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the interpretation of the second solar spectrum requires taking into account a number of physical processes that are not relevant for the intensity spectrum . some of the lines observed in the second solar spectrum , like the nai d@xmath4 and d@xmath0 , cr  i triplet at 5206 , baii d@xmath4 and d@xmath0 , mg  ii h and k , and caii h and k , are affected by quantum interference between states of different total angular momentum ( @xmath5 or @xmath1 states ) . the importance of quantum interference was first demonstrated both observationally and theoretically by . an approximate theoretical approach for treating @xmath5-state interference with partial frequency redistribution ( prd ) was proposed by @xcite . this theory was then applied to model the linear polarization profiles of the cr  i triplet at 5204 - 5208     by . an atom with a nuclear spin @xmath6 exhibits hyperfine structure splitting ( hfs ) . the quantum interference between these hyperfine structure ( @xmath1 ) states produces depolarization in the line core . nai d@xmath0 , baii d@xmath0 , and scii line at 4247  are some examples of spectral lines whose linear polarization profiles show signatures of @xmath1-state interference . the ba  ii d@xmath0 line is due to the transition from the upper @xmath7 and the lower @xmath8 fine structure levels ( see figure  [ smitha - level - diag](a ) ) . in the odd isotopes of ba , both the upper and lower levels undergo hfs due to the nuclear spin @xmath9 . this gives rise to four upper and two lower @xmath1-states ( see figure  [ smitha - level - diag](b ) ) . in modeling the baii d@xmath0 line it is necessary to account for quantum interference between the upper @xmath1-states . the odd isotopes constitute about 18%  of the total ba abundance in the sun  . the rest 82%  is from the even isotopes , which are not subject to hfs ( because @xmath10 ) .     and @xmath11 , with their hyperfine structure components ( b ) . the energy levels are not drawn to the scale.,width=377 ]    high precision spectro - polarimetric observations by using zimpol clearly revealed the existence of three distinct peaks in the linear polarization @xmath12 profiles of the ba  ii d@xmath0 line . using the last scattering approximation , explained the physical origin of these three peaks . he showed that the central @xmath2 peak arises from the even isotopes of ba , and the two side peaks from the odd isotopes . with a similar last scattering approach , the magnetic sensitivity of the ba line was explored by @xcite . the above mentioned authors however did not account for radiative transfer and prd effects . these effects were taken into account by , but the treatment was limited to the even isotopes of ba , for which hfs is absent . in @xcite we considered radiative transfer with prd in both even and odd isotopes , and the full effects of hfs with @xmath1-state interferences . here we present a brief summary of our attempts to model the baii d@xmath0 line profile observed in a quiet region close to the solar limb . further we present center - to - limb variation ( clv ) observations of this line and our initial attempts to model the same . the paper is organized as follows : in section  [ smitha - transfer - equations ] we present the polarized radiative transfer equation suitable to handle several isotopes of ba simultaneously . in section  [ smitha - obs - details ] we present the details of the observations . in section  [ smitha - modeling - pro ] we discuss the model atom and the model atmosphere used . the results are presented in section  [ smitha - results ] with concluding remarks in section  [ smitha - conclusions ] . the polarized transfer equation in a 1-d non - magnetic axisymmetric atmosphere is given by @xmath13 , \label{smitha - transfer}\end{aligned}\ ] ] in the reduced stokes vector basis . positive @xmath14 represents the linear polarization oriented parallel to the solar limb . the total opacity @xmath15 , with @xmath16 and @xmath17 being the continuum scattering and continuum absorption coefficients , respectively and @xmath18 being the wavelength averaged absorption coefficient for the @xmath19 transition . here @xmath20 and @xmath21 are the lower and upper level electronic angular momentum quantum numbers respectively . for the case of ba  ii d@xmath0 , the appropriate expression for the profile function @xmath22 is @xmath23 here the voigt profile function for the even isotopes of ba  ii is denoted as @xmath24 . this corresponds to the @xmath25 transition in the absence of hfs . @xmath26 is the profile function for the odd isotopes . it is the weighted sum of individual voigt profiles @xmath27 representing each of the @xmath28 absorption transition . the initial and the intermediate hyperfine split levels have total angular momentum quantum numbers @xmath29 and @xmath30 respectively . @xmath26 is given by @xmath31 . \label{smitha - prof - combined}\end{aligned}\ ] ] in equation  ( [ smitha - phi - g ] ) the contributions from both the @xmath32ba ( 6.6% ) and @xmath33ba ( 11.2% ) odd isotopes add upto the @xmath34 of @xmath26 . in equation  ( [ smitha - transfer ] ) , @xmath35 is the reduced total source vector for a two - level atom with an unpolarized lower level and is defined as @xmath36 showed that , for the case of ba  ii d@xmath0 any ground level polarization would be destroyed by elastic collisions with hydrogen atoms .    in equation  ( [ smitha - s - total ] ) , @xmath37 , where @xmath38 is the planck function . the line source vector @xmath39 is @xmath40 with @xmath41 here @xmath42 is a @xmath43 diagonal matrix , including the effects of hfs in odd isotopes . @xmath44 diag @xmath45 , with @xmath46 being the redistribution function components for the multipolar index @xmath47 , containing both type - ii and type - iii redistribution of @xcite . the quantum number replacement @xmath48 in equation  ( 7 ) of gives the expression for @xmath49 . we use crd functions in place of the type - iii redistribution functions as in @xcite . the contributions from the individual redistribution matrices for the odd isotopes @xmath32ba and @xmath33ba are also taken into account . @xmath50 is also a @xmath43 diagonal matrix for the even isotopes without hfs . its elements @xmath51 are the redistribution functions corresponding to the scattering transition @xmath52 . they are obtained by setting the nuclear spin @xmath10 in @xmath53 . an expression for @xmath50 in the stokes vector basis can be found in @xcite and in . we here use the angle - averaged versions of these quantities . @xcite showed that in the non - magnetic case the use of the angle - averaged redistribution matrix is sufficiently accurate for all practical purposes . we use the two branching ratios ( see * ? ? ? * ; * ? ? ? * ) @xmath54 where @xmath55 and @xmath56 are the radiative and inelastic collisional rates , respectively . @xmath57 is the elastic collision rate computed from @xcite . the depolarizing elastic collision rates are given by @xmath58 with @xmath59 . the @xmath60 is computed using @xmath61 where @xmath62 is the neutral hydrogen number density in ( @xmath63 ) , @xmath64 the temperature in kelvin , and @xmath65 the energy difference between the @xmath66 and @xmath67 fine structure levels in ( @xmath68 ) and @xmath69 is the boltzmann constant in ( @xmath68/k ) . we neglect the collisional coupling between the @xmath67 level and the metastable @xmath70 level ( see * ? ? ? . pointed out the importance of @xmath67 - @xmath70 collisions for the line center polarization of ba  ii d@xmath0 . he showed that the neglect of such collisions would lead to an overestimate of the line core polarization by @xmath71 . it was shown by that this in turn would lead to an overestimate of the microturbulent magnetic field @xmath72 by @xmath73 . our aim is to explore the roles of prd , hfs , quantum interferences , and the atmospheric temperature structure in the modeling of the triple peak structure of the ba  ii d@xmath0 linear polarization profile . we assume frequency coherent scattering in the continuum with its source vector given by @xmath74 the matrix @xmath75 is the rayleigh scattering phase matrix in the reduced basis . @xmath76 is the line thermalization parameter defined by @xmath77 . the stokes vector @xmath78 can be computed from the irreducible stokes vector @xmath79 by simple transformations . the observed polarization profiles of the ba  ii d@xmath0 line shown in this paper for @xmath80 are same as those presented in @xcite . the observations for all @xmath3 values were acquired by the eth team of stenflo on june 3 , 2008 , using their zimpol-2 imaging polarimetry system at the themis telescope on tenerife . the polarization modulation was done using ferroelectric liquid crystal modulators . the observations were recorded at various limb distances with the slit being placed parallel to the heliographic north pole . figure  [ smitha - ccd - images ] shows the ccd images of @xmath81 and @xmath2 for different @xmath3 values . these essentially represent the clv of the intensity and degree of linear polarization . in the section  [ smitha - results ] we attempt to model these clv observations . and @xmath2 observed at different limb distances on the solar disk.,title="fig:",width=211 ]   and @xmath2 observed at different limb distances on the solar disk.,title="fig:",width=219 ] the observed @xmath82 profiles of the ba  ii d@xmath0 line can be modeled using a two - stage process . this process is similar to the one described in . in this two - stage process , the intensity is first computed which is then used as an input to compute the polarization . the code that is used for computing the intensity was developed by ( * ? ? ? * referred to as the rh - code ) . to compute the intensity and polarization we need a model atom that is representative of the atom under consideration and a one - dimensional model atmosphere that mimics the solar atmosphere . for the case of ba  ii d@xmath0 , since the effects of different isotopes are to be taken into account , we construct three different atom models . out of these , two are for the odd and one for the even isotope . the atom model for the even isotope ( @xmath83ba ) is given by the five levels of figure  [ smitha - level - diag](a ) , while for the odd isotopes ( @xmath32ba and @xmath33ba ) the model is extended to include the hyperfine splitting as described by figure  [ smitha - level - diag](b ) . the hfs of the @xmath66 level and the metastable levels are neglected . the effects of hfs are taken into account only for the d@xmath0 transition in the intensity computations . for the odd isotopes , the radiative transitions between @xmath67 and the metastable levels are treated in the same way as for the even isotope . however , in the polarization computations , under the two - level atom approximation , the coupling to the metastable levels are neglected for both even and odd isotopes . we neglect the contribution from other less abundant even isotopes . the wavelengths of the six hyperfine transitions for the odd isotopes are taken from kurucz database and are also listed in table  1 of @xcite . the line strengths of these transitions are given in equation  ( [ smitha - prof - combined ] ) . model atmospheres.,width=377 ]    some of the commonly used standard 1-d solar model atmospheres are the fala , falf , falc ( see * ? ? ? * ) and the falx ( see * ? ? ? * ) . among these four models falf is the hottest and falx the coolest . however , as will be discussed below , we find that a model atmosphere that is cooler than falx is needed to fit the observed profiles . the new model , denoted @xmath84 , is obtained by reducing the temperature of the falx model by about 300 k in the height range 500  1200 km above the photosphere . the temperature structures of the standard model atmospheres and the modified model are shown in figure  [ smitha - temp - struc ] . the three atom models of ba  ii atom described in the above section give us three different sets of physical quantities ( two for the odd isotopes and one for the even isotope ) when used in the rh code to compute intensity . these quantities are the line opacity , line emissivity , continuum absorption coefficient , continuum emissivity , continuum scattering coefficient , and the mean intensity . see @xcite for the mathematical expressions that are used in the rh - code to compute these quantities for the even isotopes . for the odd isotopes , the profile functions in these expressions are replaced by @xmath26 defined by equation  ( [ smitha - prof - combined ] ) . profiles computed using different model atmospheres.,width=264 ]    the three sets of quantities obtained from three atom models are combined in the ratio of their respective isotope abundances . the combined quantities are then used as inputs to the polarization code . the @xmath2 profiles so computed for the various model atmospheres are shown in figure  [ smitha - no - fit ] . the figure shows that the @xmath2 profiles are quite sensitive to the temperature structure of the model atmosphere and that the central peak can not be reproduced from any of the standard model atmospheres . all the models produce a dip in @xmath2 at the line center . earlier studies have shown that such a central dip in @xmath2 is generally due to the effects of prd ( see for example , figure  3 of , and figure  6 of ) . the main contribution to this dip comes from the type - ii frequency redistribution . in the case of ba  ii d@xmath0 , the central dip is formed mainly through contributions from the even isotopes ( see figure  5 of * ? ? ? . however the profiles computed using only crd do not show any central dip . it is generally known that @xmath2 profiles computed with crd and prd in realistic model atmospheres differ not only in the line wings but also in the line core . this is contrary to the behaviour expected of @xmath2 profiles computed in isothermal atmospheres , where they nearly match . a study of the mechanisms that cause the differences in the line core , in the case of realistic atmospheres , would be taken up in a forthcoming publication . show that the magnitude of the @xmath2 central dip is strongly dependent on the choice of atmospheric parameters . the central dip becomes shallower for cooler models . this behavior is also evident from our figure  [ smitha - no - fit ] . such dips in @xmath2 are common in line profile modeling with prd , and are often smoothed out by applying instrumental and macro - turbulent broadening . the profiles presented in figure  [ smitha - no - fit ] have already been smeared with a gaussian function having a full width at half maximum ( fwhm ) of 70  m . though it is possible to smoothen out the dip by increasing the smearwidth , it will suppress the side peaks which are due to the odd isotopes and also disturbs the fit to the intensity profile . a smearwidth value of 70  m  has been chosen by us to optimize the fit and it is also consistent with what we expect based on the observing parameters and turbulence in the chromosphere . since all the standard model atmospheres failed to provide a fit , we had to modify the temperature structure of the falx model to get a new much cooler model . this modified model is called the @xmath84 and the details of its construction are given in @xcite . this new model atmosphere succeeds in fitting both the intensity and the linear polarization profiles . the fit to the @xmath2 profile is shown as the dashed line in figure  [ smitha - fit ] . the theoretical profile in this figure also includes the contribution from the spectrograph stray light which is 4% of the continuum intensity . this is done to mimic the observed profiles which contain the contribution from stray light . finally , the effect of microturbulent magnetic field with a strength of 2  g is included to obtain a good @xmath2 fit at the line center . profiles using the @xmath84 model atmosphere and also the comparison between profiles computed with prd and crd.,width=377 ]    the ba  ii d@xmath0 4554    line is a resonance line and is strongly influenced by the effects of prd . hence it is very important to take account of the prd effects while modeling . its importance in case of the even isotopes of ba  ii is demonstrated in . in the right panel of figure [ smitha - fit ] , we present a comparison between the observed and the theoretical @xmath2 profiles based on crd alone ( dotted line ) and on full prd ( dashed line ) for the @xmath85 model . the intensity profile can be fitted well using either prd or crd ( see * ? ? ? * ) , but the polarization profile can not be fitted with crd alone . this shows that the effects of prd are essential to model the @xmath2 profiles of the ba  ii d@xmath0 line . the left panel of figure  [ smitha - clv ] shows the clv of the observed @xmath2 profiles . the right panel of this figure shows the clv of the theoretical @xmath2 profiles computed using the @xmath84 model atmosphere . as seen from the right panel , the newly constructed model atmosphere , though successfully fits the observed profiles for @xmath80 , fails to fit the observations for @xmath86 . in fact for @xmath86 , we again obtain a central dip instead of a peak in the @xmath2 profiles . the central dip becomes deeper as @xmath87 . this behavior was noticed also for the case of na  i d@xmath0 by . thus the modification in the temperature structure of the falx model atmosphere , presented in @xcite and in this paper , helps only for @xmath80 . for other @xmath3 positions , the temperature structure has to be further modified . thus we are unable to find a single 1d model atmosphere that fits the clv observations of the ba  ii d@xmath0 line . positions . for example , when modeling the cr  i triplet in , we had to modify the temperature structure of the falf model atmosphere in the deeper layers to obtain a fit to the @xmath82 observations at @xmath88 . ] profiles and the theoretically computed clv . the theoretical profiles are computed using the @xmath84 model atmosphere.,width=377 ] in this paper we briefly describe the results of our previous paper on modeling the polarization profiles of the ba  ii d@xmath0 line by taking full account of prd , radiative transfer , and hfs effects . also , we extend the same to model the observed clv of the @xmath2 profiles . to model the clv profiles , we use the same approach that was used in @xcite . we find that some of the standard model atmospheres like the falf , falc , fala , and falx fail to reproduce the central peak , and instead give a central dip which is mainly due to prd effects . also , this central dip is quite sensitive to the temperature structure of the model atmosphere . by slightly modifying the temperature of the standard falx model atmosphere , we obtain a fit to both @xmath81 and @xmath2 profiles . this new model is called the @xmath84 . however this new model successfully fits the observations only for @xmath80 and fails at other limb distances . hence within the sample of the model atmospheres tested by us , no single one dimensional , single component model atmosphere succeeds in reproducing the observed @xmath2 profiles at all limb distances . this possibly indicates that multi - dimensional transfer effects are required to model the clv observations of this line . finally , we show that since the ba  ii d@xmath0 is a strong resonance line , it has damping wings that are governed by prd . it is therefore necessary to take full account of prd in order to correctly model the @xmath2 profile . in contrast , the intensity profiles computed with prd or crd do not differ substantially from each other through out the entire line profile . we would like to thank dr . michele bianda for useful discussions . we acknowledge the use of hydra cluster facility at the indian institute of astrophysics for computing the results presented in the paper . we would like to thank the referee for a careful reading of the manuscript and for giving useful comments and suggestions .
the ba  ii d@xmath0 line at 4554    is a good example , where the @xmath1-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 @xmath1-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 ) can not reproduce the observed wing polarization . 
 we present the observed and computed @xmath2 profiles at different limb distances . 
 the theoretical profiles strongly depend on limb distance ( @xmath3 ) and the model atmosphere which fits the limb observations fails at other @xmath3 positions . 
 [ authorguide ]