- Email: [email protected]
New findings in multiple ionization of Ne, Ar and Kr
Boom Interactions
with Materials&Atoms
Nuclear Instruments and Methods in Physics Research B 132 ( 1997) 236-240
El-SEWER
New findings in multiple ionization of Ne, Ar and Kr
Abstract
We have measured multiple ionization from heavy targets and investigated the competition of capture and pure ionization. We considered singly and doubly charged projectiles as well as the corresponding neutral ones, at energies between 25 and 50 keV/u. Recoil ions. in coincidence with electrons partially integrated in energy and solid angle were
detected. For proton impact. for the three targets studied, Ne. Ar and Kr. a strong signature of hard electron-electron interaction in the production of the doubly charged recoil ion was found. In contrast to results on Total Cross Sections for H+ impact. unexpected large contributions of degrees of ionization higher than 1 are observed. In the case of He” impact, high degrees of ionization (3 and 4 fold) dominating the recoil spectra were observed. The influence of double @ 1997 Elsevier Science B.V. capture on this latter case is discussed.
1. Introduction Multiple ionization of heavy targets by impact of charged or neutral projectiles is a physical process involving many-body Coulomb interactions. As a result. a recoil ion with a wide range of possible charges, a modified scattered projectile and the emission of electrons and radiation. are obtained. Many interesting features have been studied using heavy ions, mainly chosen because of their capability of producing highly ionized target ions [ 1,2]. Experiments using coincidence techniques and recoil spectroscopy enabled investigating
*Corresponding author. Fax: +54 944 45299: e-mail: [email protected]. ’ Also member of the Consejo National de Investigaclones Cientificas y Tknicas (CONICET). Argentina. ’ ComisiJn National de Energia AtGmica and Universidad National de Cuyo. Argentina. 0168-583X/97/$17.00 6 I997 Elsevier Science B.V. All rights reserved PIISOl68-583X(97)00453-9
many details on the production and the dynamics of these processes [3,4]. As the systems under study are complex. only numerical simulations [5] (mostly based on Classical Physics), have been extensively compared with data, and in most of the cases succesfully. Only in particular cases other model, based on the Vlasov Equation [6], have been used. Although, the theoretical understanding of the mechanisms for multiple ionization has yet not been satisfactory. It is the purpose of the present experimental study to measure details on Differential Cross Sections (DCS) for the simplest possible systems presenting multiple ionization and capture processes, with the hope that a theoretical approach would be more feasible for these cases. We have chosen singly and doubly charged projectiles colliding with Ne, Ar and Kr targets at intermediate impact energies ( 1 < rp < 3 a.u., )‘pbeing the velocity of the projectile). For these collisions previous investiga-
A. D. Gmmile:
et trl. I Nucl. Instr. and Meth. in Phw. Rrs. B 132 (1997) 236-240
tions on Total Cross Sections [7710] showed a remarkable competition between different collision channels.
2. Experiment We have measured electrons emitted in the multiple ionizing collision in coincidence with the charge state of the recoiling ion. It is well known the experimental difficulties encountered to realize this particular measurement, because of different solid angles detected. To avoid this problem and make the experiment feasible we have investigated electrons partially integrated in energy and solid angle in correspondence with recoil charge state. Recently other authors [l l] studied similar collisions but with a different setup which integrates electrons in a very large solid angle of 1.45~ srad. The experimental apparatus consists of a timeof-flight type recoil spectrometer. The recoil ions are extracted from the collision region by a weak electrostatic field, perpendicular to the beam axis. These ions are detected in coincidence with electrons extracted in the opposite direction. After extraction the electrons are selected by a thresholdpotential grid ( V,h). so that only those with energies higher than this threshold are detected. To have a potential reference, the extraction field is uniform on the beam path, and the setup is carefully aligned. The value of the threshold potential in electron-volts corresponds to the minimum electron energy selected. A weak field applied is enough to extract all the recoil ions. However, only electrons within a solid angle around 90” with respect to the beam can be extracted and detected. The size of this solid angle depends on the electron energy. The larger the ionization energy, the smaller the solid angle detected. Due to the simplicity in the detection of electrons their angular acceptance varies also with the extraction field. The higher this field, the larger the solid angle extracted. Therefore, to study DCS around 90” we have chosen to work with small extraction fields (2 V/mm). Care was taken in studying coincidence events with different extraction fields. No variation in the results were obtained for extraction fields between 0.15 and 3 V/mm
737
for single and Double Ionization (DI) in 25 keV H++Ar collisions. It is to be noted that in the present coincidence measurement ionized electrons and recoiling ions are detected, and therefore no pure capture events are recorded at all. This fact is a key point for the present experimental technique, in which one has the singly charged recoil-ion event associated only with single ionization. In the case of doubly charged impact projectiles, which may produce double capture events, these are not detected either by the present setup.
3. Proton and H impact In Fig. 1 we show data for the collision of H+ and neutral H projectiles colliding with Kr. Electrons with energies higher than el’,, are detected
-
50 keV
500 400 300
200 100
15
I.
I.
I.
I.
I
20
25
30
35
40
.I 45
Time of flight (psec) Fig. I. Recoil-ion time-of-flight spectra: (a) H impact: (b) neutral H. I’,,, is the electron threshold potential. set equal to 20 V for both (a) and (b).
238
A.D. Gonzcile: et al. I Nucl. Imtr. unti Mrth. in Phys. Rex B 132 (1997) 236-240
in coincidence with Kr charge states. Fig. l(a) is a typical time-of-flight spectrum for H+ impact. The largest peak is Kr” which has both contributions. Transfer Ionization (TI) and DI. In Fig. l(b), corresponding to H impact, the spectrum looks different, with the largest peak at Kr+. In this case the Kr’+ ions account mostly for DI, because capture is negligible for the neutral projectile. In Fig. 2 the ratios of doubly charged recoil ions with respect to the corresponding singly charged, for both H’ and H impact, are shown. For the three targets studied (Ne, Ar and Kr). we found a distinctive maximum for H+ impact. which is not observed in the case of H impact. In another article we have shown (for H’ +Ar) that
06
,
I
I
Ne
50 keV
00
II
16
.,
I
Ar
001 16
12 +L +g
08
3 04
0.0
, 1
I
I
10
this maximum also moves towards higher electron energies as the beam energy increases, being always very close to an electron energy corresponding to the velocity of the projectile. Then we have concluded [12] that these broad peaks are due to a hard projectile-electron collision followed by a hard electronelectron interaction. as a Thomaslike mechanism predicted theoretically. As expected, the effect found for Ne is weaker, but at variance with previous data from other authors [ 131, by whom the effect was observed for a He target but not for Ne. Even though we do not isolate the capture channel via measuring the scattered projectile, we measure the ionized electron associated with an event which simultaneously includes the capture of another electron. Within the formulation of Quantum Mechanics this is understood as projecting the Wave Function describing the whole state, onto a particular state, which is the final Plane Wave of the ionized electron. The whole 22active-electron Wave Function includes both TI and DI. The Projection on the final Plane Wave of the ionized electron (our detector) gives two terms which cannot be isolated from the data. Both projections are mixed on the peak observed in the case of H- impact. We want to stress that the data for the neutral H projectile give an estimate of the shape of the DI ratio. The experimental findings of Fig. 2 and the above interpretation are better understood by representing the fraction of recoil ions in each charge state normalized to the total number of coincident recoils detected. This is shown in Fig. 3, as a function of Vth, for the collision 50 keV Hi +Kr. Singly charged Kr+ presents a minimum at the same Vth as Kr2+ and Kr3 +show a maximum. At this particular range of Vth a depletion on the production of single ionization occurs, while events including electron capture are enhanced. At the same range of I/;,, no minimum is observed for a neutral H projectile.
I
100
vm Fig. 2. Ratio between doubly and singly charged recoil ions as a function of C;, in Volts for H and neutral H impact, and different targets: Ne, Ar and Kr. The filled dots represent incident H’ and the open ones, neutral H.
4. He projectiles In Fig. 4(a))(c) the energy of the electrons increases, showing different multiple ionization patterns. It is to be noted that the high contribution
600
-
25 keV/u
L n
0
q=2
*
q=3
*
015
A *A
010
q=l
A
A
.
*AA
A AA
t
1
I
I
I
10
100
I
Fig. 3. Recoil fractions for three charge states of Kr. as a function of Vlh. at the same H* impact energy as in Figs. I and 2. The fraction is defined as the area of the coincidence peak divlded by the total non-coincidence recoil counting.
of Ar’+ and At-‘+ for this low impact energy of the projectile. Data for singly charged He’ and for neutral He at the same impact energy (not shown here). have a completely different pattern, with a maximum on Ai-‘+ and on Ar+ respectively, for a wide range of V[h. The spectra in Fig. 4 suggest the promotion of highly ionizing collisions by single and double electron capture. This promotion has been previously observed for more complex reactions [14, Fig. 21 Fig. 5 shows the ratio of recoil charge states 24 with respect to singly charged Ar, for the collision “He’++Ar at 25 and 50 keV/u. The highest beam energy data resembles the maximum observed in Fig. 2 for proton impact, however in this case more collision channels contribute to the triply and doubly charged recoil ions. Dubois [9,10] reported a competition of double capture and simultaneous single ionization (DCSI), with single capture and double ionization (SCDI); both pro-
-0.5
10
1.5
20
25
3.0
Time of flight (psec) Fig. 4. Recoil-ion time-of-flight spectra. in coincidence electrons integrated in energy above eV,h.
with
ducing Ar’+ in Figs. 4 and 5. For the Total Cross Sections a cross point as a function of beam energy is placed at approximately 30 keV/u. Above this value SCDI dominates, making the two pictures on Fig. 5 one dominated by DCSI and the other by SCDI. Since the ratio Ar?+/Ar+ does not present a maximum as in proton impact, it suggests that a postcollisional effect due to the neutralization of the projectile could be present.
5. Conclusions Partial information on the emission of electrons associated with different reaction channels was obtained. We found a very different recoil pattern for collisions which lead to capture events compared to those with no capturing. The study of the ratios of recoil ions with charges 2 and 3 with respect to
240
.
“L
a
i
I
30
For Ar and Kr targets and proton energies above 100 keV (not shown in the present paper). we have found a substantial contribution from up to five-fold target ionization. These results, in which the projectile is the simplest possible, suggest the effect of collective phenomena producing multiple ionization. A theoretical approach based on collective states of plasmons. like the ones developed for studying ion-surface interactions, may be of help.
I
I
Acknowledgements
25 keV/u
.I
r-7
25
20
l
a=2
A
q=3
V
a=4
We thank partial support from the Fundacion Antorchas, Buenos Aires, Argentina.
15
References 10
[II C.L. Cocke. R.E. Olson, Phys. Rep. 205 (1991) 153. R. Darner, J. Ullrich et al.. Fourth PI H. Schmidt-B&king,
05
00
I
1
I
i
10
vt, Fig. 5. Ratio of A+ with respect to Ar . for ‘He” projectiles at two different impact energies 50 and 25 keV/u. The horizontal scale is I/,h in volts.
single ionization demonstrates an interesting enhancement which is beam energy dependent, interpreted as a Thomas-like scattering involving electron-electron interaction. For H + impact, the production of a doubly charged recoil ion was found to be larger than the corresponding singly charged one. This effect has not been observed before, neither on the Total Cross Sections [7,8], nor on DCS in the scattering angle [ 151. The relatively large production of degrees of ionization higher than 1 on DCS in collisions involving singly and doubly charged bare projectiles gives an alert sign for the comparison of experimental and theoretical DCS in these cases.
Conference on High-Energy Ion-Atom Cell.. Springer. Berlin. 1991. p. 268. J. Ullrich, R.E. Olson. R. Darner. V. Dagendorf. S. Kelbch, H. Berg, H. Schmidt-B&king, J. Phys. B 22 ( 1989) 627. [41 S. Lencinas. J. Ullrich. R. Darner et al., J. Phys. B 27 (1994) 287. R.E. Olson. J. Ullrich. H. Schmidt-B&king, Phys. Rev. A 39 (1989) 5572. [61 M. Horbatsh. J. Phys. B 25 (1992) 3797. [‘I R.D. Dubois. S.T. Manson. Phys. Rev. A 35 (1987) 2007. PI R.D. Dubois. A. Kover. Phys. Rev. A 40 (1989) 3695. [91 R.D. Dubois. Phys. Rev. A 36 (1987) 2585. UOI R.D. Dubois. Phys. Rev. A 39 (1989) 4440. [Ill Y.-S. Chung. M.E. Rudd. Phys. Rev. A 54 (1996) 4106. [I21 A.D. Gonzalez. D. Fregenal. S. Suarez. W. Wolff. H. Wolf (submitted). [I31 J. Palinkas. R. Schuch. H. Cederquist, 0. Gustafsson, Phys. Rev. Lett. 63 (1989) 2464. 1141 B. Krassig. A.D. Gonziilez. R. Koch, T.B. Quinteros. A. Skutlartz. S. Hagmann. J. Physique C 50 (1989) 159. Phys. Rev. A 48 t151 A.D. Gonzalez. E. Horsdal-Pedersen. (1993) 3689.
with Materials&Atoms
Nuclear Instruments and Methods in Physics Research B 132 ( 1997) 236-240
El-SEWER
New findings in multiple ionization of Ne, Ar and Kr
Abstract
We have measured multiple ionization from heavy targets and investigated the competition of capture and pure ionization. We considered singly and doubly charged projectiles as well as the corresponding neutral ones, at energies between 25 and 50 keV/u. Recoil ions. in coincidence with electrons partially integrated in energy and solid angle were
detected. For proton impact. for the three targets studied, Ne. Ar and Kr. a strong signature of hard electron-electron interaction in the production of the doubly charged recoil ion was found. In contrast to results on Total Cross Sections for H+ impact. unexpected large contributions of degrees of ionization higher than 1 are observed. In the case of He” impact, high degrees of ionization (3 and 4 fold) dominating the recoil spectra were observed. The influence of double @ 1997 Elsevier Science B.V. capture on this latter case is discussed.
1. Introduction Multiple ionization of heavy targets by impact of charged or neutral projectiles is a physical process involving many-body Coulomb interactions. As a result. a recoil ion with a wide range of possible charges, a modified scattered projectile and the emission of electrons and radiation. are obtained. Many interesting features have been studied using heavy ions, mainly chosen because of their capability of producing highly ionized target ions [ 1,2]. Experiments using coincidence techniques and recoil spectroscopy enabled investigating
*Corresponding author. Fax: +54 944 45299: e-mail: [email protected]. ’ Also member of the Consejo National de Investigaclones Cientificas y Tknicas (CONICET). Argentina. ’ ComisiJn National de Energia AtGmica and Universidad National de Cuyo. Argentina. 0168-583X/97/$17.00 6 I997 Elsevier Science B.V. All rights reserved PIISOl68-583X(97)00453-9
many details on the production and the dynamics of these processes [3,4]. As the systems under study are complex. only numerical simulations [5] (mostly based on Classical Physics), have been extensively compared with data, and in most of the cases succesfully. Only in particular cases other model, based on the Vlasov Equation [6], have been used. Although, the theoretical understanding of the mechanisms for multiple ionization has yet not been satisfactory. It is the purpose of the present experimental study to measure details on Differential Cross Sections (DCS) for the simplest possible systems presenting multiple ionization and capture processes, with the hope that a theoretical approach would be more feasible for these cases. We have chosen singly and doubly charged projectiles colliding with Ne, Ar and Kr targets at intermediate impact energies ( 1 < rp < 3 a.u., )‘pbeing the velocity of the projectile). For these collisions previous investiga-
A. D. Gmmile:
et trl. I Nucl. Instr. and Meth. in Phw. Rrs. B 132 (1997) 236-240
tions on Total Cross Sections [7710] showed a remarkable competition between different collision channels.
2. Experiment We have measured electrons emitted in the multiple ionizing collision in coincidence with the charge state of the recoiling ion. It is well known the experimental difficulties encountered to realize this particular measurement, because of different solid angles detected. To avoid this problem and make the experiment feasible we have investigated electrons partially integrated in energy and solid angle in correspondence with recoil charge state. Recently other authors [l l] studied similar collisions but with a different setup which integrates electrons in a very large solid angle of 1.45~ srad. The experimental apparatus consists of a timeof-flight type recoil spectrometer. The recoil ions are extracted from the collision region by a weak electrostatic field, perpendicular to the beam axis. These ions are detected in coincidence with electrons extracted in the opposite direction. After extraction the electrons are selected by a thresholdpotential grid ( V,h). so that only those with energies higher than this threshold are detected. To have a potential reference, the extraction field is uniform on the beam path, and the setup is carefully aligned. The value of the threshold potential in electron-volts corresponds to the minimum electron energy selected. A weak field applied is enough to extract all the recoil ions. However, only electrons within a solid angle around 90” with respect to the beam can be extracted and detected. The size of this solid angle depends on the electron energy. The larger the ionization energy, the smaller the solid angle detected. Due to the simplicity in the detection of electrons their angular acceptance varies also with the extraction field. The higher this field, the larger the solid angle extracted. Therefore, to study DCS around 90” we have chosen to work with small extraction fields (2 V/mm). Care was taken in studying coincidence events with different extraction fields. No variation in the results were obtained for extraction fields between 0.15 and 3 V/mm
737
for single and Double Ionization (DI) in 25 keV H++Ar collisions. It is to be noted that in the present coincidence measurement ionized electrons and recoiling ions are detected, and therefore no pure capture events are recorded at all. This fact is a key point for the present experimental technique, in which one has the singly charged recoil-ion event associated only with single ionization. In the case of doubly charged impact projectiles, which may produce double capture events, these are not detected either by the present setup.
3. Proton and H impact In Fig. 1 we show data for the collision of H+ and neutral H projectiles colliding with Kr. Electrons with energies higher than el’,, are detected
-
50 keV
500 400 300
200 100
15
I.
I.
I.
I.
I
20
25
30
35
40
.I 45
Time of flight (psec) Fig. I. Recoil-ion time-of-flight spectra: (a) H impact: (b) neutral H. I’,,, is the electron threshold potential. set equal to 20 V for both (a) and (b).
238
A.D. Gonzcile: et al. I Nucl. Imtr. unti Mrth. in Phys. Rex B 132 (1997) 236-240
in coincidence with Kr charge states. Fig. l(a) is a typical time-of-flight spectrum for H+ impact. The largest peak is Kr” which has both contributions. Transfer Ionization (TI) and DI. In Fig. l(b), corresponding to H impact, the spectrum looks different, with the largest peak at Kr+. In this case the Kr’+ ions account mostly for DI, because capture is negligible for the neutral projectile. In Fig. 2 the ratios of doubly charged recoil ions with respect to the corresponding singly charged, for both H’ and H impact, are shown. For the three targets studied (Ne, Ar and Kr). we found a distinctive maximum for H+ impact. which is not observed in the case of H impact. In another article we have shown (for H’ +Ar) that
06
,
I
I
Ne
50 keV
00
II
16
.,
I
Ar
001 16
12 +L +g
08
3 04
0.0
, 1
I
I
10
this maximum also moves towards higher electron energies as the beam energy increases, being always very close to an electron energy corresponding to the velocity of the projectile. Then we have concluded [12] that these broad peaks are due to a hard projectile-electron collision followed by a hard electronelectron interaction. as a Thomaslike mechanism predicted theoretically. As expected, the effect found for Ne is weaker, but at variance with previous data from other authors [ 131, by whom the effect was observed for a He target but not for Ne. Even though we do not isolate the capture channel via measuring the scattered projectile, we measure the ionized electron associated with an event which simultaneously includes the capture of another electron. Within the formulation of Quantum Mechanics this is understood as projecting the Wave Function describing the whole state, onto a particular state, which is the final Plane Wave of the ionized electron. The whole 22active-electron Wave Function includes both TI and DI. The Projection on the final Plane Wave of the ionized electron (our detector) gives two terms which cannot be isolated from the data. Both projections are mixed on the peak observed in the case of H- impact. We want to stress that the data for the neutral H projectile give an estimate of the shape of the DI ratio. The experimental findings of Fig. 2 and the above interpretation are better understood by representing the fraction of recoil ions in each charge state normalized to the total number of coincident recoils detected. This is shown in Fig. 3, as a function of Vth, for the collision 50 keV Hi +Kr. Singly charged Kr+ presents a minimum at the same Vth as Kr2+ and Kr3 +show a maximum. At this particular range of Vth a depletion on the production of single ionization occurs, while events including electron capture are enhanced. At the same range of I/;,, no minimum is observed for a neutral H projectile.
I
100
vm Fig. 2. Ratio between doubly and singly charged recoil ions as a function of C;, in Volts for H and neutral H impact, and different targets: Ne, Ar and Kr. The filled dots represent incident H’ and the open ones, neutral H.
4. He projectiles In Fig. 4(a))(c) the energy of the electrons increases, showing different multiple ionization patterns. It is to be noted that the high contribution
600
-
25 keV/u
L n
0
q=2
*
q=3
*
015
A *A
010
q=l
A
A
.
*AA
A AA
t
1
I
I
I
10
100
I
Fig. 3. Recoil fractions for three charge states of Kr. as a function of Vlh. at the same H* impact energy as in Figs. I and 2. The fraction is defined as the area of the coincidence peak divlded by the total non-coincidence recoil counting.
of Ar’+ and At-‘+ for this low impact energy of the projectile. Data for singly charged He’ and for neutral He at the same impact energy (not shown here). have a completely different pattern, with a maximum on Ai-‘+ and on Ar+ respectively, for a wide range of V[h. The spectra in Fig. 4 suggest the promotion of highly ionizing collisions by single and double electron capture. This promotion has been previously observed for more complex reactions [14, Fig. 21 Fig. 5 shows the ratio of recoil charge states 24 with respect to singly charged Ar, for the collision “He’++Ar at 25 and 50 keV/u. The highest beam energy data resembles the maximum observed in Fig. 2 for proton impact, however in this case more collision channels contribute to the triply and doubly charged recoil ions. Dubois [9,10] reported a competition of double capture and simultaneous single ionization (DCSI), with single capture and double ionization (SCDI); both pro-
-0.5
10
1.5
20
25
3.0
Time of flight (psec) Fig. 4. Recoil-ion time-of-flight spectra. in coincidence electrons integrated in energy above eV,h.
with
ducing Ar’+ in Figs. 4 and 5. For the Total Cross Sections a cross point as a function of beam energy is placed at approximately 30 keV/u. Above this value SCDI dominates, making the two pictures on Fig. 5 one dominated by DCSI and the other by SCDI. Since the ratio Ar?+/Ar+ does not present a maximum as in proton impact, it suggests that a postcollisional effect due to the neutralization of the projectile could be present.
5. Conclusions Partial information on the emission of electrons associated with different reaction channels was obtained. We found a very different recoil pattern for collisions which lead to capture events compared to those with no capturing. The study of the ratios of recoil ions with charges 2 and 3 with respect to
240
.
“L
a
i
I
30
For Ar and Kr targets and proton energies above 100 keV (not shown in the present paper). we have found a substantial contribution from up to five-fold target ionization. These results, in which the projectile is the simplest possible, suggest the effect of collective phenomena producing multiple ionization. A theoretical approach based on collective states of plasmons. like the ones developed for studying ion-surface interactions, may be of help.
I
I
Acknowledgements
25 keV/u
.I
r-7
25
20
l
a=2
A
q=3
V
a=4
We thank partial support from the Fundacion Antorchas, Buenos Aires, Argentina.
15
References 10
[II C.L. Cocke. R.E. Olson, Phys. Rep. 205 (1991) 153. R. Darner, J. Ullrich et al.. Fourth PI H. Schmidt-B&king,
05
00
I
1
I
i
10
vt, Fig. 5. Ratio of A+ with respect to Ar . for ‘He” projectiles at two different impact energies 50 and 25 keV/u. The horizontal scale is I/,h in volts.
single ionization demonstrates an interesting enhancement which is beam energy dependent, interpreted as a Thomas-like scattering involving electron-electron interaction. For H + impact, the production of a doubly charged recoil ion was found to be larger than the corresponding singly charged one. This effect has not been observed before, neither on the Total Cross Sections [7,8], nor on DCS in the scattering angle [ 151. The relatively large production of degrees of ionization higher than 1 on DCS in collisions involving singly and doubly charged bare projectiles gives an alert sign for the comparison of experimental and theoretical DCS in these cases.
Conference on High-Energy Ion-Atom Cell.. Springer. Berlin. 1991. p. 268. J. Ullrich, R.E. Olson. R. Darner. V. Dagendorf. S. Kelbch, H. Berg, H. Schmidt-B&king, J. Phys. B 22 ( 1989) 627. [41 S. Lencinas. J. Ullrich. R. Darner et al., J. Phys. B 27 (1994) 287. R.E. Olson. J. Ullrich. H. Schmidt-B&king, Phys. Rev. A 39 (1989) 5572. [61 M. Horbatsh. J. Phys. B 25 (1992) 3797. [‘I R.D. Dubois. S.T. Manson. Phys. Rev. A 35 (1987) 2007. PI R.D. Dubois. A. Kover. Phys. Rev. A 40 (1989) 3695. [91 R.D. Dubois. Phys. Rev. A 36 (1987) 2585. UOI R.D. Dubois. Phys. Rev. A 39 (1989) 4440. [Ill Y.-S. Chung. M.E. Rudd. Phys. Rev. A 54 (1996) 4106. [I21 A.D. Gonzalez. D. Fregenal. S. Suarez. W. Wolff. H. Wolf (submitted). [I31 J. Palinkas. R. Schuch. H. Cederquist, 0. Gustafsson, Phys. Rev. Lett. 63 (1989) 2464. 1141 B. Krassig. A.D. Gonziilez. R. Koch, T.B. Quinteros. A. Skutlartz. S. Hagmann. J. Physique C 50 (1989) 159. Phys. Rev. A 48 t151 A.D. Gonzalez. E. Horsdal-Pedersen. (1993) 3689.