In the one perturber approximation valid in the wings of a line, the quantum lineshape of Ly at the point of the profile is (Tran Minh et al. 1975):
where is the electronic density, is the energy density of the initial states, is the kinetic energy of the perturber in the initial channel, is the wave number in the initial channel, and is that in the final channel , is the Maxwellian distribution; is the reduced matrix element of the atomic dipole, and are the spin and the angular momentum of the atom in the initial channel; and are the total spin and the total angular momentum of the colliding system. The sum extends over - where is the angular momentum of the incident electron - the corresponding coupled channels and the coupled final channels and ; 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:
can be calculated in terms of hypergeometric functions by many methods with good precision. is the lower cut-off in the integration over the energies of the perturber. The 's are cylindrical Bessel functions. The lineshape can also be written:
where the cut-off , corresponding to the Debye radius, takes into account the screening of the other electrons. The contribution of the first three angular momenta, 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, is not real. This contribution will be estimated in the present work by an extrapolation method (Sect. 4). can also be decomposed into multipole contributions , with =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 -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.
© European Southern Observatory (ESO) 1998
Online publication: December 8, 1997