arXiv subjects

K. N. Nagendra

Publications and source records attributed to K. N. Nagendra.

At least 19 recordsLinked to original sources

arXiv2020

Importance of Angle-dependent Partial Frequency Redistribution in Hyperfine Structure Transitions Under Incomplete Paschen-Back Effect Regime

Angle-frequency coupling in scattering of polarized light on atoms is represented by the angle-dependent (AD) partial frequency redistribution (PRD) matrices. There are several lines in the linearly polarized solar spectrum, for which PRD combined with quantum interference between hyperfine structure states play a significant role. Here we present the solution of the polarized line transfer equation including the AD-PRD matrix for scattering on a two-level atom with hyperfine structure splitting (HFS) and an unpolarized lower level. We account for the effects of arbitrary magnetic fields (including the incomplete Paschen-Back effect regime) and elastic collisions. For exploratory purposes we consider a self-emitting isothermal planar atmosphere and use atomic parameters that represent an isolated Na\,{\sc i} D$_2$ line. For this case we show that the AD-PRD effects are significant for field strengths below about 30G, but that the computationally much less demanding approximation of angle-averaged (AA) PRD may be used for stronger fields.

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arXiv2020

Resonance Line Polarization in Spherically Symmetric Moving Media: a Parametric Study

In the present paper we consider the problem of resonance line polarization formed in the spherically symmetric expanding atmospheres. For the solution of the concerned polarized transfer equation we use the comoving frame formulation, and apply the Accelerated Lambda Iteration (ALI) method. We restrict ourselves to the non-relativistic regime of velocities wherein mainly Doppler shift effects are significant. For our studies, we consider the scattering on a two-level atom, including the effects of partial frequency redistribution (PFR). We present the dependence of linearly polarized profiles on different atmospheric and atomic parameters.

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arXiv2019

Polarized Line Formation in Arbitrary Strength Magnetic Fields: the case of a two-level atom with hyperfine structure splitting

Quantum interference effects together with partial frequency redistribution (PFR) in line scattering produce subtle signatures in the so called Second Solar Spectrum (the linearly polarized spectrum of the Sun). These signatures are modified in the presence of arbitrary strength magnetic fields via the Hanle, Zeeman, and Paschen-Back effects. In the present paper we solve the problem of polarized line formation in a magnetized atmosphere taking into account scattering in a two-level atom with hyperfine structure splitting together with PFR. To this end we incorporate the collisionless PFR matrix derived in Sowmya et al. (2014) in the polarized transfer equation. We apply the scattering expansion method to solve this transfer equation. We study the combined effects of PFR and Paschen-Back effect on polarized line profiles formed in an isothermal one-dimensional planar atmosphere. For this purpose, we consider the cases of D$_2$ lines of Li\,{\sc i} and Na\,{\sc i}.

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arXiv2015

Polarized scattering with Paschen-Back effect, hyperfine structure, and partial frequency redistribution in magnetized stellar atmospheres

$F$-state interference significantly modifies the polarization produced by scattering processes in the solar atmosphere. Its signature in the emergent Stokes spectrum in the absence of magnetic fields is depolarization in the line core. In the present paper, we derive the partial frequency redistribution (PRD) matrix that includes interference between the upper hyperfine structure states of a two-level atom in the presence of magnetic fields of arbitrary strengths. The theory is applied to the Na I D$_2$ line that is produced by the transition between the lower $J=1/2$ and upper $J=3/2$ states which split into $F$ states because of the coupling with the nuclear spin $I_s=3/2$. The properties of the PRD matrix for the single-scattering case is explored, in particular, the effects of the magnetic field in the Paschen--Back regime and their usefulness as a tool for the diagnostics of solar magnetic fields.

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arXiv2015

Polarized scattering of light for arbitrary magnetic fields with level-crossings from the combination of hyperfine and fine structure splittings

Interference between magnetic substates of the hyperfine structure states belonging to different fine structure states of the same term influences the polarization for some of the diagnostically important lines of the Sun's spectrum, like the sodium and lithium doublets. The polarization signatures of this combined interference contain information on the properties of the solar magnetic fields. Motivated by this, in the present paper, we study the problem of polarized scattering on a two-term atom with hyperfine structure by accounting for the partial redistribution in the photon frequencies arising due to the Doppler motions of the atoms. We consider the scattering atoms to be under the influence of a magnetic field of arbitrary strength and develop a formalism based on the Kramers--Heisenberg approach to calculate the scattering cross section for this process. We explore the rich polarization effects that arise from various level-crossings in the Paschen--Back regime in a single scattering case using the lithium atomic system as a concrete example that is relevant to the Sun.

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arXiv2015

Blend lines in the polarized spectrum of the Sun

Blend lines form an integral part of the theoretical analysis and modeling of the polarized spectrum of the Sun. Their interaction with other spectral lines needs to be explored and understood before we can properly use the main spectral lines to diagnose the Sun. They are known to cause a decrease in the polarization in the wings of the main line on which they superpose, or in the polarization of the continuum, when they are assumed to be formed either under the local thermodynamic equilibrium (LTE) conditions, or when their intrinsic polarizability factor is zero. In this paper, we describe the theoretical framework to include the blend lines formed under non-LTE conditions, in the radiative transfer equation, and the numerical techniques to solve it. The properties of a blend line having an intrinsic polarization of its own and its interaction with the main line are discussed. The results of our analysis show that the influence of the blend lines on the main spectral lines, though small in the present context, is important and needs to be considered when interpreting the polarized spectral lines in the second solar spectrum.

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arXiv2014

The quantum interference effects in the Sc II 4247 A line of the Second Solar Spectrum

The Sc II 4247 A line formed in the chromosphere is one of the lines well known, like the Na I D_2 and Ba II D_2, for its prominent triple peak structure in Q/I and the underlying quantum interference effects governing it. In this paper, we try to study the nature of this triple peak structure using the theory of F-state interference including the effects of partial frequency redistribution (PRD) and radiative transfer (RT). We compare our results with the observations taken in a quiet region near the solar limb. In spite of accounting for PRD and RT effects it has not been possible to reproduce the observed triple peak structure in Q/I. While the two wing PRD peaks (on either side of central peak) and the near wing continuum can be reproduced, the central peak is completely suppressed by the enhanced depolarization resulting from the hyperfine structure splitting. This suppression remains for all the tested widely different 1D model atmospheres or for any multi-component combinations of them. While multidimensional radiative transfer effects may improve the fit to the intensity profiles, they do not appear capable of explaining the enigmatic central Q/I peak. This leads us to suspect that some aspect of quantum physics is missing.

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arXiv2014

The role of quantum interference and partial redistribution in the solar Ba II D2 4554 A line

The Ba II D2 line at 4554 A is a good example, where the F-state interference effects due to the odd isotopes produce polarization profiles, which are very different from those of the even isotopes that do not exhibit F-state interference. It is therefore necessary to account for the contributions from the different isotopes to understand the observed linear polarization profiles of this line. In this paper we present radiative transfer modeling with partial frequency redistribution (PRD), which is shown to be essential to model this line. This is because complete frequency redistribution (CRD) cannot reproduce the observed wing polarization. We present the observed and computed Q/I profiles at different limb distances. The theoretical profiles strongly depend on limb distance (μ) and the model atmosphere which fits the limb observations fails at other μ positions.

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arXiv2014

Effect of cross-redistribution on the resonance scattering polarization of O {\sc i} line at 1302 Å\,

Oxygen is the most abundant element on the Sun after Hydrogen and Helium. The intensity spectrum of resonance lines of neutral Oxygen namely O {\sc i} (1302, 1305 and 1306 Å\,) has been studied in the literature for chromospheric diagnostics. In this paper we study the resonance scattering polarization in the O {\sc i} line at 1302 Å\, using two-dimensional radiative transfer in a composite atmosphere constructed using a two-dimensional magneto-hydrodynamical snapshot in the photosphere and columns of the one-dimensional FALC atmosphere in the chromosphere. The methods developed by us recently in a series of papers to solve multi-dimensional polarized radiative transfer have been incorporated in our new code POLY2D which we use for our analysis. We find that multi-dimensional radiative transfer including XRD effects is important in reproducing the amplitude and shape of scattering polarization signals of the O {\sc i} line at 1302 Å\,.

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arXiv2014

Center to limb observations and modeling of the Ca I 4227 A line

The observed center-to-limb variation (CLV) of the scattering polarization in different lines of the Second Solar Spectrum can be used to constrain the height variation of various atmospheric parameters, in particular the magnetic fields via the Hanle effect. Here we attempt to model non-magnetic CLV observations of the $Q/I$ profiles of the Ca I 4227 A line recorded with the ZIMPOL-3 at IRSOL. For modeling, we use the polarized radiative transfer with partial frequency redistribution with a number of realistic 1-D model atmospheres. We find that all the standard FAL model atmospheres, used by us, fail to simultaneously fit the observed ($I$, $Q/I$) at all the limb distances ($μ$). However, an attempt is made to find a single model which can provide a fit at least to the CLV of the observed $Q/I$ instead of a simultaneous fit to the ($I$, $Q/I$) at all $μ$. To this end we construct a new 1-D model by combining two of the standard models after modifying their temperature structures in the appropriate height ranges. This new combined model closely reproduces the observed $Q/I$ at all the $μ$, but fails to reproduce the observed rest intensity at different $μ$. Hence we find that no single 1-D model atmosphere succeeds in providing a good representation of the real Sun. This failure of 1-D models does not however cause an impediment to the magnetic field diagnostic potential of the Ca I 4227 A line. To demonstrate this we deduce the field strength at various $μ$ positions without invoking the use of radiative transfer.

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arXiv2014

Polarized light scattering with Paschen-Back effect, level-crossing of fine structure states and partial frequency redistribution

The quantum interference between the fine structure states of an atom modifies the shapes of the emergent Stokes profiles in the Second Solar Spectrum. This phenomenon has been studied in great detail both in the presence and absence of magnetic fields. By assuming a flat-spectrum for the incident radiation, the signatures of this effect have been explored for arbitrary field strengths. Even though the theory which takes into account the frequency dependence of the incident radiation is well developed, it is restricted to the regime in which the magnetic splitting is much smaller than the fine structure splitting. In the present paper, we carry out a generalization of our scattering matrix formalism including the effects of partial frequency redistribution (PRD) for arbitrary magnetic fields. We test the formalism using available benchmarks for special cases. In particular we apply it to the Li\,{\sc i} 6708\,Å D$_1$ and D$_2$ line system, for which observable effects from the Paschen-Back regime are expected in the Sun's spectrum.

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arXiv2013

Polarized Line Formation in Multi-dimensional Media. IV. A Fourier Decomposition Technique To Formulate The Transfer Equation With Angle-Dependent Partial Frequency Redistribution

To explain the linear polarization observed in spatially resolved structures in the solar atmosphere, the solution of polarized radiative transfer (RT) equation in multi-dimensional (multi-D) geometries is essential. For strong resonance lines partial frequency redistribution (PRD) effects also become important. In a series of papers we have been investigating the nature of Stokes profiles formed in multi-D media including PRD in line scattering. For numerical simplicity so far we restricted our attention to the particular case of PRD functions which are averaged over all the incident and scattered directions. In this paper we formulate the polarized RT equation in multi-D media that takes into account Hanle effect with angle-dependent PRD functions. We generalize here to the multi-D case, the method of Fourier series expansion of angle-dependent PRD functions originally developed for RT in 1D geometry. We show that the Stokes source vector $\bm{S}=(S_I,S_Q,S_U)^T$ and the Stokes vector $\bm{I}=(I,Q,U)^T$ can be expanded in terms of infinite sets of components ${\tilde{\bm{\mathcal{S}}}}^{(k)}$, ${\tilde{\bm{\mathcal{I}}}}^{(k)}$ respectively, $k\in[0,+\infty)$. We show that the components ${\tilde{\bm{\mathcal{S}}}}^{(k)}$ become independent of the azimuthal angle ($φ$) of the scattered ray, whereas the components ${\tilde{\bm{\mathcal{I}}}}^{(k)}$ remain dependent on $φ$ due to the nature of RT in multi-D geometry. We also establish that ${\tilde{\bm{\mathcal{S}}}}^{(k)}$ and ${\tilde{\bm{\mathcal{I}}}}^{(k)}$ satisfy a simple transfer equation, which can be solved by any iterative method like an Approximate Lambda Iteration (ALI) or a Biconjugate-Gradient type projection method provided we truncate the Fourier series to have a finite number of terms.

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arXiv2013

Polarized Line Formation in Multi-dimensional Media. V. Effects of Angle-Dependent Partial Frequency Redistribution

The solution of polarized radiative transfer equation with angle-dependent (AD) partial frequency redistribution (PRD) is a challenging problem. Modeling the observed, linearly polarized strong resonance lines in the solar spectrum often requires the solution of the AD line transfer problems in one-dimensional (1D) or multi-dimensional (multi-D) geometries. The purpose of this paper is to develop an understanding of the relative importance of the AD PRD effects and the multi-D transfer effects and particularly their combined influence on the line polarization. This would help in a quantitative analysis of the second solar spectrum (the linearly polarized spectrum of the Sun). We consider both non-magnetic and magnetic media. In this paper we reduce the Stokes vector transfer equation to a simpler form using a Fourier decomposition technique for multi-D media. A fast numerical method is also devised to solve the concerned multi-D transfer problem. The numerical results are presented for a two-dimensional medium with a moderate optical thickness (effectively thin), and are computed for a collisionless frequency redistribution. We show that the AD PRD effects are significant, and can not be ignored in a quantitative fine analysis of the line polarization. These effects are accentuated by the finite dimensionality of the medium (multi-D transfer). The presence of magnetic fields (Hanle effect) modifies the impact of these two effects to a considerable extent.

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arXiv2013

Multi-dimensional radiative transfer to analyze Hanle effect in Ca {\sc ii} K line at 3933 Å\,

Radiative transfer (RT) studies of the linearly polarized spectrum of the Sun (the second solar spectrum) have generally focused on the line formation, with an aim to understand the vertical structure of the solar atmosphere using one-dimensional (1D) model atmospheres. Modeling spatial structuring in the observations of the linearly polarized line profiles requires the solution of multi-dimensional (multi-D) polarized RT equation and a model solar atmosphere obtained by magneto-hydrodynamical (MHD) simulations of the solar atmosphere. Our aim in this paper is to analyze the chromospheric resonance line Ca {\sc ii} K at 3933 Å using multi-D polarized RT with Hanle effect and partial frequency redistribution in line scattering. We use an atmosphere which is constructed by a two-dimensional snapshot of the three-dimensional MHD simulations of the solar photosphere, combined with columns of an 1D atmosphere in the chromosphere. This paper represents the first application of polarized multi-D RT to explore the chromospheric lines using multi-D MHD atmospheres, with PRD as the line scattering mechanism. We find that the horizontal inhomogeneities caused by MHD in the lower layers of the atmosphere are responsible for strong spatial inhomogeneities in the wings of the linear polarization profiles, while the use of horizontally homogeneous chromosphere (FALC) produces spatially homogeneous linear polarization in the line core.

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arXiv2013

An efficient decomposition technique to solve angle-dependent Hanle scattering problems

Hanle scattering is an important diagnostic tool to study weak solar magnetic fields. Partial frequency redistribution (PRD) is necessary to interpret the linear polarization observed in strong resonance lines. Usually angle-averaged PRD functions are used to analyze linear polarization. However it is established that angle-dependent PRD functions are often necessary to interpret polarization profiles formed in the presence of weak magnetic fields. Our aim is to present an efficient decomposition technique, and the numerical method to solve the concerned angle-dependent line transfer problem. Together with the standard Stokes decomposition technique we employ Fourier expansion over the outgoing azimuth angle to express in a more convenient form, the angle-dependent PRD function for the Hanle effect. It allows the use of angle-dependent frequency domains of Bommier to solve the Hanle transfer problem. Such an approach is self-consistent and accurate compared to a recent approach where angle-averaged frequency domains were used to solve the same problem. We show that it is necessary to incorporate angle-dependent frequency domains instead of angle-averaged frequency domains to solve the Hanle transfer problem accurately, especially for the Stokes U parameter. The importance of using angle-dependent domains has been highlighted by taking the example of Hanle effect in the case of line transfer with vertical magnetic fields in a slab atmosphere. We have also studied the case of polarized line formation when micro-turbulent magnetic fields are present. The difference between angle-averaged and angle-dependent solutions is enhanced by the presence of micro-turbulent fields.

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arXiv2013

Modeling the quantum interference signatures of the Ba II D2 4554 A line in the second solar spectrum

Quantum interference effects play a vital role in shaping the linear polarization profiles of solar spectral lines. The Ba II D2 line at 4554 A is a prominent example, where the F-state interference effects due to the odd isotopes produce polarization profiles, which are very different from those of the even isotopes that have no F-state interference. It is therefore necessary to account for the contributions from the different isotopes to understand the observed linear polarization profiles of this line. Here we do radiative transfer modeling with partial frequency redistribution (PRD) of such observations while accounting for the interference effects and isotope composition. The Ba II D2 polarization profile is found to be strongly governed by the PRD mechanism. We show how a full PRD treatment succeeds in reproducing the observations, while complete frequency redistribution (CRD) alone fails to produce polarization profiles that have any resemblance with the observed ones. However, we also find that the line center polarization is sensitive to the temperature structure of the model atmosphere. To obtain a good fit to the line center peak of the observed Stokes Q/I profile, a small modification of the FALX model atmosphere is needed, by lowering the temperature in the line-forming layers. Because of the pronounced temperature sensitivity of the Ba II D2 line it may not be a suitable tool for Hanle magnetic-field diagnostics of the solar chromosphere, because there is currently no straightforward way to separate the temperature and magnetic-field effects from each other.

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arXiv2012

Polarized line formation with J-state interference in the presence of magnetic fields: A heuristic treatment of collisional frequency redistribution

An expression for the partial frequency redistribution (PRD) matrix for line scattering in a two-term atom, which includes the J-state interference between its fine structure line components is derived. The influence of collisions (both elastic and inelastic) and an external magnetic field on the scattering process is taken into account. The lower term is assumed to be unpolarized and infinitely sharp. The linear Zeeman regime in which the Zeeman splitting is much smaller than the fine structure splitting is considered. The inelastic collision rates between the different levels are included in our treatment. We account for the depolarization caused by the collisions coupling the fine structure states of the upper term, but neglect the polarization transfer between the fine structure states. When the fine structure splitting goes to zero, we recover the redistribution matrix that represents the scattering on a two-level atom (which exhibits only m-state interference --- namely the Hanle effect). The way in which the multipolar index of the scattering atom enters into the expression for the redistribution matrix through the collisional branching ratios is discussed. The properties of the redistribution matrix are explored for a single scattering process for an L=0 to 1 to 0 scattering transition with S=1/2 (a hypothetical doublet centered at 5000 A and 5001 A). Further, a method for solving the Hanle radiative transfer equation for a two-term atom in the presence of collisions, PRD, and J-state interference is developed. The Stokes profiles emerging from an isothermal constant property medium are computed.

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arXiv2012

Polarized line transfer with F-state interference in a non-magnetic medium: Partial frequency redistribution effects in the collisionless regime

Quantum interference phenomena manifests itself in several ways in the polarized solar spectrum formed due to coherent scattering processes. One such effect arises due to interference between the fine structure (J) states giving rise to multiplets. Another effect is that which arises due to interference between the hyperfine structure (F) states. We extend the redistribution matrix derived for the J-state interference to the case of F-state interference. We then incorporate it into the polarized radiative transfer equation and solve it for isothermal constant property slab atmospheres. The relevant transfer equation is solved using a polarized approximate lambda iteration (PALI) technique based on operator perturbation. An alternative method derived from the Neumann series expansion is also proposed and is found to be relatively more efficient than the PALI method. The effects of PRD and the F-state interference on the shapes of the linearly polarized Stokes profiles are discussed. The emergent Stokes profiles are computed for hypothetical line transitions arising due to hyperfine structure splitting (HFS) of the upper J=3/2 and lower J=1/2 levels of a two-level atom model with nuclear spin I_s=3/2. We confine our attention to the non-magnetic scattering in the collisionless regime.

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