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TABLE IV: PARTIAL TRANSFER CROSS SECTIONS IN Ti*t~ He COLLISIONS.

Table 13 IV: PARTIAL TRANSFER CROSS SECTIONS IN Ti*t~ He COLLISIONS.

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CCC results for the excitations from the helium metastable states due to the unitarity of the close coupling formalism (as soon as convergence is established). Cross sections for excitation from the metastable He levels to n=2-4 levels are estimated to be accurate to better than 10%. An exception to this is the excitation of F states, which are likely to be systematically too high at the intermediate energies due to lack of allowance for excitation of G and higher L states. In our assessment of the relevant cross sections we shall give a relatively great weight to the new CCC data. Im nie frirttaar nanainaratinn nf thn tranratinnl Antn aimilarhs ac in [Al Int pie atarte nt
CCC results for the excitations from the helium metastable states due to the unitarity of the close coupling formalism (as soon as convergence is established). Cross sections for excitation from the metastable He levels to n=2-4 levels are estimated to be accurate to better than 10%. An exception to this is the excitation of F states, which are likely to be systematically too high at the intermediate energies due to lack of allowance for excitation of G and higher L states. In our assessment of the relevant cross sections we shall give a relatively great weight to the new CCC data. Im nie frirttaar nanainaratinn nf thn tranratinnl Antn aimilarhs ac in [Al Int pie atarte nt
Fig. 2. Excitation cross sections for singlet-singlet transitions in He. Theory: boxes - [15], solid circles - [16], solid triangles - [17], open triangles - [18], open circles - [20], dotted curves- [22], solid curves- CBE cross sections and dot-dashed curves - a dipole model Eq.(22), present work, Dashed curve in Fig. 2f represents the pure exchange excitation cross section, present work (see text).
Fig. 2. Excitation cross sections for singlet-singlet transitions in He. Theory: boxes - [15], solid circles - [16], solid triangles - [17], open triangles - [18], open circles - [20], dotted curves- [22], solid curves- CBE cross sections and dot-dashed curves - a dipole model Eq.(22), present work, Dashed curve in Fig. 2f represents the pure exchange excitation cross section, present work (see text).
Cross sections for single ionization of rare gas ions by electron impact have been reviewed. New reliable experimental data are presently available for neon (q=4, 5, 6, 7) and for krypton (q=11) ions. These data have been fitted with standard simple formula. The fitting coefficients are given for all the possible ion charge states for neon, argon and for q<11 for krypton ions. Scaling of the neon data provides additional information on the highest charge states of argon (q211) for which no experimental results are presently available.
Cross sections for single ionization of rare gas ions by electron impact have been reviewed. New reliable experimental data are presently available for neon (q=4, 5, 6, 7) and for krypton (q=11) ions. These data have been fitted with standard simple formula. The fitting coefficients are given for all the possible ion charge states for neon, argon and for q<11 for krypton ions. Scaling of the neon data provides additional information on the highest charge states of argon (q211) for which no experimental results are presently available.
Table 1:Threshold energies (eV) for the ionisation of electrons in the outermost sub- shell of tungsten ions, calculated by the MCDF-code of Grant et al. [11, 12].
Table 1:Threshold energies (eV) for the ionisation of electrons in the outermost sub- shell of tungsten ions, calculated by the MCDF-code of Grant et al. [11, 12].
FIG. 1. Hypothetical DR resonance at Een = 5eV with a given strength S (see text). Experimental rate data with 5% statistical uncertainty at the maximum are simulated (open circles with error bars). From the experiment the transverse and longitudinal tem- peratures of the electron beam are deduced and thus the distribution function Eq.(13) is inferred. The original DR resonance is convoluted with this distribution (solid line).
FIG. 1. Hypothetical DR resonance at Een = 5eV with a given strength S (see text). Experimental rate data with 5% statistical uncertainty at the maximum are simulated (open circles with error bars). From the experiment the transverse and longitudinal tem- peratures of the electron beam are deduced and thus the distribution function Eq.(13) is inferred. The original DR resonance is convoluted with this distribution (solid line).
FIG. 3. Dielectronic recombination of He*+ ions in the range of An = 2 transitions. Adopted from [32].
FIG. 3. Dielectronic recombination of He*+ ions in the range of An = 2 transitions. Adopted from [32].
FIG. 5. Dielectronic recombination cross sections of O"*+ [24]; a) experiment, b) theory.
FIG. 5. Dielectronic recombination cross sections of O"*+ [24]; a) experiment, b) theory.
At low center-of-mass energies An = 0 transitions are observed Li-like ions are particularly attractive for electron-ion collision studies. Their electronic structure is relatively simple and therefore allows particularly reliable theoretical calculations, and yet, the structure is sufficiently complex to support a wide variety of processes and elec- tronic couplings. Thus, Li-like ions have received much attention with respect to DR and also with respect to electron-impact ionization [19,26]. Experimental results for DR are presently available for the following Li-like ions: B?+ [37], C3* [34,37], N** [37,38], O°+ [34,37], F&* [38], Ne’+ [39], Sit!+ [38,40,41], Cl’4+ [40,41], Art5+ [12,39], Cut [42], Au76+ [43,44], and U89+ [44,45]. rf . Az ~
At low center-of-mass energies An = 0 transitions are observed Li-like ions are particularly attractive for electron-ion collision studies. Their electronic structure is relatively simple and therefore allows particularly reliable theoretical calculations, and yet, the structure is sufficiently complex to support a wide variety of processes and elec- tronic couplings. Thus, Li-like ions have received much attention with respect to DR and also with respect to electron-impact ionization [19,26]. Experimental results for DR are presently available for the following Li-like ions: B?+ [37], C3* [34,37], N** [37,38], O°+ [34,37], F&* [38], Ne’+ [39], Sit!+ [38,40,41], Cl’4+ [40,41], Art5+ [12,39], Cut [42], Au76+ [43,44], and U89+ [44,45]. rf . Az ~
resonances with an energy spread reduced by more than an order of magnitude. Cross sections and rates for DR of Li-like ions in low charge states involving An = 0 transi- tions are strongly influenced by external electric fields in the interaction region. This is partic- ularly true for the highest Rydberg states contributing to the measured rates. Fig. 9 shows 3 calculations [38] for different external electric fields (0 V/cm, 5 V/cm, and 10 V/cm). Compar- ison with the experimental data suggests that external fields of about 5 V/cm were present in this measurement at Aarhus. Apparently, the lower Rydberg states are not influenced by such fields. Also in the preceding experiments at ORNL, the size of the electric field in the collision region was unknown. Therefore, the convoluted theoretical rates were calculated for different fields [46] and compared to the experimental data. Fields as high as 625 V/cm were invoked to find a satisfactory level of agreement between theory and experiment. In a sense, the ORNL experiment could be used to determine the external electric field in the collision region assuming that theory is capable to calculate the DR rates correctly. Thus, apart from the determination of the electric field in the collision region and the electron-beam energy-distribution function characteristic for this particular experiment, these measurements could provide little informa- tion on the DR process. They did, however, spark the enormous experimental development in the field of electron-ion collisions making use of accelerator techniques. the field of electron-ion collisions making use of accelerator techniques.
resonances with an energy spread reduced by more than an order of magnitude. Cross sections and rates for DR of Li-like ions in low charge states involving An = 0 transi- tions are strongly influenced by external electric fields in the interaction region. This is partic- ularly true for the highest Rydberg states contributing to the measured rates. Fig. 9 shows 3 calculations [38] for different external electric fields (0 V/cm, 5 V/cm, and 10 V/cm). Compar- ison with the experimental data suggests that external fields of about 5 V/cm were present in this measurement at Aarhus. Apparently, the lower Rydberg states are not influenced by such fields. Also in the preceding experiments at ORNL, the size of the electric field in the collision region was unknown. Therefore, the convoluted theoretical rates were calculated for different fields [46] and compared to the experimental data. Fields as high as 625 V/cm were invoked to find a satisfactory level of agreement between theory and experiment. In a sense, the ORNL experiment could be used to determine the external electric field in the collision region assuming that theory is capable to calculate the DR rates correctly. Thus, apart from the determination of the electric field in the collision region and the electron-beam energy-distribution function characteristic for this particular experiment, these measurements could provide little informa- tion on the DR process. They did, however, spark the enormous experimental development in the field of electron-ion collisions making use of accelerator techniques. the field of electron-ion collisions making use of accelerator techniques.
FIG. 12. Dielectronic and radiative recombination rates of Au’®+ ions. The solid line is a theory curve composed of different independent calculations which were added and convoluted with the spatial energy distribution function at the ESR cooler (see text). Adopted from [44]. new measurement has a much better energy resolution than the previous one, however, t counting rates were not sufficient yet to study also DR for An = 1 transitions of the 2s cx electron. This was possible, though, in the older Ar'®+ experiment at GSI and it was a possible in several experiments with Li-like ions at the TSR storage ring (see below). DR along the Li-like ion sequence has been studied for atomic numbers up to Z=92. T more examples for measurements involving 2s — 2p transitions are shown in the next figur While the lowest excitation energies of Ar!>* are at 31.87eV and 35.04eV, respectively, t corresponding numbers for Cu?*+ are already 55.15eV and 80.77eV [47]. Thus, also the « perimental separation of single Rydberg resonances associated with the lowest core excitatic becomes easier. Fig.11 shows DR rates for An = 0 core transitions of Cu**t ions. The rates energies beyond about 22eV were multiplied by a factor 10 so that the many resolved Rydbe states can be seen all at a time. These data were obtained at the TSR.
FIG. 12. Dielectronic and radiative recombination rates of Au’®+ ions. The solid line is a theory curve composed of different independent calculations which were added and convoluted with the spatial energy distribution function at the ESR cooler (see text). Adopted from [44]. new measurement has a much better energy resolution than the previous one, however, t counting rates were not sufficient yet to study also DR for An = 1 transitions of the 2s cx electron. This was possible, though, in the older Ar'®+ experiment at GSI and it was a possible in several experiments with Li-like ions at the TSR storage ring (see below). DR along the Li-like ion sequence has been studied for atomic numbers up to Z=92. T more examples for measurements involving 2s — 2p transitions are shown in the next figur While the lowest excitation energies of Ar!>* are at 31.87eV and 35.04eV, respectively, t corresponding numbers for Cu?*+ are already 55.15eV and 80.77eV [47]. Thus, also the « perimental separation of single Rydberg resonances associated with the lowest core excitatic becomes easier. Fig.11 shows DR rates for An = 0 core transitions of Cu**t ions. The rates energies beyond about 22eV were multiplied by a factor 10 so that the many resolved Rydbe states can be seen all at a time. These data were obtained at the TSR.
FIG. 14. Dielectronic recombination cross sections of Ar!5+ ions with An = 1 transitions from the L-shell. Expansion of Fig. 13 in the range near the series limits. Adopted from (12).
FIG. 14. Dielectronic recombination cross sections of Ar!5+ ions with An = 1 transitions from the L-shell. Expansion of Fig. 13 in the range near the series limits. Adopted from (12).
FIG. 15. Dielectronic recombination cross sections of Cu2®+ ions with An = 1 transitions from the L-shell in the range of 1s?3é3é’ resonances. In the experimental spectrum the participating terms are indicated. The solid line results from the theoretical calculation. Adopted from [42].
FIG. 15. Dielectronic recombination cross sections of Cu2®+ ions with An = 1 transitions from the L-shell in the range of 1s?3é3é’ resonances. In the experimental spectrum the participating terms are indicated. The solid line results from the theoretical calculation. Adopted from [42].
FIG. 17. Experimental and theoretical dielectronic recombination cross sections of Si ions with An = 1 and An = 2 transitions from the K-shell. Adopted from [40]. 7114
FIG. 17. Experimental and theoretical dielectronic recombination cross sections of Si ions with An = 1 and An = 2 transitions from the K-shell. Adopted from [40]. 7114
FIG. 18. Experimental (a) and theoretical (b and c) dielectronic recombination rates of metastable (c) and ground-state (b) F°+ ions with An = 0 transitions from the L-shell. Adopted from [49]. Experimental results for DR of Be-like ions are presently available for C?+ [48,49], N?* [48] O*t [48,49], F>+ [48,49]. None of these experiments was carried out at a storage ring. Thu: metastable ions in the parent beam could not be avoided. In fact, a statistical population o the 2s?15 ground state (25%) and the 2s2p?P metastable states (75%) had to be invoked i the comparison of theory and experiment [49]. The data are restricted to An = 0 transition of one of the L-shell electrons of the parent ion core. The dominant DR channel from th ground state leads to 2s2p (1P)né intermediate resonant states. For metastable 2s2p°P paren ions the dominant DR channels lead to 2p? (?P)né states and, with much smaller probabilit; to 2s2p(1P)né states. The interpretation of the experiments is complicated by the fact tha lifetimes of the intermediate resonant states may be comparable to the flight times of the ion between interaction region and charge state analyzer. An example for studies on Be-like ions is given in Fig.18. The experimental data for. F°+ An example for studies on Be-like ions is given in Fig.18. The experimental data for. F°*
FIG. 18. Experimental (a) and theoretical (b and c) dielectronic recombination rates of metastable (c) and ground-state (b) F°+ ions with An = 0 transitions from the L-shell. Adopted from [49]. Experimental results for DR of Be-like ions are presently available for C?+ [48,49], N?* [48] O*t [48,49], F>+ [48,49]. None of these experiments was carried out at a storage ring. Thu: metastable ions in the parent beam could not be avoided. In fact, a statistical population o the 2s?15 ground state (25%) and the 2s2p?P metastable states (75%) had to be invoked i the comparison of theory and experiment [49]. The data are restricted to An = 0 transition of one of the L-shell electrons of the parent ion core. The dominant DR channel from th ground state leads to 2s2p (1P)né intermediate resonant states. For metastable 2s2p°P paren ions the dominant DR channels lead to 2p? (?P)né states and, with much smaller probabilit; to 2s2p(1P)né states. The interpretation of the experiments is complicated by the fact tha lifetimes of the intermediate resonant states may be comparable to the flight times of the ion between interaction region and charge state analyzer. An example for studies on Be-like ions is given in Fig.18. The experimental data for. F°+ An example for studies on Be-like ions is given in Fig.18. The experimental data for. F°*
Discrepancies between theory and experiment have also been reported for DR of C+. Th experiment was assumed to have an electron-energy spread as low as about 0.04eV and unde this condition the measured data [9] exceed theoretical predictions [51] by more than a fact three. The experimental data are displayed in Fig.19 together with theoretical predictions fc DR of 1s?2s?2p ?P ground-state parent ions. The theoretical cross sections are displayed i 0.01eV wide bins. The resonance strength contained in each of the bins is divided by the bi width to provide an average cross section in that particular energy bin. The discrepancy « theory and experiment cannot be rationalized by electric field mixing. However, the possibl presence of metastable states in the parent ion beam has not been considered in the calcul tions. The poor quality of the available experimental data on Ct did not stimulate furthe theoretical effort.
Discrepancies between theory and experiment have also been reported for DR of C+. Th experiment was assumed to have an electron-energy spread as low as about 0.04eV and unde this condition the measured data [9] exceed theoretical predictions [51] by more than a fact three. The experimental data are displayed in Fig.19 together with theoretical predictions fc DR of 1s?2s?2p ?P ground-state parent ions. The theoretical cross sections are displayed i 0.01eV wide bins. The resonance strength contained in each of the bins is divided by the bi width to provide an average cross section in that particular energy bin. The discrepancy « theory and experiment cannot be rationalized by electric field mixing. However, the possibl presence of metastable states in the parent ion beam has not been considered in the calcul tions. The poor quality of the available experimental data on Ct did not stimulate furthe theoretical effort.
FIG. 21. Experimental and theoretical dielectronic recombination rates of Fe®+ ions with An = 0 and An = 1 transitions from the M-shell. Adopted from [55].
FIG. 21. Experimental and theoretical dielectronic recombination rates of Fe®+ ions with An = 0 and An = 1 transitions from the M-shell. Adopted from [55].
FIG. 22. Experimental (solid line) and theoretical (dotted line) DR rates of Fe!** ions with An = 1 and An = 2 transitions from the L-shell. Adopted from [55]. An = 2 transitions from the L shell
FIG. 22. Experimental (solid line) and theoretical (dotted line) DR rates of Fe!** ions with An = 1 and An = 2 transitions from the L-shell. Adopted from [55]. An = 2 transitions from the L shell
FIG. 23. Experimental recombination rates of U?8+ ions near zero center-of-mass energy [57-60]. Energies are negative for v. < v; and positive for ve > v;.
FIG. 23. Experimental recombination rates of U?8+ ions near zero center-of-mass energy [57-60]. Energies are negative for v. < v; and positive for ve > v;.
FIG. 25. Experimental and theoretical electron-detachment cross sections of D~ ions. Adopted from [61]. At ASTRID the detachment of D~ ions by electron impact was studied [61] in a center- of-mass energy range from 0 to 20eV. Cross sections for H~ ions have been available from inclined-beams experiments already for two and a half decades [62]. Motivations for the new measurements were (a) the possible access to very low collision energies in a merged-beams arrangement and (b) the possibility to study resonance features in the detachment cross section with high energy resolution. Very low energies were not accessible in the previous experiments. The detachment threshold law for H~ (or D7) is of particular interest since the 3 outgoing particles are two electrons and one neutral atom. This is very different from the situation des-
FIG. 25. Experimental and theoretical electron-detachment cross sections of D~ ions. Adopted from [61]. At ASTRID the detachment of D~ ions by electron impact was studied [61] in a center- of-mass energy range from 0 to 20eV. Cross sections for H~ ions have been available from inclined-beams experiments already for two and a half decades [62]. Motivations for the new measurements were (a) the possible access to very low collision energies in a merged-beams arrangement and (b) the possibility to study resonance features in the detachment cross section with high energy resolution. Very low energies were not accessible in the previous experiments. The detachment threshold law for H~ (or D7) is of particular interest since the 3 outgoing particles are two electrons and one neutral atom. This is very different from the situation des-
Previous crossed-beams experiments studying electron-impact ionization of Li-like ions [19] were restricted to charge states below q = 7. Within this restricted range very detailed cross section measurements are available [66,67] for all ions except Bet [64] and Ne’+ [65]. The data clearly show the presence of indirect ionization mechanisms as described by Eqs.(6)-(9). The observed resonances are associated with K — L and K — M excitations. The intermediate resonant states decay by photon or electron emission. Emission of two electrons leads to net single ionization of the parent ion. With increasing ion charge state the probability for photoemission is expected to go up and hence, the role of indirect processes in the ionization
Previous crossed-beams experiments studying electron-impact ionization of Li-like ions [19] were restricted to charge states below q = 7. Within this restricted range very detailed cross section measurements are available [66,67] for all ions except Bet [64] and Ne’+ [65]. The data clearly show the presence of indirect ionization mechanisms as described by Eqs.(6)-(9). The observed resonances are associated with K — L and K — M excitations. The intermediate resonant states decay by photon or electron emission. Emission of two electrons leads to net single ionization of the parent ion. With increasing ion charge state the probability for photoemission is expected to go up and hence, the role of indirect processes in the ionization
FIG. 27. Experimental electron-impact ionization cross sections of Fe!®+ ions. The open circles are from crossed-beams measurements [69], the dots with statistical error bars are from a storage-ring experiment [20]. The dashed line is a semi-empirical description of direct ionization based on the Lotz formula [73]. Adopted from [20].
FIG. 27. Experimental electron-impact ionization cross sections of Fe!®+ ions. The open circles are from crossed-beams measurements [69], the dots with statistical error bars are from a storage-ring experiment [20]. The dashed line is a semi-empirical description of direct ionization based on the Lotz formula [73]. Adopted from [20].
FIG. 28. Experimental and theoretical electron-impact ionization cross sections of Fe!>+ ions. Expanded from Fig. 27. The dotted line is a calculation by Badnell and Pindzola [20], the dashed line is from Chen et al. [70]. Adopted from [20].
FIG. 28. Experimental and theoretical electron-impact ionization cross sections of Fe!>+ ions. Expanded from Fig. 27. The dotted line is a calculation by Badnell and Pindzola [20], the dashed line is from Chen et al. [70]. Adopted from [20].
Table If. The number of the target electrons N and the minimal ionization energies I, eV, for neutral atoms
Table If. The number of the target electrons N and the minimal ionization energies I, eV, for neutral atoms
Fig. 1. MI cross sections of Ne, Mg, Ar, Fe, Cu and Ga atoms. Symbols correspond to the experimental data, the dashed curves in Figs. la and If represent the semiempirical formalism [6] and the solid curves are the present result, Eq. (6). (a): crosses - [28], open triangles - [29], solid triangles - [30], boxes - [31], solid circles - [32], open circles ~ [26]; (b) : solid circles - [15]; (c) : crosses - [33], open circles - [32], solid circles - [16], triangles - [34]; (a): circles - [17]; (e) : circles - [18]; (f): circles - [19].
Fig. 1. MI cross sections of Ne, Mg, Ar, Fe, Cu and Ga atoms. Symbols correspond to the experimental data, the dashed curves in Figs. la and If represent the semiempirical formalism [6] and the solid curves are the present result, Eq. (6). (a): crosses - [28], open triangles - [29], solid triangles - [30], boxes - [31], solid circles - [32], open circles ~ [26]; (b) : solid circles - [15]; (c) : crosses - [33], open circles - [32], solid circles - [16], triangles - [34]; (a): circles - [17]; (e) : circles - [18]; (f): circles - [19].
Fig. 2. MI cross sections of Kr, In, Xe and U atoms. Symbols correspond to the experimental.data, the solid curves are the present result, Eq. (6). (a): crosses - [33], open circles - [26], triangles - [34]; theory: dashed curves - semiempirical formula [7] for n = 6, 7 and 8; (b) : open circles - [19]; theory - dashed curve - semiempirical formalism [6]; (c) : crosses - [33], solid circles - [32], open circles - [26], triangles - [34]; (d) : crosses - [33], triangles - [34]; (e) : crosses and solid circle - [33]; (f): circles - [35].
Fig. 2. MI cross sections of Kr, In, Xe and U atoms. Symbols correspond to the experimental.data, the solid curves are the present result, Eq. (6). (a): crosses - [33], open circles - [26], triangles - [34]; theory: dashed curves - semiempirical formula [7] for n = 6, 7 and 8; (b) : open circles - [19]; theory - dashed curve - semiempirical formalism [6]; (c) : crosses - [33], solid circles - [32], open circles - [26], triangles - [34]; (d) : crosses - [33], triangles - [34]; (e) : crosses and solid circle - [33]; (f): circles - [35].
Table IV. Maximum values of triple- and quadruple-ionization cross sections o,, cm’, and the corresponding incident electron energies E,"", eV, Eq. (8) in comparison with experimental data for neutral targets
Table IV. Maximum values of triple- and quadruple-ionization cross sections o,, cm’, and the corresponding incident electron energies E,"", eV, Eq. (8) in comparison with experimental data for neutral targets
Fig. 4. MI cross sections of positive ions. Circles correspond to the experimental data, the solid curves are the present results, Eq. (6). (a): [22]; (b): [22]; (¢): [22]; (a): [25]; (e): [38]; (): [4].
Fig. 4. MI cross sections of positive ions. Circles correspond to the experimental data, the solid curves are the present results, Eq. (6). (a): [22]; (b): [22]; (¢): [22]; (a): [25]; (e): [38]; (): [4].
revious work on Si?t+- H collisions [4] is extended, for the Si*+ projectile, in a number of ways. The energy ange is enlarged to include 0.2 — 100 keV/u. The n=6 capture states are included at the projectile in order tc nprove the accuracy of the calculated transfer cross section to n=5 states. At the hydrogen center, we include 1e n=2 states of hydrogen. Except for these changes, we keep closely to the prescriptions of the calculation: 1 ref. [4]. The calculated partial transfer cross sections are given in Table I. The results of the calculations are ummarized in figure 1. TABLE I: PARTIAL TRANSFER CROSS SECTIONS TO Si*+ né STATES IN Si**— H COLLISIONS.
revious work on Si?t+- H collisions [4] is extended, for the Si*+ projectile, in a number of ways. The energy ange is enlarged to include 0.2 — 100 keV/u. The n=6 capture states are included at the projectile in order tc nprove the accuracy of the calculated transfer cross section to n=5 states. At the hydrogen center, we include 1e n=2 states of hydrogen. Except for these changes, we keep closely to the prescriptions of the calculation: 1 ref. [4]. The calculated partial transfer cross sections are given in Table I. The results of the calculations are ummarized in figure 1. TABLE I: PARTIAL TRANSFER CROSS SECTIONS TO Si*+ né STATES IN Si**— H COLLISIONS.
by the population of the n=3 states. The population of the n=4 states is weaker by about a factor of 3. The calculated population of the n=5 states is again weaker by about the same factor. The undulations of these latter cross sections, when displayed over energy, should be seen as an artefact of the calculations. One would expect that these undulations would vanish in a calculation with a much larger basis set. Out of the n=3 states in the projectile, the 3d state is the one which, in slow collisions, is populated much stronger than the others, cf. table I. A detailed, quantitative analysis of this effect is outside the scope of this investigation. We merely note that the initial 1s state of H crosses those states, which correlate to the Si?+ 3d states, at the favourable interatomic separation of about 7 a.u. The avoided crossing with the other n=3 states occurs at smaller separations (4 a.u. for the 3p state). Furthermore, the coupling with the 3d states in slow collisions is favoured since it conserves the nodal structure of the electronic wavefunction. Finally, rotational coupling plays a role here since the initial state correlates to the 3po state of the united atom. This rotational coupling leads to substantial population of m=1 and m=2 states within the 3d manifold.
by the population of the n=3 states. The population of the n=4 states is weaker by about a factor of 3. The calculated population of the n=5 states is again weaker by about the same factor. The undulations of these latter cross sections, when displayed over energy, should be seen as an artefact of the calculations. One would expect that these undulations would vanish in a calculation with a much larger basis set. Out of the n=3 states in the projectile, the 3d state is the one which, in slow collisions, is populated much stronger than the others, cf. table I. A detailed, quantitative analysis of this effect is outside the scope of this investigation. We merely note that the initial 1s state of H crosses those states, which correlate to the Si?+ 3d states, at the favourable interatomic separation of about 7 a.u. The avoided crossing with the other n=3 states occurs at smaller separations (4 a.u. for the 3p state). Furthermore, the coupling with the 3d states in slow collisions is favoured since it conserves the nodal structure of the electronic wavefunction. Finally, rotational coupling plays a role here since the initial state correlates to the 3po state of the united atom. This rotational coupling leads to substantial population of m=1 and m=2 states within the 3d manifold.
TABLE II: PARTIAL ELECTRON TRANSFER CROSS SECTIONS (IN CM’) TO PROJECTILE né STATES IN Ni!°+ H COLLISIONS. ENERGIES ARE IN keV/u. are rather loosely bound in comparison to the electron states in the projectile core. Hence the core is expected to play only a minor role here. As in the previous work [5] we take only 1s ground state of hydrogen at the target center. At the projectile center, we take representations of the 3d and the n=4...8 states. These 77 states are constructed from 145 hydrogenic states. The occupied states up to the 3p states are excluded, the pseudostates which result from the diagonalization of the Ni potential have been kept in the calculation. The calculation at the lowest energy has been done without the n=8 states at the projectile. At the target, only the is taken.
TABLE II: PARTIAL ELECTRON TRANSFER CROSS SECTIONS (IN CM’) TO PROJECTILE né STATES IN Ni!°+ H COLLISIONS. ENERGIES ARE IN keV/u. are rather loosely bound in comparison to the electron states in the projectile core. Hence the core is expected to play only a minor role here. As in the previous work [5] we take only 1s ground state of hydrogen at the target center. At the projectile center, we take representations of the 3d and the n=4...8 states. These 77 states are constructed from 145 hydrogenic states. The occupied states up to the 3p states are excluded, the pseudostates which result from the diagonalization of the Ni potential have been kept in the calculation. The calculation at the lowest energy has been done without the n=8 states at the projectile. At the target, only the is taken.
FIG. 4: Partial transfer and excitation cross section in Si*+— He collisions. For the investigation of Sit+— He collisions, we adopt the same states at the silicon center that are used in th description of the Si*+— H system, see the earlier discussion. The two-electron charge transfer configuration are then constructed by taking anti-symmetrized products between these states and the ground state of the He” ion. Besides the helium ground state, also representations of the helium !2S and !2P excited states are include in the basis. The partial transfer cross sections from the calculations are given in table III. We include als the calculated excitation cross section to the n=2 states in helium since excitation cross sections of helium it collisions with multiply charged ions are still very scarce. Figure 4 shows the partial transfer and excitatio cross sections, summed over £ quantum numbers. In comparison to the Sit*— H system, cf. figure 1, the population of states in the n=3 shell is more promi- nent in slow Si*+— He collisions. The tighter binding of the electron in helium favours the capture into tighter bound final states. This is also reflected in the £ dependence of capture to the n=3 shell. In the present system, the more tightly bound 3s and 3p states are favoured in slow collisions, in contrast to the Sitt~ H system where the 3d state is clearly favoured.
FIG. 4: Partial transfer and excitation cross section in Si*+— He collisions. For the investigation of Sit+— He collisions, we adopt the same states at the silicon center that are used in th description of the Si*+— H system, see the earlier discussion. The two-electron charge transfer configuration are then constructed by taking anti-symmetrized products between these states and the ground state of the He” ion. Besides the helium ground state, also representations of the helium !2S and !2P excited states are include in the basis. The partial transfer cross sections from the calculations are given in table III. We include als the calculated excitation cross section to the n=2 states in helium since excitation cross sections of helium it collisions with multiply charged ions are still very scarce. Figure 4 shows the partial transfer and excitatio cross sections, summed over £ quantum numbers. In comparison to the Sit*— H system, cf. figure 1, the population of states in the n=3 shell is more promi- nent in slow Si*+— He collisions. The tighter binding of the electron in helium favours the capture into tighter bound final states. This is also reflected in the £ dependence of capture to the n=3 shell. In the present system, the more tightly bound 3s and 3p states are favoured in slow collisions, in contrast to the Sitt~ H system where the 3d state is clearly favoured.
TABLE III: PARTIAL TRANSFER AND EXCITATION CROSS SECTIONS IN Si4+— He COLLISIONS. THE TRANS- FER CROSS SECTIONS ARE CALCULATED FOR né STATES WITH n=3-5, THE EXCITATION CROSS SECTIONS FOR n=2.
TABLE III: PARTIAL TRANSFER AND EXCITATION CROSS SECTIONS IN Si4+— He COLLISIONS. THE TRANS- FER CROSS SECTIONS ARE CALCULATED FOR né STATES WITH n=3-5, THE EXCITATION CROSS SECTIONS FOR n=2.
TABLE V: PARTIAL TRANSFER CROSS SECTIONS TO Ti3+ STATES IN Ti4+- H, COLLISIONS. for re" — 2 [14]. AS expected, these total transfer cross secuions agree well among each other. In figure 8 we show the ratio of calculated total tramsfer cross sections for molecular and atomic hydroget targets. This ratio is seen to be close to one at low energies but it increases at the higher energies. This feature o the cross section ratio has been observed before for other systems, see ref. [11] and references given therein. ] reflects the large transfer probability between a multiply charged ion and atomic hydrogen. The second electro: in molecular hydrogen and the slight ground-state energy shift between atomic and molecular hydrogen has onl avery minor effect in slow collisions.
TABLE V: PARTIAL TRANSFER CROSS SECTIONS TO Ti3+ STATES IN Ti4+- H, COLLISIONS. for re" — 2 [14]. AS expected, these total transfer cross secuions agree well among each other. In figure 8 we show the ratio of calculated total tramsfer cross sections for molecular and atomic hydroget targets. This ratio is seen to be close to one at low energies but it increases at the higher energies. This feature o the cross section ratio has been observed before for other systems, see ref. [11] and references given therein. ] reflects the large transfer probability between a multiply charged ion and atomic hydrogen. The second electro: in molecular hydrogen and the slight ground-state energy shift between atomic and molecular hydrogen has onl avery minor effect in slow collisions.
TABLE VI: PARTIAL TRANSFER CROSS SECTIONS IN Fe’+— H, COLLISIONS.
TABLE VI: PARTIAL TRANSFER CROSS SECTIONS IN Fe’+— H, COLLISIONS.
shell. The n distribution of electron transfer should be reliable except for the highest n shell considered in a given study. The & distribution within a given n shell will become less reliable for the higher n shells. Boh VEE CELE Ye REE UH MEE OULAUIUREANIAR VV AUER EAR CE EA VAL FE ORAWER WU EAR IARI BO AVA AUIE ENIE CEE LEAR EEWA FO OLE The studies performed in this work are examples of what can be done for other systems of interest if the need arises. There are however clear limitations on which systems can be pursued in a feasible and meaningful manner. The existence of an easy level structure in the charge transfer system is prerequisite for the theoretical description and the use in plasma diagnostics alike. Hence charge transfer systems with only one electron outside an inert core have been considered here. A second electron has been alowed for in the target, in the case of collisions with helium or molecular hydrogen targets. One could also treat systems with a second electron in the projectile ion in a similar way. All slowly colliding systems must be treated in a case-by-case fashion. On the other hand, we have indicated that the n distribution may be interpolated between equi-charged systems if the charge state is sufficiently high.
shell. The n distribution of electron transfer should be reliable except for the highest n shell considered in a given study. The & distribution within a given n shell will become less reliable for the higher n shells. Boh VEE CELE Ye REE UH MEE OULAUIUREANIAR VV AUER EAR CE EA VAL FE ORAWER WU EAR IARI BO AVA AUIE ENIE CEE LEAR EEWA FO OLE The studies performed in this work are examples of what can be done for other systems of interest if the need arises. There are however clear limitations on which systems can be pursued in a feasible and meaningful manner. The existence of an easy level structure in the charge transfer system is prerequisite for the theoretical description and the use in plasma diagnostics alike. Hence charge transfer systems with only one electron outside an inert core have been considered here. A second electron has been alowed for in the target, in the case of collisions with helium or molecular hydrogen targets. One could also treat systems with a second electron in the projectile ion in a similar way. All slowly colliding systems must be treated in a case-by-case fashion. On the other hand, we have indicated that the n distribution may be interpolated between equi-charged systems if the charge state is sufficiently high.
FIG. 1. Scaled cross sections for 11S —>n'S transitions in He induced by proton impact. The curve is analytic fit, Eqs. (11)-(13), of the experimental data (symbols, Ref. [6]).
FIG. 1. Scaled cross sections for 11S —>n'S transitions in He induced by proton impact. The curve is analytic fit, Eqs. (11)-(13), of the experimental data (symbols, Ref. [6]).
FIG. 3. Scaled cross sections for proton-impact induced I'S —> n'D transitions in He. The experimental data (Ref. [6]) are shown by symbols. The curve is analytic fit to the data by Eqs. (21)-(23).
FIG. 3. Scaled cross sections for proton-impact induced I'S —> n'D transitions in He. The experimental data (Ref. [6]) are shown by symbols. The curve is analytic fit to the data by Eqs. (21)-(23).
The Al’*/H system has been studied previously by Phaneuf et al. [8], Kirby and Heil [9] and Gargaud et al. [6]. In view of our findings on the importance of rotational coupling, it is scarcely surprising that the calculations of Kirby and Heil, who neglect rotational coupling, differ significantly from those of Gargaud et al. who take both radial and rotational coupling into account.
The Al’*/H system has been studied previously by Phaneuf et al. [8], Kirby and Heil [9] and Gargaud et al. [6]. In view of our findings on the importance of rotational coupling, it is scarcely surprising that the calculations of Kirby and Heil, who neglect rotational coupling, differ significantly from those of Gargaud et al. who take both radial and rotational coupling into account.
Fig. 2: Al5+/H charge transfer cross-sections (10°!®cm2) as a function of centre of mass energy. The full and long broken curves refer respectively to our calculations with and without rotational coupling [6], the chain curve to the calculations of Kirby and Heil [9], the short broken curve to the calculation of Phaneuf et al. [8]. The experimental result of Phaneuf et al. is designated by the full triangle. Fig 1: Adiabatic potential energy curves of Al H+?
Fig. 2: Al5+/H charge transfer cross-sections (10°!®cm2) as a function of centre of mass energy. The full and long broken curves refer respectively to our calculations with and without rotational coupling [6], the chain curve to the calculations of Kirby and Heil [9], the short broken curve to the calculation of Phaneuf et al. [8]. The experimental result of Phaneuf et al. is designated by the full triangle. Fig 1: Adiabatic potential energy curves of Al H+?
The Ar®+/H system might appear a simpler one for model potential calculations but there is still some difficulty in separating the contributions of the individual channels to the energy gain spectra [10, 11]. Hansen et al.[12, 13] have pointed out how the energy profile pertaining to a specific state depends on both kinematic and differential cross sections. We present in table 2 and figure 5 our theoretical results and compare them to the absolute data of Crandall er al. {14]. Unfornutanely there is no detailed experimental data on how the Ar®+/H electron capture cross sections vary with collision energy. We believe nevertheless that our theoretical calculations are of good quality.
The Ar®+/H system might appear a simpler one for model potential calculations but there is still some difficulty in separating the contributions of the individual channels to the energy gain spectra [10, 11]. Hansen et al.[12, 13] have pointed out how the energy profile pertaining to a specific state depends on both kinematic and differential cross sections. We present in table 2 and figure 5 our theoretical results and compare them to the absolute data of Crandall er al. {14]. Unfornutanely there is no detailed experimental data on how the Ar®+/H electron capture cross sections vary with collision energy. We believe nevertheless that our theoretical calculations are of good quality.
Table 2 : Art®/H charge exchange cross-sections (10 “!6 cm?) as a function of energy. Fig 5: Ar*®/H total and partial charge exchange cross-sections (10 “!6 cm?) as a function of ion energy (eV/amu).
Table 2 : Art®/H charge exchange cross-sections (10 “!6 cm?) as a function of energy. Fig 5: Ar*®/H total and partial charge exchange cross-sections (10 “!6 cm?) as a function of ion energy (eV/amu).
Figure 6 : Comparison of our theoretical Art®/He charge transfer cross-sections (10 ~!© cm?) with the experimental data of Andersson et al. [16] and the theoretical data of Hansen et al. [13] as a function of ion energy (eV/amu). Andersson ef al. : total (m) ; 4p (A) ; 4s (@); 3d (#). Hansen ef al. : (- - - ). Our results (------- ). Table 3 : Ar*6/He charge exchange cross-sections (10 -!6 cm?) as a function of energy.
Figure 6 : Comparison of our theoretical Art®/He charge transfer cross-sections (10 ~!© cm?) with the experimental data of Andersson et al. [16] and the theoretical data of Hansen et al. [13] as a function of ion energy (eV/amu). Andersson ef al. : total (m) ; 4p (A) ; 4s (@); 3d (#). Hansen ef al. : (- - - ). Our results (------- ). Table 3 : Ar*6/He charge exchange cross-sections (10 -!6 cm?) as a function of energy.
Table 5 : Sit/He charge exchange cross-sections (10°16 cm?) as a function of energy (eV).
Table 5 : Sit/He charge exchange cross-sections (10°16 cm?) as a function of energy (eV).
Fig 8 : SiH3* adiabatic potential energy curves (a.u.) as a function of internuclear distance (ag). The Si3+/H system is the odd man out of this compilation. Even though model potential techniques could probably be used to provide a crude estimate of the cross section, this system is typically a three-electron problem for which the techniques used to reduce two-electron systems such as Ar®*/He to an effective one-electron problem are not appropriate. An ab initio configuration-interaction with an effective potential to reproduce the (2s22p®) core has been used in this case [17]. The electron capture process is dominated by an avoided crossing of the Landau-Zener type. We present in figure 8 the adiabatic potential energies and in table 5 cross-sections for capture to the (3s2)!§, (3s3p)3P, (3s3p)!P of Si2*.
Fig 8 : SiH3* adiabatic potential energy curves (a.u.) as a function of internuclear distance (ag). The Si3+/H system is the odd man out of this compilation. Even though model potential techniques could probably be used to provide a crude estimate of the cross section, this system is typically a three-electron problem for which the techniques used to reduce two-electron systems such as Ar®*/He to an effective one-electron problem are not appropriate. An ab initio configuration-interaction with an effective potential to reproduce the (2s22p®) core has been used in this case [17]. The electron capture process is dominated by an avoided crossing of the Landau-Zener type. We present in figure 8 the adiabatic potential energies and in table 5 cross-sections for capture to the (3s2)!§, (3s3p)3P, (3s3p)!P of Si2*.
Fig 7 : SiH?* adiabatic potential energy curves (a.u.) of as a function of internuclear distance (ag).
Fig 7 : SiH?* adiabatic potential energy curves (a.u.) of as a function of internuclear distance (ag).
Table 6 : Sit4/H charge exchange cross-sections (10716 cm?) as a function of energy.
Table 6 : Sit4/H charge exchange cross-sections (10716 cm?) as a function of energy.
Being ideally suited for the treatment by model potential techniques [16], this system is an interesting bench- mark system. Two electron capture channels are probable (figure 9) Capture to the 3d state which involves a complex crossing network of 2, I] and A states, dominates for collison energies above 20 eV /amu while capture to the 4s channel dominates at eV energies (table 6). At very low energies, the system exhibits a Langevin effect. Recent merged beam measurements {19] have confirmed the existence of this phenomenon (figure 10).
Being ideally suited for the treatment by model potential techniques [16], this system is an interesting bench- mark system. Two electron capture channels are probable (figure 9) Capture to the 3d state which involves a complex crossing network of 2, I] and A states, dominates for collison energies above 20 eV /amu while capture to the 4s channel dominates at eV energies (table 6). At very low energies, the system exhibits a Langevin effect. Recent merged beam measurements {19] have confirmed the existence of this phenomenon (figure 10).
The advanced adiabatic method, also known as the hidden crossing method, utilizes the strong localization of the nonadiabatic transitions at low collision energies between two quasi- molecular states to those internuclear distances where respective quasimolecular electronic data. Careful comparison with other more robust methods for a set of fundamental channels allows one to assess the range of reliability of the CTMC results, and to extend with some reasonable level of confidence to other similar systems. Previous works have, for example, tabulated state selective charge transfer cross sections for C®+ and O%+ with atomic hydrogen [4] and He?+ with H in both the ground and n=2 states of hydrogen [5], and total electron loss cross sections for Fe ions colliding with H, H2, and He [5]. The basic CTMC method has been described by Abrines and Percival [6] and Olson and
The advanced adiabatic method, also known as the hidden crossing method, utilizes the strong localization of the nonadiabatic transitions at low collision energies between two quasi- molecular states to those internuclear distances where respective quasimolecular electronic data. Careful comparison with other more robust methods for a set of fundamental channels allows one to assess the range of reliability of the CTMC results, and to extend with some reasonable level of confidence to other similar systems. Previous works have, for example, tabulated state selective charge transfer cross sections for C®+ and O%+ with atomic hydrogen [4] and He?+ with H in both the ground and n=2 states of hydrogen [5], and total electron loss cross sections for Fe ions colliding with H, H2, and He [5]. The basic CTMC method has been described by Abrines and Percival [6] and Olson and
Figure 2: Quasimolecular, adiabatic electronic energies for the o states for the Ne'°t + H collision system expressed in terms of the effective principal quantum number in the united atom limit, Negs = V1/./—2Ewes. Only those ionic curves of Ne®*(n) + Ht that are coupled radially with the initial state, Ne'°* + H(1s),via the Q type transitions (filled circles), are shown. (Hollow circles, localization of the S superseries; “crossings” of the curves are the isolated Landau-Zener avoided crossings [15], which do not belong to the Q supersertes). eigenenergies cross [13]. Since, according to Wigner-Neumann noncrossing rule, the stat of the same symmetry (that radially couple) cannot cross for real R, the localization of t strong transitions is shifted to the plane of complex internuclear distances R, where the a abatic electronic Hamiltonian is no longer Hermitian. When R is extended formally to t plane of complex R, the eigenvalues and eigenfunctions of the instantaneous electronic ad batic Hamiltonian H(R) of a two-center—one-electron system are analytic functions. This i: consequence of the analyticity of H(R). The only singularities have the nature of branch poit that connect pairwise different branches of the single, multivalued eigenenergy function E(1 If R is real, these branches coincide with the electronic eigenenergies.
Figure 2: Quasimolecular, adiabatic electronic energies for the o states for the Ne'°t + H collision system expressed in terms of the effective principal quantum number in the united atom limit, Negs = V1/./—2Ewes. Only those ionic curves of Ne®*(n) + Ht that are coupled radially with the initial state, Ne'°* + H(1s),via the Q type transitions (filled circles), are shown. (Hollow circles, localization of the S superseries; “crossings” of the curves are the isolated Landau-Zener avoided crossings [15], which do not belong to the Q supersertes). eigenenergies cross [13]. Since, according to Wigner-Neumann noncrossing rule, the stat of the same symmetry (that radially couple) cannot cross for real R, the localization of t strong transitions is shifted to the plane of complex internuclear distances R, where the a abatic electronic Hamiltonian is no longer Hermitian. When R is extended formally to t plane of complex R, the eigenvalues and eigenfunctions of the instantaneous electronic ad batic Hamiltonian H(R) of a two-center—one-electron system are analytic functions. This i: consequence of the analyticity of H(R). The only singularities have the nature of branch poit that connect pairwise different branches of the single, multivalued eigenenergy function E(1 If R is real, these branches coincide with the electronic eigenenergies.
note that the small fluctuations in the otherwise smooth dependence on n of these cross sections arises from the statistical uncertainty in the Monte Carlo sampling in the CTMC procedure, rather than from some physically significant process. In particular, for the largest cross sections, this uncertainty is very small (on the order of less than five percent) but for the smallest cross sections, it can be much larger. Fitting a smooth curve through the points displayed readily demonstrates that above n=4, the well known scaling of the cross section as 1/n* is obtained. Thus, these results give a point of reference for the low n-levels while results for the higher n-levels may be obtained from this scaling. Also readily seen from the figures is the fact that above about 20 keV/u, the distribution of n-levels populated, and the magnitude of the n-level cross sections does depend much on the ionic charge state of the neon ion, or the impact energy in the range 50 to 1000
note that the small fluctuations in the otherwise smooth dependence on n of these cross sections arises from the statistical uncertainty in the Monte Carlo sampling in the CTMC procedure, rather than from some physically significant process. In particular, for the largest cross sections, this uncertainty is very small (on the order of less than five percent) but for the smallest cross sections, it can be much larger. Fitting a smooth curve through the points displayed readily demonstrates that above n=4, the well known scaling of the cross section as 1/n* is obtained. Thus, these results give a point of reference for the low n-levels while results for the higher n-levels may be obtained from this scaling. Also readily seen from the figures is the fact that above about 20 keV/u, the distribution of n-levels populated, and the magnitude of the n-level cross sections does depend much on the ionic charge state of the neon ion, or the impact energy in the range 50 to 1000
Figure 9: A detailed comparison of the HC and CTMC results for charge transfer in collisions of Ne** (q=7-10) with hydrogen. Here the solid curves indicate the HC results and the dotted curves indicated the CTMC results. Neon ion charge states are indicated by the labels.
Figure 9: A detailed comparison of the HC and CTMC results for charge transfer in collisions of Ne** (q=7-10) with hydrogen. Here the solid curves indicate the HC results and the dotted curves indicated the CTMC results. Neon ion charge states are indicated by the labels.
Figure I. Cross sections for one-electron capture in AP*-H(1s) collisions. &, experimental [3]; — theory, [3]; ---, theory [4]. (from [3]) coupling calculations [3], [4]. Cross sections for one-electron capture in Al’*-H) collisions, whic have also been measured over the same energy range [3] are also dominated by the Al’('S) produc channel. In this case, cross sections are only weakly dependent on energy rising from 4.9 x 10°" on at 5.0 keV to 5.2 x 10°° cm’ at 30 keV. TES meaairementc have alen heen carnmed nit [5S] far the nnewelaectran crantiure nrnceccec involving collisions of metallic ions X* with H, H, and He have been summarised in recent review [1], [2]. Data for many of the fusion relevant species are still not extensive. At velocities V<lai (corresponding to 25keV amu’) electron capture may take place very effectively through avoidec crossings involving the potential energy curves which describe the initial and final molecular systems These avoided crossings occur at internuclear separations R, ~ (q-1)/AE. The energy defect AE characterises each collision product channel which results in products X’(n, 1) and Y(n,’ 1’) i specific excited states. In some cases, electron capture may be dominated by a limited number o excited product channels. Identification of these product channels may be carried out using th technique of translational energy spectroscopy (TES) which also allows product channels involvin; any metastable primary ions present to be detected and identified. A few measurements of this typ: have been carried out for metallic species in this laboratory. In one-electron capture by Al’’ ions in collisions with H atoms in the range 5-30 keV, our TE!
Figure I. Cross sections for one-electron capture in AP*-H(1s) collisions. &, experimental [3]; — theory, [3]; ---, theory [4]. (from [3]) coupling calculations [3], [4]. Cross sections for one-electron capture in Al’*-H) collisions, whic have also been measured over the same energy range [3] are also dominated by the Al’('S) produc channel. In this case, cross sections are only weakly dependent on energy rising from 4.9 x 10°" on at 5.0 keV to 5.2 x 10°° cm’ at 30 keV. TES meaairementc have alen heen carnmed nit [5S] far the nnewelaectran crantiure nrnceccec involving collisions of metallic ions X* with H, H, and He have been summarised in recent review [1], [2]. Data for many of the fusion relevant species are still not extensive. At velocities V<lai (corresponding to 25keV amu’) electron capture may take place very effectively through avoidec crossings involving the potential energy curves which describe the initial and final molecular systems These avoided crossings occur at internuclear separations R, ~ (q-1)/AE. The energy defect AE characterises each collision product channel which results in products X’(n, 1) and Y(n,’ 1’) i specific excited states. In some cases, electron capture may be dominated by a limited number o excited product channels. Identification of these product channels may be carried out using th technique of translational energy spectroscopy (TES) which also allows product channels involvin; any metastable primary ions present to be detected and identified. A few measurements of this typ: have been carried out for metallic species in this laboratory. In one-electron capture by Al’’ ions in collisions with H atoms in the range 5-30 keV, our TE!
Energy change spectra obtained from the TES measurements [5] of one-electron capture by Fe™ ions in H and He at 12 keV are shown in Fig 2. For He, the main product channels (corresponding to peaks A, B, C and D) are correlated with metastable rather than ground state primary ions. Peaks A, B and C appear to arise as a result of capture of a 3d electron by metastable Fe*’ (3p°3d°)‘G, “P and (3d°4s)°D ions. Peak D can be correlated with 3d and 4s electron capture by metastable Fe’ (3d"4s)°D primary ions. In the case of Fe’’-H collisions, although the situation is less clear, it again seems likely that the collision channels comprising the incompletely resolved peaks A, B and C in Fig 2 are attributable to “G, “P and °D metastable rather than ground state primary ions. A large number of product channels leading to Fe’’(3d°) 4s, 4p formation appear to be involved. In the case of one-electron capture by Fe’ ions in H and He, the energy change spectra obtained For He, the peaks A, B, C and D which comprise more than 96% of the Fe*” products can be correlated with collision channels involving Fe“"(3d*)°D ground state primary ions which, as showr in Table 1, have a high statistical weight. Peak A can be correlated with the 4s capture channel Fe*"(3d*)4s°D + He"(1s) while Peak B can be identified with the three 3d capture channels Fe’'(3d°)'G, P, D + He’(1s). Peaks C and D can also be correlated with the 3d capture channel: Fe*(3d°)‘F + He'(1s) and Fe*’(3d°)°S + He'(1s) respectively. The minor peak E which comprise: no more than 4% of the total Fe” yield, can be correlated with metastable primary ion collisior channels. Pr: 2 aa eee ig ar: ie as s oe a: ‘s' —" veges
Energy change spectra obtained from the TES measurements [5] of one-electron capture by Fe™ ions in H and He at 12 keV are shown in Fig 2. For He, the main product channels (corresponding to peaks A, B, C and D) are correlated with metastable rather than ground state primary ions. Peaks A, B and C appear to arise as a result of capture of a 3d electron by metastable Fe*’ (3p°3d°)‘G, “P and (3d°4s)°D ions. Peak D can be correlated with 3d and 4s electron capture by metastable Fe’ (3d"4s)°D primary ions. In the case of Fe’’-H collisions, although the situation is less clear, it again seems likely that the collision channels comprising the incompletely resolved peaks A, B and C in Fig 2 are attributable to “G, “P and °D metastable rather than ground state primary ions. A large number of product channels leading to Fe’’(3d°) 4s, 4p formation appear to be involved. In the case of one-electron capture by Fe’ ions in H and He, the energy change spectra obtained For He, the peaks A, B, C and D which comprise more than 96% of the Fe*” products can be correlated with collision channels involving Fe“"(3d*)°D ground state primary ions which, as showr in Table 1, have a high statistical weight. Peak A can be correlated with the 4s capture channel Fe*"(3d*)4s°D + He"(1s) while Peak B can be identified with the three 3d capture channels Fe’'(3d°)'G, P, D + He’(1s). Peaks C and D can also be correlated with the 3d capture channel: Fe*(3d°)‘F + He'(1s) and Fe*’(3d°)°S + He'(1s) respectively. The minor peak E which comprise: no more than 4% of the total Fe” yield, can be correlated with metastable primary ion collisior channels. Pr: 2 aa eee ig ar: ie as s oe a: ‘s' —" veges
Figure 4. Cross sections for (a) one-electron capture }90q by H” (b) one-electron capture 2001q by He** and (c) two-electron capture 2900q by He** in collisions with Mg. Full symbols [7]; Open symbols [14]. (from [7]).
Figure 4. Cross sections for (a) one-electron capture }90q by H” (b) one-electron capture 2001q by He** and (c) two-electron capture 2900q by He** in collisions with Mg. Full symbols [7]; Open symbols [14]. (from [7]).
Figure 5. Cross sections for single and double ionization of Mg. ®, [7]; O,[14]; o1 , 02, cross sections for single and double ionization by electrons [11]; P, Born approximation calculations for (3s + 2p) removal by H” ions [16]. (from [7]). lower energies, 3s capture is dominant [15]. In the case of one-electron capture by He’” ions, a high energy ‘bulge’ in the total cross section, which is reflected by the energy dependence of the individual contributions 29612 and 290)3, also seems likely to arise through electron capture from inner subshells. In the case of two-electron capture by He” ions, the dominant contribution at high energies can be seen to arise mainly from 2963 and ‘29Goa.
Figure 5. Cross sections for single and double ionization of Mg. ®, [7]; O,[14]; o1 , 02, cross sections for single and double ionization by electrons [11]; P, Born approximation calculations for (3s + 2p) removal by H” ions [16]. (from [7]). lower energies, 3s capture is dominant [15]. In the case of one-electron capture by He’” ions, a high energy ‘bulge’ in the total cross section, which is reflected by the energy dependence of the individual contributions 29612 and 290)3, also seems likely to arise through electron capture from inner subshells. In the case of two-electron capture by He” ions, the dominant contribution at high energies can be seen to arise mainly from 2963 and ‘29Goa.
The cross sections 2961, for one-electron capture by He” ions in Figure 6 show that while the simple charge transfer contribution 261: is dominant at low energies, the transfer ionization contributions 20614 for q = 2 become very important as the impact energy increases. There is also evidence of broad maxima in the cross sections for q => 4. A qualitative description of the measured values of 29014 in both Fe and Cu has been given by Patton et al [8] and Shah et al [9] in terms of a model in which electron capture of either 4s, 3d or 3p target electrons takes place together with the removal of additional electrons through binary collisions. The pure ionization cross sections 29621 for He” impact in Figure 6 can be seen to exhibit a behaviour very similar to the corresponding values for H” impact. UPOSs SCCTIONS 29000q 101 CWOrCICOCLIOM CAPlUle TI CUMISIUIS UL TIR LUIS WILT De ARI NYE AUS LO, ] are shown in Figure 7. Over the energy range shown, cross sections 29692 for simple charg ansfer are exceeded by the transfer ionization contribution 29003 in Cu and by both 29003 and 2900. 1 . In the latter case, even 2960s exceeds 2962 at energies above about 50 keV amu’. Shah et al [17] [8] have also considered the measured q state distributions in Fe and Cu resultin 9m one-electron capture by H’ ions in terms of a model based on an independent electro scription of multiple ionization [18]. The outer electron subshell structure in iro s)’(3p)°(3d)°(4s)’ is rather different from the (3s)’(3p)°(3d)'"4s structure in copper. However, i th cases, electron removal in the energy range considered is believed to involve primarily the 4 id 3d subshells. The probability of one-electron capture and simultaneous ionization is the pressed as a product of an electron capture probability P, and an ionization probability P, for th moval of n electrons from the target where n > 0 and, in this case, n = (q+1). Then the electro ipture cross section
The cross sections 2961, for one-electron capture by He” ions in Figure 6 show that while the simple charge transfer contribution 261: is dominant at low energies, the transfer ionization contributions 20614 for q = 2 become very important as the impact energy increases. There is also evidence of broad maxima in the cross sections for q => 4. A qualitative description of the measured values of 29014 in both Fe and Cu has been given by Patton et al [8] and Shah et al [9] in terms of a model in which electron capture of either 4s, 3d or 3p target electrons takes place together with the removal of additional electrons through binary collisions. The pure ionization cross sections 29621 for He” impact in Figure 6 can be seen to exhibit a behaviour very similar to the corresponding values for H” impact. UPOSs SCCTIONS 29000q 101 CWOrCICOCLIOM CAPlUle TI CUMISIUIS UL TIR LUIS WILT De ARI NYE AUS LO, ] are shown in Figure 7. Over the energy range shown, cross sections 29692 for simple charg ansfer are exceeded by the transfer ionization contribution 29003 in Cu and by both 29003 and 2900. 1 . In the latter case, even 2960s exceeds 2962 at energies above about 50 keV amu’. Shah et al [17] [8] have also considered the measured q state distributions in Fe and Cu resultin 9m one-electron capture by H’ ions in terms of a model based on an independent electro scription of multiple ionization [18]. The outer electron subshell structure in iro s)’(3p)°(3d)°(4s)’ is rather different from the (3s)’(3p)°(3d)'"4s structure in copper. However, i th cases, electron removal in the energy range considered is believed to involve primarily the 4 id 3d subshells. The probability of one-electron capture and simultaneous ionization is the pressed as a product of an electron capture probability P, and an ionization probability P, for th moval of n electrons from the target where n > 0 and, in this case, n = (q+1). Then the electro ipture cross section
Fig.2. Cross sections for OVIII light emission ; + : resulting from electron capture O”™ +Li collisions. Circles, squares, diamonds and triangles denote 7-6, 8-7, 9-8 , 10-9 and 11-10 transitions respectively. Open symbols: theory [4], connected by lines to guide the eye; closed symbols: experiment (9-8) by Wolfrum et al [6]). (From Olson et al [4]).
Fig.2. Cross sections for OVIII light emission ; + : resulting from electron capture O”™ +Li collisions. Circles, squares, diamonds and triangles denote 7-6, 8-7, 9-8 , 10-9 and 11-10 transitions respectively. Open symbols: theory [4], connected by lines to guide the eye; closed symbols: experiment (9-8) by Wolfrum et al [6]). (From Olson et al [4]).
One might wonder whether the nearly exclusive dependence of the capture cross section on the charge state q also holds for lower q-values. Such a core-independence would e.g. allow to use cross sections for He-like C** projectiles in place of those for bare Be** projec- tiles, which are difficult to determine experimentally in view of toxic risks. For q=5 the question of core-independence has been investigated in some more detail by Hoekstra et al [8]. Fig.3 shows cross sections for the (n=7— n=6) transition in the fourfold charged ions of B, C and N resulting from collisions of the fivefold charged bare, H-like and He-like ions of B, C and N respectively. The experimentally determined cross sections are com- pared with theoretical ones, obtained from CTMC calculations for bare B® projectiles. Again one can see that the intuitive expectation of nearly core-independent capture and emission cross sections is fulfilled. The next step of sophistication is a test of Especific capture cross sections. These are needed for plasma diagnostics purposes since due to strong electric and magnetic fields in fusion plasmas eventually various Estates are mixed, resulting in branching ratios for line emission which strongly deviate from those of atoms in a field-free region. The comparisons shown above for visible light emission - summed over all transitions between certain n-levels - represent in fact a test for mainly the higher-/ levels, since these contrib- ute most due to their large branching ratios for transitions to the next lower n-level anc due to the more or less statistical cross sections. The strongest core-effect however car be expected for the low-/ levels due to the deeper core penetration of the correspondinc elctron orbits. Since the low-/ levels in ions with a He-like core are sufficiently shifted, the corresponding optical transitions can be distinguished spectroscopically. This allowed us to compare experimentally determined /-specific emission cross sections for He-like pro- jectiles with theoretical ones for bare projectiles [8]. In fig.4 such a comparison is showr for N° and B* projectiles respectively. Good agreement between theoretical predictions and experimental results is observed for all lines investigated. This clearly indicates that
One might wonder whether the nearly exclusive dependence of the capture cross section on the charge state q also holds for lower q-values. Such a core-independence would e.g. allow to use cross sections for He-like C** projectiles in place of those for bare Be** projec- tiles, which are difficult to determine experimentally in view of toxic risks. For q=5 the question of core-independence has been investigated in some more detail by Hoekstra et al [8]. Fig.3 shows cross sections for the (n=7— n=6) transition in the fourfold charged ions of B, C and N resulting from collisions of the fivefold charged bare, H-like and He-like ions of B, C and N respectively. The experimentally determined cross sections are com- pared with theoretical ones, obtained from CTMC calculations for bare B® projectiles. Again one can see that the intuitive expectation of nearly core-independent capture and emission cross sections is fulfilled. The next step of sophistication is a test of Especific capture cross sections. These are needed for plasma diagnostics purposes since due to strong electric and magnetic fields in fusion plasmas eventually various Estates are mixed, resulting in branching ratios for line emission which strongly deviate from those of atoms in a field-free region. The comparisons shown above for visible light emission - summed over all transitions between certain n-levels - represent in fact a test for mainly the higher-/ levels, since these contrib- ute most due to their large branching ratios for transitions to the next lower n-level anc due to the more or less statistical cross sections. The strongest core-effect however car be expected for the low-/ levels due to the deeper core penetration of the correspondinc elctron orbits. Since the low-/ levels in ions with a He-like core are sufficiently shifted, the corresponding optical transitions can be distinguished spectroscopically. This allowed us to compare experimentally determined /-specific emission cross sections for He-like pro- jectiles with theoretical ones for bare projectiles [8]. In fig.4 such a comparison is showr for N° and B* projectiles respectively. Good agreement between theoretical predictions and experimental results is observed for all lines investigated. This clearly indicates that
FIG.4._ Comparison of experimental line — emission cross sections obtained with He- like N** projectiles colliding on Li with theo- retical CTMC results calculated for bare BY projectiles (From Hoekstra et al [8])
FIG.4._ Comparison of experimental line — emission cross sections obtained with He- like N** projectiles colliding on Li with theo- retical CTMC results calculated for bare BY projectiles (From Hoekstra et al [8])
for targets with similar electron binding energies, whereby these "neighbouring" cross sec: tions can again be checked by comparing experimentally and theoretically determined val. ues . Such a comparison is shown in fig.6, in which cross sections for Cvi(n=7—> n=6 emission following collisions of C** on various targets are plotted as a function of the initia electron binding energy. Again the results of -CTMC calculations are in good agreemen with available experimental results and therefore one can trust that the extrapolated emis sion cross section of roughly 4.10% cm? for C®+He’ collisions is describing the reality i ee ee ce Fig.6. Cross sections for the emission of visible light at the n=7—-» n=6 transtion in C* resulting from C* collisions on various targets, i.e. as a function of the target ionization potential. (a) for a collision energy of 4 keViamu, (b) for 40 keViamu. Closed symbols represent experimental results from Wolfrum et al [6], Hoekstra et al [13] and Bliek et al [12], and open circles CTMC calculations [12] (from Bliek et al. [12}).
for targets with similar electron binding energies, whereby these "neighbouring" cross sec: tions can again be checked by comparing experimentally and theoretically determined val. ues . Such a comparison is shown in fig.6, in which cross sections for Cvi(n=7—> n=6 emission following collisions of C** on various targets are plotted as a function of the initia electron binding energy. Again the results of -CTMC calculations are in good agreemen with available experimental results and therefore one can trust that the extrapolated emis sion cross section of roughly 4.10% cm? for C®+He’ collisions is describing the reality i ee ee ce Fig.6. Cross sections for the emission of visible light at the n=7—-» n=6 transtion in C* resulting from C* collisions on various targets, i.e. as a function of the target ionization potential. (a) for a collision energy of 4 keViamu, (b) for 40 keViamu. Closed symbols represent experimental results from Wolfrum et al [6], Hoekstra et al [13] and Bliek et al [12], and open circles CTMC calculations [12] (from Bliek et al. [12}).
Fig.7. VUV spectrum of N** (3 21) transitions obtained from 879 eV/amu N* collisions on He
Fig.7. VUV spectrum of N** (3 21) transitions obtained from 879 eV/amu N* collisions on He
Table 4. Emission cross sections o,,, in the collisions Ar®™ (3s? 'S) + He (H,) - Ar®* (3s’nl 7L) + He’* (H,") at 60 keV (H,) and 120 keV (He).
Table 4. Emission cross sections o,,, in the collisions Ar®™ (3s? 'S) + He (H,) - Ar®* (3s’nl 7L) + He’* (H,") at 60 keV (H,) and 120 keV (He).
Table 6. Excitation cross sections o (nl) (10°? cm?) of 3s*nl and 3s3pnl states for Ar®™ + He (H,) ~ Ar* + He* (H,*) at 120 and 60 keV respectively.
Table 6. Excitation cross sections o (nl) (10°? cm?) of 3s*nl and 3s3pnl states for Ar®™ + He (H,) ~ Ar* + He* (H,*) at 120 and 60 keV respectively.
Fig. 2 Schematic energy level diagram of Fe VIII
Fig. 2 Schematic energy level diagram of Fe VIII
Figure 1. Scaled electron capture cross sections, 0/q, as a function of scaled energy, E/q’” for Si** + H collisions, for 4<q<14. The solid curve is the analytical fit from Eq.(2). See Table II for fitting coefficients.
Figure 1. Scaled electron capture cross sections, 0/q, as a function of scaled energy, E/q’” for Si** + H collisions, for 4<q<14. The solid curve is the analytical fit from Eq.(2). See Table II for fitting coefficients.
Figure 2. Scaled electron capture cross sections, o/q, as a function of scaled energy, E/q'” for Ti?+ + H collisions, for 4<qs<22. The solid curve is the analytical fit from Eq.(2). See Table ll for fitting coefficients.
Figure 2. Scaled electron capture cross sections, o/q, as a function of scaled energy, E/q'” for Ti?+ + H collisions, for 4<qs<22. The solid curve is the analytical fit from Eq.(2). See Table ll for fitting coefficients.
Figure 3. Scaled electron capture cross sections, o/q, as a function of scaled energy, E/q’” for Nit* + H collisions, for 4<q528 except q=14. The solid curve is the analytical fit from Eq.(2). See Table I for fitting coefficients.
Figure 3. Scaled electron capture cross sections, o/q, as a function of scaled energy, E/q’” for Nit* + H collisions, for 4<q528 except q=14. The solid curve is the analytical fit from Eq.(2). See Table I for fitting coefficients.
Figure 4. Scaled electron capture cross sections, o/q, as a function of scaled energy, E/q'” for Mo** + H collisions, for 4<q530. The solid curve is the analytical fit from Eq.(2). See Table LU for fitting coefficients.
Figure 4. Scaled electron capture cross sections, o/q, as a function of scaled energy, E/q'” for Mo** + H collisions, for 4<q530. The solid curve is the analytical fit from Eq.(2). See Table LU for fitting coefficients.
Figure 5. Scaled electron capture cross sections, o/q, as a function of scaled energy, E/q'” for W* + H collisions, for 4<q<30 and q=74. The solid curve is the analytical fit from Eq.(2). See Table II for fitting coefficients. i
Figure 5. Scaled electron capture cross sections, o/q, as a function of scaled energy, E/q'” for W* + H collisions, for 4<q<30 and q=74. The solid curve is the analytical fit from Eq.(2). See Table II for fitting coefficients. i
Figure 6. Scaled electron capture cross sections, o/q, as a function of scaled energy, E/q'” for Sit*+ H collisions, for 2<q<9. The solid curve is the analytical fit from Eq.(2). See Table II for fitting coefficients.
Figure 6. Scaled electron capture cross sections, o/q, as a function of scaled energy, E/q'” for Sit*+ H collisions, for 2<q<9. The solid curve is the analytical fit from Eq.(2). See Table II for fitting coefficients.
Figure 7. Scaled electron capture cross sections, 0/q, as a function of scaled energy, E/q'” for Sit* + He collision for q=11, 13 and 14. The solid curve is the analytical fit from eq.(2). See Table Il for fitting coefficients.
Figure 7. Scaled electron capture cross sections, 0/q, as a function of scaled energy, E/q'” for Sit* + He collision for q=11, 13 and 14. The solid curve is the analytical fit from eq.(2). See Table Il for fitting coefficients.
Figure 8. Doubly scaled electron capture cross sections, (Ip/I,)’o/q, as a function of doubly scaled energy, (I,/1,E/q'” for Sit’ + (H,H,,He) collisions. (M1) H, (CQ) H, (OQ) He. The solid line corresponds to the analytical fit of Fig. 1.
Figure 8. Doubly scaled electron capture cross sections, (Ip/I,)’o/q, as a function of doubly scaled energy, (I,/1,E/q'” for Sit’ + (H,H,,He) collisions. (M1) H, (CQ) H, (OQ) He. The solid line corresponds to the analytical fit of Fig. 1.
Figure 9. Doubly scaled electron capture cross sections, (I,/I,)’o/q, as a function of doubly scaled energy, (I,/I,)°E/q" for Mot + (H,H, collisions. (Wi) H,(O)) H,. The solid line corresponds to the analytical fit of Fig. 4.
Figure 9. Doubly scaled electron capture cross sections, (I,/I,)’o/q, as a function of doubly scaled energy, (I,/I,)°E/q" for Mot + (H,H, collisions. (Wi) H,(O)) H,. The solid line corresponds to the analytical fit of Fig. 4.
Figure 10. Doubly scaled electron capture cross sections, (Ip/I,)’0/q, as a function of doubly scaled energy, (Iy/I,)°E/ for W* + (H,H,) collisions. (@) H, (QO) H,. The solid line corresponds to the analytical fit of Fig. 5.
Figure 10. Doubly scaled electron capture cross sections, (Ip/I,)’0/q, as a function of doubly scaled energy, (Iy/I,)°E/ for W* + (H,H,) collisions. (@) H, (QO) H,. The solid line corresponds to the analytical fit of Fig. 5.
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