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Astron. Astrophys. 334, 1136-1144 (1998)

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2. Description of present H- [FORMULA] collision calculations

The two-centre atomic orbital close coupling method (TCAO) (Bates & McCarroll 1958) has been successfully exploited for treating ion-atom collisions in the intermediate energy range (from 1 keV to 200 or 300 keV) over the past two decades. In this energy range, the velocities of the projectile and the active atomic electron are of the same order of magnitude, and thus three reaction processes compete, i.e. excitation, capture and ionization, which makes the TCAO close-coupling method a natural choice to describe the collision process dynamics (Kuang & Lin 1996a).

Within the semi-classical impact parameter approximation, the time dependent wave function [FORMULA] is expanded in terms of bound atomic orbitals plus continuum states including the plane wave electronic translational factor. The time dependent electronic wave function is given by:

[EQUATION]

where particle A is the incident proton, i.e. the projectile, and particle B is the proton of the initial hydrogen atom, i.e. the target. [FORMULA] and [FORMULA] are the transition amplitudes for the occupation of the atomic states [FORMULA] and [FORMULA] whose respective eigenenergies are [FORMULA] and [FORMULA]. We have first performed a traditional symmetric TCAO close-coupling calculation (S). The basis set (TCAO-S) includes 26 states on each centre of this symmetric collision, allowing us to describe all the sublevels from [FORMULA] to [FORMULA]. Furthermore it includes other bound states and pseudostates to take the continuum into account.

Fritsch & Lin (1982) have shown that a TCAO calculation with 22 states on each centre reproduces correctly all the first [FORMULA] molecular states. However, the TCAO(S) close-coupling method introduces spurious oscillatory structures in the cross sections (Slim & Ermolaev 1994, Kuang & Lin 1996a and b), that are absent in the experimental data (Park et al. 1976, Detleffsen et al. 1994). These structures, due to the representation of the continuum by short range pseudostates, disappear when the basis set includes pseudo-continuum states at only one centre. Thus, to obtain a better agreement for the excitation cross sections and the polarization fraction, we have performed an asymmetric TCAO close-coupling calculation (A), in which all the pseudostates on the projectile centre have been removed. The basis (TCAO-A) includes 26 states on the target centre B and only 19 bound states on the projectile A (Balanca et al. 1997). The total cross sections from a state i to a state f are easily obtained for capture and direct excitation:

[EQUATION]

The differential cross sections are given by:

[EQUATION]

where the scattering amplitude is:

[EQUATION]

[FORMULA] is the mass of the proton, v the relative velocity, [FORMULA] is the difference between the initial and final magnetic quantum numbers, [FORMULA] is the Bessel function of integer order [FORMULA], while [FORMULA] and [FORMULA] is the limit of the integration of the coupled equations ([FORMULA]). [FORMULA] is the transition amplitude from a state i of B to a state j of A or B for an impact parameter b with initial conditions [FORMULA] and [FORMULA]. The transition amplitude must include the Coulomb phase factor as shown by Dubois et al. (1993). In the case of capture to a state j of A, the amplitude is given by:

[EQUATION]

For direct excitation, the expression is similarly obtained by changing amplitude [FORMULA] into [FORMULA]. These differential cross sections yields the excited atom angular distribution of velocities. Since the thermal velocities of the hydrogen atoms before excitation are very small compared to the incoming proton velocities, conservation of the momenta leads us to draw the diagram of the protons and excited atoms velocities as shown in Fig. 1 for direct excitation: the hydrogen atoms move perpendicularly to the direction of the scattered protons. For capture, the relative position of the scattered protons and the excited atoms are exchanged.

[FIGURE] Fig. 1. Diagram of velocities

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© European Southern Observatory (ESO) 1998

Online publication: June 2, 1998

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