*Astron. Astrophys. 324, 344-356 (1997)*
## Appendix A
Since the formulation and notations used in the present paper are
somewhat different from those in Stenflo (1994), it may be difficult
for the reader to find from that monograph the expressions for the
coefficients in Eq. (9) and the exponent
*r* in Eq. (11) for the coefficients . As
these quantities are key components of the general theory of polarized
Raman scattering with quantum interferences, we give the explicit
expressions here.
Let the labels *m* and *n* in be
represented numerically by the differences and
between the *J* quantum numbers of the
excited and initial states, while the labels *a* and *f* are
represented by the *J* quantum numbers of states *a* and
*f*. For economy of notation we keep the name *c* for the
function and define
Then
For symmetry reasons
Due to this symmetry the same off-diagonal expressions occur twice
when summing over both *m* and *n* in Eq. (9). Since in the
somewhat different formulation in Stenflo (1994) the double sum has
been transformed to single sums, the expressions for the differently
defined off-diagonal coefficients there differ by a factor of two from
those here. Both definitions however lead to the identical results for
.
Another symmetry is that
(symmetry with respect to reversal of the scattering direction).
The symmetries (A18) and (A19) for are also
valid for of Eq. (11).
The above algebraic expressions for ,
together with the two symmetry relations, cover all the cases that can
occur for electric dipole transitions. When a transition is not
allowed, its oscillator strength is zero, which means that its
factor will not contribute to the sum in Eq.
(9).
The exponent *r* that determines the sign of the expression
(11) for can be written as a sum of the
contributions from the various transitions that are part of
. Thus
depends exclusively on the difference
and between the
angular momentum quantum numbers *L* and *J* of the upper
level *u* and lower level . We may thus
write
According to Stenflo (1994) the explicit expressions for all the
various cases that can occur for electric dipole transitions are
© European Southern Observatory (ESO) 1997
Online publication: May 26, 1998
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