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Astron. Astrophys. 324, 344-356 (1997)

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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 [FORMULA] in Eq. (9) and the exponent r in Eq. (11) for the coefficients [FORMULA]. 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 [FORMULA] be represented numerically by the differences [FORMULA] and [FORMULA] 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

[EQUATION]

Then

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

For symmetry reasons

[EQUATION]

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 [FORMULA].

Another symmetry is that

[EQUATION]

(symmetry with respect to reversal of the scattering direction). The symmetries (A18) and (A19) for [FORMULA] are also valid for [FORMULA] of Eq. (11).

The above algebraic expressions for [FORMULA], 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 [FORMULA] factor will not contribute to the sum in Eq. (9).

The exponent r that determines the sign of the expression (11) for [FORMULA] can be written as a sum of the contributions from the various transitions that are part of [FORMULA]. Thus

[EQUATION]

[FORMULA] depends exclusively on the difference [FORMULA] and [FORMULA] between the angular momentum quantum numbers L and J of the upper level u and lower level [FORMULA]. We may thus write

[EQUATION]

According to Stenflo (1994) the explicit expressions for all the various cases that can occur for electric dipole transitions are

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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