## 2. Quantum theory of line polarizability## 2.1. IntroductionFor allowed electric dipole transitions the quantum-mechanical
amplitude for scattering from initial substate are the polarization vectors of the radiation
field, the frequency of the scattered radiation,
the energy difference between the upper and
final levels, and the damping constant (which is
the sum of the radiative and collisional contributions). The summation
is done over all the intermediate states The radiation coherency matrix that describes the transformation from the polarization state of the incident to that of the scattered radiation is given by (Stenflo 1994 ), where the symbols and
stand for tensor product and complex
conjugation, respectively. For simplicity we have here assumed (as we
will do in the rest of the paper) that there is no atomic polarization
in the initial state Note that the initial, intermediate, and final states may have
arbitrary Zeeman splittings, and that in Eq.
(1) contains the Zeeman displacements of the magnetic sublevels
To obtain the scattering Mueller matrix that describes how a Stokes 4-vector is transformed by the scattering process, we form where is a purely mathematical transformation matrix without physical contents, given explicitly in Stenflo (1994 ). The constant of proportionality is determined by the normalization condition for . We may expand in terms of its multipolar components (Stenflo 1996 ): where index In the weak-field limit the effect of the Zeeman splitting on the
shapes of the absorption line profiles can be disregarded. This means
that all the matrix components of can be
assumed to have the same frequency profile, which therefore can be
factorized out from as a common scalar. The
remaining, frequency-independent matrix that describes the
polarization properties of the scattering process is in its normalized
form the so-called where, in the case of zero magnetic fields,
is the classical, Rayleigh scattering phase matrix. By definition and
normalization and . While
for classical scattering the values of these
coefficients generally depend on the total angular momentum quantum
numbers of the initial, intermediate, and final states When the magnetic field is weak but not zero it influences the
scattering process via the If we disregard the here unimportant circular polarization, the phase matrix that results from Eq. (1) can in the case of zero magnetic field be written as where is the Rayleigh phase matrix that is valid for the case of classical dipole scattering. thus represents the fraction of scattering processes that occurs as classical dipole scattering, while the remaining fraction, , represents unpolarized, isotropic scattering. ## 2.2. Raman scattering with quantum interferencesIn the present paper the terms The scattering theory developed in the present paper is valid for general Raman scattering. The other terms will be used when addressing special cases. Let us now in what follows for convenience of notation let
The coefficients , which are frequency dependent (see below), contain not only the squared terms but also the interference terms between the various intermediate states of different total angular momenta. Eq. (7) allows us to define a phase matrix of the form (5) if we let If more than one intermediate state contributes to the transition
from state The coefficients can more explicitly be given as (cf. Stenflo 1994 ), where Here the exponent The off-diagonal terms in Eq. (9) represent quantum mechanical interferences between states with different values of and . For such terms cancel out, so that we get Quantum interferences between excited states of different total
angular momenta can only be neglected when the line opacity becomes
much smaller than the continuum opacity at distances from the resonant
frequencies that are comparable to the fine structure splitting. In
many cases this condition is not at all satisfied, e.g. in the case of
the Ca II H and K lines at 3965 and 3933 Å , the
Na I D The diagonal coefficients are given by where the brackets denote 6- ## 2.3. Polarizability for an entire multipletTo obtain the polarizability for an entire
multiplet one has to extend the sums in Eq. (2) over all the possible
initial and final states of the multiplet, i.e., one has to add
together all the possible fluorescent or Raman scattering
contributions within the multiplet. Thereby one needs to account for
the multiplicity () of these states, which
represents the number of magnetic substates for a given where is given by Eqs. (1)-(3) in the case
of arbitrary Zeeman splitting, and by Eq. (7) in the case of weak
magnetic fields. The above as well as the following expressions are
valid for multiplets produced by fine-structure splitting. For a
hyperfine structure multiplet the expressions remain the same if we
replace If we introduce a phase matrix as in Eq. (5) with the same normalization as before but now representing the whole multiplet, and define then the polarizability coefficients become In the general case when the incident radiation is not spectrally flat one has to apply the intensity as an additional weight to the nominator and denominator of Eq. (17). Also in Eq. (16) needs to be convolved with a Gaussian to account for the thermal and turbulent Doppler broadening before it is inserted in the nominator and denominator of Eq. (17) to form . Fig. 1 shows the energy level diagrams for two multiplets that we will be dealing with, fine structure multiplet No. 318 of neutral iron, and the hyperfine multiplet of the Ba II 4554 Å line. The solid, dashed and dotted lines show some of the different possible combinations of fluorescent scattering transitions in these multiplets.
If we sum up all the allowed fluorescent combinations with Eq.
(16), then Eq. (17) gives us a polarizability
that can have a very complex wavelength dependence, as shown by
Fig. 2, in which the two multiplets of Fig. 1 are
represented by the two lower panels. The upper panels represent
respectively scattering in the Na I D
## 2.4. Interference and spectroscopic stabilityThe principle of spectroscopic stability provides us with a
powerful tool to check the correctness of our algorithms to compute
the polarizability. In the present context we use this principle to
determine what must happen in the limit of vanishing fine (or
hyperfine) structure splitting. The fine structure is physically due
to the electron spin, so the polarizability in
the limit of vanishing splitting is simply obtained by letting the
spin quantum number Let us use this principle to check the results of Fig. 2. The 1 Na I multiplet represents a transition for which . With in Eq. (14) we find that . The solid curve indeed approaches unity asymptotically when moving either to smaller or larger wavelengths. A transition is the quantum-mechanical analog to classical dipole-type scattering, for which of course always has its classical value, unity. The 1 Ca II 2
Ca II transition corresponds to .
This gives the asymptotic value , in agreement
with the solid curve. Similarly, as the 318 Fe I
multiplet is an In the hyperfine case of the Ba II 4554 Å line
we have to let the nuclear spin The dashed curves in Fig. 2 show the solutions for that are obtained if all the quantum interference (off-diagonal) terms in Eq. (9) are omitted. The asymptotic behavior is then entirely different and violates the principle of spectroscopic stability. The oscillator strengths The principle of spectroscopic stability can be shown to be obeyed also for scattering from each separate initial state, before summing in Eq. (16) over (but after summing over the final states). This is an even more restrictive condition that the algebraic expressions have to satisfy. © European Southern Observatory (ESO) 1997 Online publication: May 26, 1998 |