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

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3. Results

All the following results have been obtained in a TCAO(A) calculation for the excitation cross sections and a TCAO(S) calculation for the electron capture cross sections, since these approaches are the most accurate in each case, as explained above.

3.1. Excitation and electron capture cross sections to n = 2 and 3

Total excitation and electron capture cross sections to [FORMULA] obtained from the present close-coupling calculations are shown in Fig. 2. We would like to point out that the electron capture process is of the same order of magnitude as the direct excitation process for energies lower than 25 keV and that it decreases rapidly when the collision energy E increases. The same behaviour is found for [FORMULA]. This is not surprising since for [FORMULA] keV, the collision velocity is lower or equal to the classical orbiting electron velocity, which induces equal probabilities for the two processes. For higher energies, the collision is very fast and the capture probability decreases.

[FIGURE] Fig. 2. Total excitation and capture to n=3. Theory: excitation TCAO(A) (full line) and capture TCAO(S) (dotted line). Experiment: excitation [FORMULA] Park et al. 1976.

In spite of small discrepancies for the lower energies, due to the lack of [FORMULA] orbitals in the basis set (Kuang & Lin 1996b), our calculated excitation cross sections are in good agreement with the experimental measurements carried out by Park et al. (1976) and with the more recent theoretical results of Kuang & Lin (1996b).

The calculated cross sections for capture and excitation into the [FORMULA] and [FORMULA] levels are compared with the available experimental data (Morgan et al. 1973, Morgan et al. 1980 and Barnett 1990 extracted from McLaughlin et al. 1997, Detleffsen et al. 1994 and Higgins et al. 1996) in Figs. 3 and 4. The agreement is good over this energy range. The results for capture and excitation into the [FORMULA], [FORMULA] and [FORMULA] levels are shown in Figs. 5 and 6. The agreement with the experiments is also very good. Nevertheless we want to point out the presence of oscillations in the excitation cross sections around 20 keV partly due to the ionization process which becomes more significant at these energies and must carefully be taken into account. These oscillations would disappear if a larger basis set were used as shown by Kuang & Lin (1996b). This problem does not exist for capture.

[FIGURE] Fig. 3. Capture to n=2. Theory: TCAO(S) 2s (dotted line) and 2p (full line). Experiment: 2s [FORMULA] Morgan et al. 1980, 2p [FORMULA] Morgan et al. 1973 and [FORMULA] Barnett 1990.

[FIGURE] Fig. 4. Direct excitation to n=2. Theory: TCAO(A) 2s (dotted line) and 2p (full line). Experiment: 2s [FORMULA] Barnett 1990 and [FORMULA] Higgins et al. 1996, 2p [FORMULA] Barnett 1990 and [FORMULA] Detleffsen et al. 1994.

[FIGURE] Fig. 5. Capture to n=3. Theory: TCAO(S) 3s (dotted line), 3p (full line) and 3d (broken line). Experiment: 3s [FORMULA] Hughes et al. 1992.

[FIGURE] Fig. 6. Excitation to n=3. Theory: TCAO(A) 3s (dotted line), 3p (full line) and 3d (broken line). Experiment: 3p [FORMULA] Detleffsen et al. 1994.

3.2. Balmer [FORMULA] emission

The cross section for Balmer [FORMULA] emission, [FORMULA], is given by the expression:


The various [FORMULA] cross sections coefficients are proportional to the radiative decay to the [FORMULA] level, so that 0.1184 is the branching ratio for the [FORMULA] decay relative to the [FORMULA] decay.

In Fig. 7 our results are compared with the available experimental data of Donnelly et al. (1991) and Detleffsen et al. (1994).

[FIGURE] Fig. 7. Balmer H [FORMULA] cross section. Theory: TCAO(A) Direct excitation (full line), TCAO(S) Capture (dotted line). Experiment: [FORMULA] Donnelly et al. 1991, [FORMULA] Detleffsen et al. 1994.

We note that the data of Donnelly et al. (1991) were not corrected for cascade effects, estimated to be up to [FORMULA]. Considering the experimental uncertainties in the Balmer [FORMULA] predictions, there is a satisfactory agreement between our theoretical close coupling calculations and the experimental data. Such agreement of the experiment with the close coupling methods was recently pointed out by McLaughlin et al. (1997), who also tested perturbative methods and concluded that the perturbative methods strongly underestimate the cross section for Balmer [FORMULA] emission in the 0-200 keV energy range. This conclusion justifies the use of close-coupling methods in spite of the large computational effort needed.

3.3. Differential cross sections

Fig. 8 shows for capture and excitation the reduced differential cross sections [FORMULA] = [FORMULA] of the different sublevels of [FORMULA] for a 1 keV collision energy. All of them are maximum for an angle [FORMULA] which varies from [FORMULA] to [FORMULA] degree. The reduced differential cross sections for excitation and cature into [FORMULA], not shown here, have a similar behaviour with a maximum for the same angle. This maximum angle [FORMULA] varies with energy as shown in Fig. 9. Results for excitation of [FORMULA] at 400 eV (Gaussorgues & Salin 1971) are also reported. One can see that the deflection angle is relatively large at the lowest values of the energy E.

[FIGURE] Fig. 8. Reduced differential cross sections [FORMULA] [FORMULA] [FORMULA] for capture (left) and direct excitation (right).

[FIGURE] Fig. 9. Variation of [FORMULA] versus energy. [FORMULA] this work; [FORMULA] Gaussorgues & Salin 1971.

Due to the Doppler effect, the intensity of the light emitted in a direction [FORMULA] at a given wavelength [FORMULA] in the near wings of H [FORMULA] is directly related to the number of excited atoms in a given direction [FORMULA] from the direction of the incident beam and thus to the differential cross section.

3.4. Polarization fraction of the Lyman [FORMULA], Lyman [FORMULA] and Balmer [FORMULA] lines

For comparison with the experiments, we consider excitation of unpolarized atoms by unidirectional and monoenergetic protons. Since we are interested in the polarization of radiation emitted from the excited states, the density matrix of these states has to be used. Since the collision duration is small, we assume that the photon is emitted long after the excitation process is completed, so that the excitation and the decay processes can be decoupled. We also consider as negligible the other processes that could modify the statistical equilibrium of the levels. The density matrix of the excited atoms is thus directly related to the scattering amplitudes for excitation of the [FORMULA] magnetic sublevels. If we choose as quantization axis the incident proton beam direction, the density matrix is diagonal and the diagonal elements [FORMULA] are proportional to the excitation cross sections [FORMULA] from the ground state [FORMULA].

It is useful to introduce the irreducible tensorial representation of the atomic density matrix (Blum 1981):


where the bracket is a (3j) angular momentum recoupling coefficient. The excited atomic states anisotropy can be directly measured by the alignment parameter, proportional to the [FORMULA] ratio (Fano & Macek 1973):


which depends on the relative population of the [FORMULA] magnetic substates. [FORMULA] is defined as:


for a radiative dipole [FORMULA] transition (the brackett is a {6j} coefficient). For a [FORMULA] transition, [FORMULA] is given in terms of the cross sections [FORMULA] for excitation or capture into the sublevel nlm:


and for a [FORMULA], [FORMULA] is given by:


The [FORMULA], [FORMULA] and [FORMULA] levels calculated alignments following excitation and capture by the proton beam are very similar at the lower energies (see Figs. 10 a,b,c respectively), but whereas the negative alignment of the levels excited during a charge exchange process regularly decreases when the energy increases, the alignment of the directly excited levels increases until very small negative value are reached.

[FIGURE] Fig. 10. Alignment [FORMULA]. Theory: excitation (full line), capture (dotted line). Experiment: [FORMULA] Hippler 1993.

Fig. 10a presents our calculated [FORMULA] compared with the recent experimental data of Hippler (1993) who measured the alignment of the H(2p) level produced either by excitation or by charge transfer. Our results agree with the experimental data, particularly for the lower energies where the alignment is positive. At low incident energies ([FORMULA] 5 keV), [FORMULA] is close to the maximum value of [FORMULA] expected from the rotational coupling theory (Hippler et al. 1988, Hippler (1993)). This result demonstrates that the rotational coupling between the [FORMULA] and [FORMULA] orbitals is the dominant mechanism for H(2p) excitation at these energies: the rotational coupling mechanism only populates the [FORMULA] atomic substates. When the energy increases, the alignment decreases and the mechanisms for H([FORMULA]) excitation become more complicated. Direct long range couplings as well as ionization become more important.

The linear polarization degree for observation in the direction [FORMULA] from the proton beam direction is defined as:


where [FORMULA] ([FORMULA]) is the light emissivity when the electric vector is parallel (perpendicular) to the plane defined by the proton beam and the photon travel direction. The emissivities [FORMULA] and [FORMULA] can be written as the sum of the emissivities for each of the [FORMULA] transitions which contribute to the line and are expressed in terms of the [FORMULA] and [FORMULA] atomic density matrix tensorial elements (Blum 1981):



[FORMULA] accounts for the depolarization due to the fine structure interaction (the hyperfine structure is negligible at the considered densities).


where j denotes a fine structure level of l and [FORMULA] the atomic spin.

If we introduce [FORMULA], the polarization fraction obtained for [FORMULA] degrees, the observed polarization in the direction [FORMULA] is given by:


Explicit expressions of the polarization fraction [FORMULA] for the Lyman [FORMULA], Lyman [FORMULA] and Balmer [FORMULA] lines are the following:


The present results for the [FORMULA] polarization fraction after capture or direct excitation to the [FORMULA] and [FORMULA] levels are shown in Figs. 11 a, b, c. We observe first that [FORMULA] has the same order of magnitude for the two processes at the lower energies, but decreases much faster with energy for capture than for excitation. However, we want to emphasize that, due to the Doppler shift, the capture and the excitation processes contribute to different regions of the line profile.

[FIGURE] Fig. 11. Polarization as a function of energy a [FORMULA], b [FORMULA] and c [FORMULA]. Theory: excitation (full line), capture (dotted line). Experiment: a [FORMULA] Hippler et al. 1988 and _bf c [FORMULA] Werner & Schartner 1996.

The comparison with the available experimental data shows that for Lyman [FORMULA], our results agree reasonably well with the measurements of Hippler et al. 1988. We also find a good agreement for H [FORMULA] with the experimental data from Werner & Schartner (1996) obtained at large energies (E [FORMULA] 40 keV). At these energies, capture gives no contribution and the experiment yields the polarized emission from directly excited atoms. As for the excitation cross sections (see Sect. 3.1), we note the presence of oscillations in the polarization fraction of the radiation emitted by atoms after direct excitation (without capture). These oscillations, particularly around 20 keV, have no physical significance and a more realistic and smooth polarization fraction variation might be obtained after averaging the calculated values. We want to emphasize that the H [FORMULA] line polarization fraction is very small for energies larger than 100 keV and indeed recent measurements by Werner & Schartner (1996) have shown that it becomes negative beyond 200 keV.

From the differential cross sections (see Sect. 3.3) we can define a differential polarization fraction giving the polarization of the emitted line due to excited atoms deflected either in direction [FORMULA] or [FORMULA] depending whether capture or direct excitation occurs. Our results (Balanca et al. 1997) show that after some oscillations at smaller angles, the polarization fraction is almost constant mainly around the deflection angle maximum.

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© European Southern Observatory (ESO) 1998

Online publication: June 2, 1998