The collected data for ionisation are listed in Table 1.
We are particularly interested here in pure single ionisation -reaction (I), though cross sections for
transfer ionisation (σti) H+ + He(1s2) → H + He2+ + e- (V) double ionisation (σdi) H+ + He(1s2) → H+ + He2+ + 2e-(VI) and total free electron production (σ- = σsi + σti + 2σdi), are also addressed.
3.1. Experimental data
Experimental data for ionisation have been discussed in detail by Rudd et al. (1985) [2] and by Gilbody (1992) [3].
However, only few experimental methods can provide accurate data for pure single ionisation, especially at low energy collisions (< 15 keV). Very accurate measurements (within 10%) for individual channels (I), (V) and (VI) have been performed by the Belfast group [12, 13] using the cross beam coincidence counting (CBCC) technique. Experi-mental studies based on the parallel plate capacitor (PPC)
technique provide cross sections for the total free electron production (see for example in [14]), including contributions from the channels (I), (V), and (VI). Large discrepancies (up to 50%) between several investigators using the above technique have been encountered in the literature. Fig. 1 presents the detailed measurements for free electron production performed by Rudd et al. (1983) [15] in comparison with the pure single ionisation cross sections of Shah et al. (1989) [13]. As expected the former cross sections are slightly larger than the latter ones in the energy range from 15 to 200 keV, but they are too large for lower collision e for pure single ionisation. The coincidence counting measurements of Afrosimov et al. (1969) [16] for pure single ionisation (not shown) deviate considerably from the previous data. The recommended data by Rudd et al.
(1985) [2] for σ- are close to the values of Rudd et al. (1983) [15], while the recommended data by Barnett (1990) [17] for σsi are considerably larger than the values of Rudd et al.
(1983) [15] and Shah et al. (1989) [13] at collision energies E < 40 keV, where transfer and double ionisation are of minor importance for pure single ionisation. The coincidence counting measurements of Afrosimov et al.
(1969) [16] for pure single ionisation (not shown) deviate considerably from the previous data. The recommended data by Rudd et al. (1985) [2] for σ- are close to the values of Rudd et al. (1983) [15], while the recommended data by Barnett (1990) [17] for σsi are considerably larger than the values of Rudd et al. (1983) [15] and Shah et al. (1989) [13]
at energy collisions E < 40 keV.
3.2. Calculated data
A considerable theoretical effort has been achieved during the last fifteen years in evaluating cross sections for proton collisions with helium atoms. Elaborate quantum-mechanical and classical calculations have been developed in order to obtain a better physical insight of the collision processes (I), (V), and (VI). CTMC studies for ionisation processes have been early performed by McDowell and co-workers [21] and by Zaifman and Maor (1986) [22]. Four-body CTMC calculations of Wetmore and Olson (1988) [23], where electron-electron correlation is ignored, and of Montemayor and Schiwietz (1989) [24], where a radial correlation between the two electrons is included, give improved results for pure single ionisation cross sections at collision energies higher than 100 keV. Still, transfer ionisation is not satisfactory described. A considerable improvement has been proposed by Cohen (1996) [25] using the so-called Quasi-classical trajectory Monte Carlo (QTMC) method, which emphasises the stability of the helium atom.
Fig. 1 includes our results, based on two different types of potentials (see Section 2.1), and the ones obtained by Cohen (1996) [25] for pure single ionisation (I). A better
tions when pure Coulomb interactions are considered. This is in agreement with the general observation that Coulomb potential, being better justified physically, should be used in CTMC calculations [26]. In the present case, the initial electronic radial distribution according to the model potential presents a cut-off at a distance between the electron and the He+ core smaller than the one corresponding to the pure Coulomb case [10]. Consequently, the electron is restricted into a smaller collision volume; contributions to ionisation cross sections coming from large impact parameters, which are important at low energies, are therefore suppressed. Our three-body CTMC calculations reproduce quite successfully the form of experimental single ionisation cross section; the position of the maximum is located at 80 keV, being close to the experimental one at 100 keV. At this energy region, our results are about 25% higher than the data of Shah et al. (1989) [16]. They are in good agreement with experimental data in the energy region around 20 keV, but smaller by ~30% at lower energies. We note that our results are near to those of Schultz and Olson (1988) [27] (not shown) at the higher collision energies
Figure 1. Single ionisation cross sections as a function of collision energy. Experimental data: full circles, Shah et al. (1989) [16];
open circles Rudd et al. (1983) [18]. Compiled data: crosses, Rudd et al. (1985) [2] and Barnett (1990) [20]. CTMC calculations: black solid line, present work using Coulomb potential between e- and He+ core; black dashed line, present work using model potential
considered here. Quantum-mechanical calculations, based on several approximations, have been extensively performed for evaluating the pure single ionisation cross section. Fig. 2 compares results obtained by
– Continuum Distorted Wave Eikonal Initial State (CDW-EIS) approximation of Fainstein et al. (1987) [28],
– Close Coupling approach of Slim et al. (1991) [29]
and Chen and Msezane (1994) [30],
– Two Channel Plane Wave Born Approximation (2-CPWBA) of Das and Malik (1997) [31]
– Multi-Electron Hidden Crossing theory (MEHC) of Krstić et al. (1998) [32].
Also included in Fig. 2 are our 3CTMC results for the pure Coulomb case and experimental data of Shah et al.
(1989) [16] and Rudd et al. (1983) [18]. Calculations of Sahoo et al. (2000) [33] using the impact parameter Born approximation (IPBA, not shown) are in agreement with the measurements of Rudd et al. [18]. The above quantum-mechanical calculations agree with the measurements in different energy regions, reflecting the range of validity of each approximation.
Table 1. Ionization data for H+ + He ground state collisions
Figure 2. Single ionisation cross sections as a function of collision energy. Experimental data: as in Fig. 1. Theory: dash dot line, CDW-EIS calculations from Fainstein et al. (1987) [28]; dashed line, CC calculations from Chen and Msezane (1994) [30]; short dash line, 2-CPWBA calculations from Das and Malik (1997) [31];
short dash-dot line, CCAO calculations from Slim et al. (1991) [29], solid line MEHC theory from Krstic et al (1998) [32], black solid line, 3CTMC calculations, present work.
Table 2. Excitation data for H+ + He ground state collisions
SYMBOLS AND ACRONYMS USED IN TABLES 1 AND 2 Quantities
σe: Single excitation cross section.
σsi: Single ionization cross section.
σsec: Single electron capture cross section.
σti: Transfer ionization cross section.
σdi: Double ionization cross section.
σαsi: Apparent single ionisation cross section (σsec + σsi).
σ−: Total free electron production cross section (σsi + σti + 2σdi).
σ+: Total positive ion production cross section (σ- + σsec + 2σdec).
dσe/dΩ: Differential cross section for single excitation.
dσsi/dΩ: Differential cross section for single ionization.
dσsec/dΩ: Differential cross section for single electron capture.
d2σsi/dΩdΕ: Differential cross section for single ionization.
d2σdi/dΩdΕ: Differential cross section for double ionization.
Pi(b): Ionization probability.
Methods: Experimental (E), Theoretical (T) (E)
CBCC: Cross Beam Coincidence Counting measurement.
CC: Coincidence Counting measurement.
EL: Energy Loss measurement.