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Astron. Astrophys. 329, 792-798 (1998)

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2. Theory

In the one perturber approximation valid in the wings of a line, the quantum lineshape of Ly [FORMULA] at the point [FORMULA] of the profile is (Tran Minh et al. 1975):

[EQUATION]

where [FORMULA] is the electronic density, [FORMULA] is the energy density of the initial states, [FORMULA] is the kinetic energy of the perturber in the initial channel, [FORMULA] is the wave number in the initial channel, and [FORMULA] is that in the final channel [FORMULA], [FORMULA] is the Maxwellian distribution; [FORMULA] is the reduced matrix element of the atomic dipole, [FORMULA] and [FORMULA] are the spin and the angular momentum of the atom in the initial channel; [FORMULA] and [FORMULA] are the total spin and the total angular momentum of the colliding system. The sum [FORMULA] extends over [FORMULA] - where [FORMULA] is the angular momentum of the incident electron - the corresponding coupled channels [FORMULA] and the coupled final channels [FORMULA] and [FORMULA] ; [FORMULA] is connected to kinetic energies of the perturber in the initial and final states i and f. The radial wave functions G are solutions of the coupled differential equations.

The details of the formalism and the approximations used (dipolar approximation for the potential, exact resonance, no-quenching approximation, etc) were discussed in the previous papers as well as its classical limit (Tran Minh et al. 1980). The lineshape finally reads:

[EQUATION]

where

[EQUATION]

can be calculated in terms of hypergeometric [FORMULA] functions by many methods with good precision. [FORMULA] is the lower cut-off in the integration over the energies of the perturber. The [FORMULA] 's are cylindrical Bessel functions. The lineshape can also be written:

[EQUATION]

where the cut-off [FORMULA], corresponding to the Debye radius, takes into account the screening of the other electrons. The contribution of the first three angular momenta, [FORMULA] 0, 1 and 2, which was shown to be the major contribution in the far wings, cannot be provided by the exact resonance method since in these cases, [FORMULA] is not real. This contribution will be estimated in the present work by an extrapolation method (Sect. 4). [FORMULA] can also be decomposed into multipole contributions [FORMULA], with [FORMULA] =1, 2, and 3 for the dipole, quadrupole and polarization potentials (Tran Minh et al. 1980); but when these higher multipoles contribute significantly to the e [FORMULA] -H potential, correlations between the atomic electron and the perturber become important and it is not sufficient to use a multipolar expansion for the potential. In particular, exchange effects become important.

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© European Southern Observatory (ESO) 1998

Online publication: December 8, 1997
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