Astron. Astrophys. 358, 373-377 (2000)

3. Collision rates of the Zeeman multiplet

The results presented in the following paragraphs have been obtained from a quantum description of the collision using very accurate potential energy curves for NaH.

3.1. Interatomic potentials

The molecular states correlated to the Na()+ H() asymptote are the attractive state and the repulsive state. The Gaussian basis sets employed for their calculation are general contractions based on atomic natural orbitals. The basis sets include for hydrogen and sodium the primitive basis set and of Sadlej contracted to and respectively. The total number of contracted Gaussian functions was 41.

To obtain the best accuracy for the potential energy, multireference configuration interaction (MRCI) wave functions were constructed using multiconfiguration self-consistent field (MCSCF) active space (Werner & Knowles 1985, 1988, Knowles & Werner 1985, 1988) with core and valence orbitals (). The MRCI wave functions accounted for more than 4 millions of configurations which were internally contracted to 250 thousands. All the calculations have been performed with the MOLPRO code ¹. Our results compare well with previous results of Sachs & Hinze (1974) at the scale of the figure (Fig. 1).

In order to test the basis set and the active space, the minimum of the attractive state was determined. A fit of the first vibrational levels yielded the harmonic frequency and the anharmonicity constant . The vibrational levels were obtained from numerical integration of the radial Schrödinger equation using the Numerov method (Johnson 1977). The calculated values of these spectroscopic constants are given in Table 1 for comparison with the experimental results (Huber & Herzberg 1979) and the theoretical results obtained by Sachs & Hinze (1974). Our results agree very well with the experiment and improve the previous theoretical results.

Table 1. Calculated and experimental spectroscopic constants for the state of

3.2. Dynamics of the collision

The quantum mechanical formulation of the collision is that given by Mies (1973) and generalized by Launay and Roueff (1977). A sodium atom with angular momentum J collides with an H atom with angular momentum . We couple J and to obtain the total angular momentum j of the two atoms. Owing to the invariance of the interaction potential V under rotations of the total system, the total angular momentum and its space fixed projection are conserved during the collision. It is convenient to use scattering channel states which describe the asymptotic fragments with relative angular momentum l. The total wave function is expanded in terms of these channel states, the expansion coefficients are the radial amplitudes which satisfy the usual coupled radial equations (Spielfiedel et al. 1991). These radial equations which describe the relative motion of the two atoms depend on the electrostatic interaction potential at each internuclear distance R. The asymptotic form of the radial equations define the T-matrix elements (Spielfiedel et al. 1991).

In the particular case of the Na atom in its ground state () colliding with an H atom in its ground state (), the channels for a given angular momentum are the following: ; ; ; .

From the general equations (Spielfiedel et al. 1991), it can be shown that these channels are uncoupled and that the radial amplitudes are just given by:

µ is the reduced mass of the colliding system, is the wave number defined by where E is the kinetic energy. is the adiabatic interatomic potential corresponding to for channel 1 and for channels 2, 3, 4. The asymptotic form of the radial function F is just proportional to where is the phase shift. From we obtain the S-matrix element and the T-matrix element .

3.3. Relaxation constant of the electronic ground state

As inelastic cross sections are negligible, it can be shown (Spielfiedel et al. 2000), that the relaxation constants of the electronic ground state J is given by:

where is the perturber (H atom) density, v is the relative velocity and the cross section has the following expression:

with:

In the particular case studied here, and (uncoupled T-matrix elements). The coefficients generalize the Grawert factors of Reid (1973) defined for a perturber with angular momentum. In the case under study, and and .

3.4. Results

The typical variations of the phase shifts and with l (Fig. 2) exhibit rapid oscillations for the lower values of l. Such behaviour is usual and may be explained by successive increments of of the phase shift (Child 1974). One also notices a stationary phase point around of the phase shifts characteristic of phase shifts in attractive potentials.

Fig. 2. Variation of the phase shift versus l for the and potentials: eV (full line: ; dotted line: )).

Fig. 3 shows the energy dependence of the cross section . An important feature is the appearance of oscillations in the low energy range. These oscillations are due to resonances in the attractive part of the molecular potential. These oscillations disappear after averaging over the velocities (see Fig. 4). The variation of the rate coefficient with the temperature is very smooth and can be fitted by the following expression:

Fig. 3. Depolarization cross section as function of energy

Fig. 4. Relaxation rate as function of temperature .

© European Southern Observatory (ESO) 2000

Online publication: June 26, 2000
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