field.
Context. The second solar spectrum resulting from coherent scattering is a main tool for diagnostics of turbulent magnetic fields on the Sun. Scattering on diatomic molecules plays an important role in forming this spectrum and even dominates in some spectral regions. Aims. In a magnetic field electronic states of a molecule are often perturbed via the Paschen-Back effect. Sometimes this perturbation can completely change the spectrum, not only quantitatively, but even qualitatively. Here we calculate molecular scattering properties taking into account the Paschen-Back effect. Methods. Starting with the Hund's case (a) wave functions as a basis we obtain with the perturbation theory wave functions of the intermediate Hund's case (a-b) in a magnetic field. Using new, perturbed values of the Landé factors and transition amplitudes we calculate the Mueller matrix for coherent scattering at diatomic molecules in the intermediate Hund's case (a-b) and look for the effects that can be caused by the Paschen-Back effect. Results. We have found a considerable deviation from the Zeeman regime and discuss here the quantitative and qualitative effects on observed polarization signals for the CN B 2 Σ − X 2 Σ and MgH B 2 Σ − X 2 Σ systems as examples.
In many wavelength regions molecular lines dominate the second solar spectrum that results from coherent scattering. Scattering polarization is modified by magnetic fields via the Hanle effect. This allows us to explore the magnetic field regime with weak field strengths and mixed polarities, which is not seen with the Zeeman effect and thus contains complementary information. Molecular lines are particularly well suited to diagnose such turbulent fields because they exhibit a broad range of magnetic sensitivities within narrow spectral regions. Thus, it is possible to employ the technique of the differential Hanle effect, i.e. to obtain field strengths by observing polarization ratios in various lines. We have identified one R- and one P-triplet of C$_2$ at 5140 Å and 5141 Å, respectively, that satisfy all conditions to be used in the differential Hanle effect. Based on these lines we have developed a model that can diagnose turbulent magnetic fields using the Hanle effect. The tool is sensitive over a broad range of magnetic field strengths from a few Gauss up to several hundred Gauss. This tool has allowed us to find a significant Hanle depolarization of C$_2$ lines in quiet Sun observations, which corresponds to a magnetic field strength of $15\pm3$ G.
Recent advances in the computation of the Zeeman splitting of molecular lines have paved the way for their use as diagnostics of solar and stellar magnetic fields. A systematic study of their diagnostic capabilities had not been carried out so far, however. Here we investigate how molecular lines can be used to deduce the magnetic and thermal structure of sunspots, starspots and cool stars. First, we briefly describe the Stokes radiative transfer of Zeeman-split molecular lines. Then, we compute Stokes spectra of TiO, OH, CH and FeH lines and investigate their diagnostic capabilities. We also compare the synthetic profiles with observations. Spectra of TiO, OH and FeH are found to be interesting diagnostics of sunspot magnetic fields. This is also true for cool stars, where, however, the OH Stokes $V$ profiles may require very high S/N data to be reliably employed. Finally we investigate the potential of various molecular bands for high-contrast imaging of the solar surface. The violet CN and CH bands turn out to be most promising for imaging the photosphere, the TiO bands are excellent for imaging sunspot umbrae, while the UV OH band can be used for imaging both the photosphere and sunspots.
Deciphering and understanding the small-scale magnetic activity of the quiet solar photosphere should help to solve many of the key problems of solar and stellar physics, such as the magnetic coupling to the outer atmosphere and the coronal heating. At present, we can see only approximately 1 per cent of the complex magnetism of the quiet Sun, which highlights the need to develop a reliable way to investigate the remaining 99 per cent. Here we report three-dimensional radiative transfer modelling of scattering polarization in atomic and molecular lines that indicates the presence of hidden, mixed-polarity fields on subresolution scales. Combining this modelling with recent observational data, we find a ubiquitous tangled magnetic field with an average strength of approximately 130 G, which is much stronger in the intergranular regions of solar surface convection than in the granular regions. So the average magnetic energy density in the quiet solar photosphere is at least two orders of magnitude greater than that derived from simplistic one-dimensional investigations, and sufficient to balance radiative energy losses from the solar chromosphere.
Analysis of the Hanle effect in solar molecular lines allows us to obtain empirical information on hidden, mixed-polarity magnetic fields at subresolution scales in the (granular) upflowing regions of the `quiet' solar photosphere. Here we report that collisions seem to be very efficient in depolarizing the rotational levels of MgH lines. This has the interesting consequence that in the upflowing regions of the quiet solar photosphere the strength of the hidden magnetic field cannot be sensibly larger than 10 G, assuming the simplest case of a single valued microturbulent field that fills the entire upflowing photospheric volume. Alternatively, an equally good theoretical fit to the observed scattering polarization amplitudes can be achieved by assuming that the rate of depolarizing collisions is an order of magnitude smaller than in the previous collisionally dominated case, but then the required strength of the hidden field in the upflowing regions turns out to be unrealistically high. These constraints reinforce our previously obtained conclusion that there is a vast amount of hidden magnetic energy and unsigned magnetic flux localized in the (intergranular) downflowing regions of the quiet solar photosphere. Comment: 13 pages, 4 figures, ApJ Letters (2005), in press
The authors derive in a coherent manner the master equation for the
density matrix of an atom interacting with a bath of perturbers and
photons, in the presence of a weak magnetic field. This paper has been
inspired by astrophysical purposes: the interpretation of line
polarization induced by anisotropic excitation of the levels, eventually
modified by the local magnetic field (the Hanle effect); the
polarization can be due to scattering of the incident anisotropic
radiation, as in solar prominences, or to impact polarization, as in
solar flares. The physical conditions are then those of numerous
astrophysical media: any directions of polarization and magnetic field,
two-level atom approximation not valid, weak radiation field, weak
density of perturbers. The master equation for the atomic density matrix
has been derived in the framework of the impact approximation.
As expected, the semi-classical theory of atom-electron and atom-ion collisions ( tealt{b11 Se62}; see also tealt{b11 SB96}) leads to incorrect results when it is applied to impact polarization computations. This is due to the fact that, near the threshold, the Delta M=± 1 transitions between Zeeman sublevels remain open, even though the momentum conservation law tep{b11 PS58} implies that they should be forbidden at the exact threshold. An approximate model of momentum transfer is proposed, which corrects this behavior, leading to a fairly good agreement with observed impact polarization for several target atoms. The agreement is improved also for the total cross-section, because the Delta M=± 1 cross-sections were previously overestimated. Comparison with experimental results are shown, for electron-atom impact polarization and total cross-section. As a second step, energy transfer is also taken into account, by splitting the collision process in two halves: the first half is modeled within the initial conditions, while the second half is modeled within the final conditions.
The collisional broadening theory of Anstee & O'Mara for s-p and p-s
transitions by collisions with atomic hydrogen is extended to p-d and
d-p transitions. Width cross-sections for collisional broadening of
atomic lines corresponding to p-d and d-p transitions by neutral
hydrogen atoms are tabulated as a function of effective principal
quantum numbers for the upper and lower states, for a relative velocity
of 10^4ms^-1. The cross-sections vary with velocity as v^-alpha, and
results for the parameter alpha are similarly tabulated. The
cross-sections are tested by application to synthesis of suitable strong
solar lines. The derived abundances are consistent with meteoritic
values.
Approximations in Brueckner's theory of spectral line broadening by collisions with neutral hydrogen atoms relevant to a solar-type
atmosphere are discussed, and a modified theory for s-p transitions is presented. The theory utilizes explicit expressions for the interatomic interaction energy between a hydrogen
atom in its ground state and general m = 0, ± 1 p states, derived from second-order perturbation theory without exchange, allowing the removal of the Lindholm-Foley average
over m states in the original Brueckner model. Approximate upper and lower bounds for the linewidth of the sodium D lines are derived, and these values are contrasted with available theoretical, experimental and solar empirical results.
The removal of the Lindholm–Foley average is shown to reduce the D-state linewidths by about 30 per cent, and an analysis of the interatomic separations important in the line broadening cross-section
for the D lines has shown that there is little atomic overlap at the separations that are important.
Quiescent prominences
It is found that Heii 4686 is emitted in the same cold region of ≲ 10000 K as hydrogen, metal and neutral helium emission lines. This conclusion is based on the finding that the observed width of λ4686 is the same as the calculated width of λ4686. The calculated width is derived from the observed widths of hydrogen and metallic lines. The large intensity of Heii 4686 in ≲ 10000 K can be explained by the ionization of Heii due to the UV radiation below 228 Å that comes from the corona and the transition region.
Loop prominences
The very broad width (30 to 50 km s−1) of λ4686 for two post-flare loop prominences shows that the Heii line is emitted in hot regions different from regions of hydrogen and metal emission. From the widths of the Balmer lines and many metallic lines the kinetic temperature for one loop is found to be 16000 K in one part and 7600 K in another part. The electron densities are 1012.0 cm−3 and less than 1011.0 cm−3 respectively.
Chromosphere
The intensity of λ4686 in the chromosphere can be interpreted in terms of a temperature of ≲ 10000 K with the ionization due to UV radiation. But, since observations of the width of λ4686 are not available, a definitive conclusion for the chromosphere cannot be reached.