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Wake effects on the transmission offast H2+ -ions through thin carbon foils
Nuclear Inst~ments Nosh-Holland
and Methods in Physics Research B 93 ( 1994) 1l-1 3 Bum tntsraotion8 Materials & Atoms
with
Wake effects on the transmission of fast Hz -ions through thin carbon foils Mario
M. Jakas
D~~a~t~~ent~ de F&X Funda~entai y Expe~i~e~tal~ Universidad de La Lagma, 382Oj La ,kguna, Ten&@, spain Received 21 January 1994 and in revised form 25 March 1994
According to Monte Carlo calculations the transmission of 0.1-1.0 MeV/amu H: molecular ions through thin carbon foils show a particularly high sensitivity upon the presence of wake forces. Our results also disagree with previous evidences pointing towards a nearly universal curve for the transmission yield as a function of dwell-time in the foil. Existing experiments however do not actually allow us to check for the validity of our calculations.
We have calculated the transmission yield of energetic H: molecular ions through thin carbon foils. Our Monte Carlo code is similar to that in ref. [ I 1. Calculations in this Note, however, have for the first time included the wake forces. Our Monte Carlo also includes an advanced technique, described in details elsewhere [ 3 1, which speeds up simulations to the point of rendering calculations of this type quite affordable even for a small personal computer. For the sake of completeness, we briefly mention that our MC follows the motion of the two nuclei in the H! molecule as one “reduced” particle subjected to the Coulomb repulsion, energy loss straggling, multiple scattering with target atom and wake forces. Energy loss is purposely excluded since it cannot modify the relative motion of the nuclei. Similarly, one assumes that right upon penetration the nuclei become dissociated, i.e., the binding electron is lost. Whereas capture of an electron into a binding state can occur only at the exit of the foil. The wake forces used in our calculations are obtained from an approximate expression given in ref. [2]. Although such forces slightly differ from exact calculations [ 41, they seem acceptable at this first attempt to include wake forces in transmission yield calculations. In order to cancel the probability of the electronic capture out of the transmission yield, it is customary to introduce the relative transmission yield, i.e., Y/2&, where Y is the transmission yield and, #0 is the neutral fraction for protons at same velocity. The relative transmission yield, therefore, can be seen as the transmission yield if the electron capture into the binding state occur with a probability of one. In Fig. 1 we show the results of our MC calculations both with- and without wake forces. As one can clearly see, by including the wake forces it removes the apparent insensitivity with the bombarding energy
showed by the transmission yield in the cases where wake forces are not present. For 0.1 MeV/amu bombarding energy, the yields obtained including wake forces become larger than those where wake forces are not. Whereas for 1.O and 0.4 MeV/amu the yields go well below the no-wake counte~a~. An interesting result however appears in the case of 0.4 Mev/amu. Here the transmission yield begins to increase around 10 fs, reaching the values of no-wake
Dwell
Time
(fs)
Fig. 1. Relative transmission yield for Hz -ions as a function of dwell-time in carbon foils. Experiments (crosses); Monte Carlo calculations: Without wake forces (full symbols); with wake forces (open symbols). Bombarding energies are in MeV/amu.
0168-583X/94/$07.00 @ 1994 - Elsevier Science B.V. All rights reserved SSDIO168-583X(94)00189-3
M.M. Jakas / Nucl. Instr. and Meth. in Phys. Res. B 93 (1994) 11-13
12
3
0
(a)
E = 0.1
(b)
E = 1 .O MeV/amu
2
Internuclear
MeV/amu
4
distance
J 8
6
(a.u.)
Fig. 2. Interatomic potential for proton fragments during passage through the foil, i.e. wake plus screened Coulomb potential. The internuclear axis is parallel-to- (continuous line), and perpendicular-to-the-beam (long-dashed line), respectively. Initial distribution of internuclear distance (short-dashed line). cases at approximately 20 fs. Although the origin of such an increase has not been investigated in detail, it seems related to the fact that at large dwell-times the multiple scattering gets more developed and so the occurrence of wake forces becomes less relevant. In this regard, such a behavior might also appear in the case of 1.O MeV/amu. However, as multiple scattering results smaller with increasing energy such an effect cannot be seen within dwell-times smaller than 30 fs. The strong influence of the wake forces upon transmission yields can be readily understood by looking at the interatomic potential of the nuclei. In Figs. 2a and 2b we plot the potential energy of the two nuclei as a function of the internuclear position. For a comparison, the initial distribution of internuclear separations is also plotted in the same figures. We can see that for molecules oriented perpendicular-to-the-beam the wake forces do not appreciably change the Coulomb repulsion. However for those nuclei which have their axis aligned with the beam direction the wake potential exhibits a strong dependence with bombarding energy. In fact, for 1.0 MeV/amu the wake potential translates into a much stronger re-
pulsion between the nuclei, as the nearest minimum appears far apart from the initial distribution of the molecules. For 0.1 MeV/amu, however, a significant part of the incoming nuclei find themselves into the potential well. A well which, on the other hand, also occurs along lateral displacements starting from wellaligned positions. Accordingly, one can expect that for 0.1 MeV/amu molecules the transmission yields should benefit from the wake forces. Whereas at 1.0 MeV/amu the wake potential, as mentioned before, actually adds up a repulsive force, thus making more difftcult for the nuclei to reach the region in the configuration space where recombination is possible. One can be tempted to think that as number of molecules which are subjected to the wake “attraction” is pretty small, i.e. lo%, then the wake-effect cannot be expected to be greater than 10%. This reasoning however is in error. First, although it is true that only a small fraction of the incoming molecules result within the “well”, it does not imply that nuclei sitting off the well will be less influenced by wake forces. For a nucleus initially off the well it will be accelerated towards regions of smaller potential. In this way its kinetic energy will get increased and consequently, the possibility of recombination reduced. Second, one must recall that the transmission of molecules relates to an exceedingly particular sub-set of trajectories, i.e. those leading to bound clusters. Therefore, any mechanism which can modify such histories may have a dramatic consequences upon recombination, even though it amounts to a small perturbation upon the whole set of trajectories. For example, the fact that at 0.1 MeV/amu the wake-well becomes both deeper and closer to the origin compared to those of higher energies appears to make a substantial difference here. It is evident that at 0.1 MeV/amu the particles can be more easily “confined” within the walls of the wake-well. Then, the population of low relative-velocity increases due to particles which are either approaching to or bouncing off the potential wall. In addition these nuclei become nearly “trapped” within distances where the binding potential becomes larger (in absolute sense) and therefore recombination is highly favored. Notice that this is not possible at higher energies, because most of the nuclei initiate penetration sitting far away from the bottom of the wake-well. Then, as mentioned above they will be subjected to an additional force where otherwise, i.e. with no wake, there was only the nearly vanishing force in the tail of the Coulomb repulsion. It is rather striking however, that the experiments appear closer to calculations obtained without wake forces rather than to those where the wake is “on”. Moreover, the sensitivity with energy resulting from
MM. Jakas I Nucl. Instr. and Meth. in Phys. Res. B 93 (1994) 11-13
our calculation is not observed in the experiments, which, as mentioned elsewhere [ 51 appear to follow a “universal curve”. It should be mentioned that the expe~ments shown in Fig. 1 cover a range of bombarding energy between 0.4 and 1.0 MeV/amu, while the ones at much larger dwell times are for 0.5 , 0.75 and 1.0 MeV/amu, respectively. Moreover, the wake model used in our MC is the same as the one previously used to calculate the energy-angle spectra of dissociated fragments 161, for which calculations compare remarkably well with experiments. The discrepancy between our calculations and the experiments pointed out above does not necessarily imply a contradiction. In fact, observe that at large dwell-times only high-energy data are available and, deviations between these high energy data and our calculations are not conclusive. This is particularly so, after considering the difficulties associated with measuring such low transmission yields. Second, the results for 0.4 MeV/amu do not reach large dwelltimes where wake effects become apparent. In the same manner as the low energy data in ref. [ 71 are all for dwell-times smaller or equal to 4 fs where, again, calculations show nothing but a small sensitivity upon the occurrence of wake forces. One can start speculating about the validity of several assumptions entering our calculations. This however, would not be worth doing in the light of current experimental data, It is evident that new measurements of transmission yields at large dwell-times in carbon foils, i.e., 10 fs or more, and bombarding energies within the range of 0.1 to 0.5 MeV/amu appear as crucial in order to determine whether or not our current understanding of wake-forces and transmission of molecules are both adequate.
13
Acknowledgements
The author would like to thank the A. von Humboldt Foundation for providing financial support during earlier stage of this work. Discussions with Dr. N.E. Capuj are grateful acknowledged.
References
[l] D. Zajfman, Phys. Rev. A 42 (1990) 5374. 121Z. Vager and D. Gemmel, Phys. Rev. Lett. 37 (1976) 1352. [ 31 M.M. Jakas and N.E. Capuj, Nucl. Instr. and Meth. B 93 (this issue) ( 1994) 14. [4] P. Echenique, R. Ritchie and W. Brandt, Phys. Rev. B 20 (1979) 2567. [51 N. Cue, N.V. de Castro-Faria, M.J. Gaillard, J.-C. Poizat, J. Remillieux, D.S. Gemmel and I. Plesser, Phys. Rev. Lett. 45 (1980) 613. [ 61 A. Faibis, R. Kaim, I. Plesser and Z. Vager, Nucl. Instr. and Meth. 170 ( 1980) 99. [7] T.R. Fox, K. Lamb and R.Levi-Seti, NucLInstr. and Meth. 194 (1982) 285. See also R. Levi-Seti, K. Lamb and T.R. Fox, ibid. p. 281.
and Methods in Physics Research B 93 ( 1994) 1l-1 3 Bum tntsraotion8 Materials & Atoms
with
Wake effects on the transmission of fast Hz -ions through thin carbon foils Mario
M. Jakas
D~~a~t~~ent~ de F&X Funda~entai y Expe~i~e~tal~ Universidad de La Lagma, 382Oj La ,kguna, Ten&@, spain Received 21 January 1994 and in revised form 25 March 1994
According to Monte Carlo calculations the transmission of 0.1-1.0 MeV/amu H: molecular ions through thin carbon foils show a particularly high sensitivity upon the presence of wake forces. Our results also disagree with previous evidences pointing towards a nearly universal curve for the transmission yield as a function of dwell-time in the foil. Existing experiments however do not actually allow us to check for the validity of our calculations.
We have calculated the transmission yield of energetic H: molecular ions through thin carbon foils. Our Monte Carlo code is similar to that in ref. [ I 1. Calculations in this Note, however, have for the first time included the wake forces. Our Monte Carlo also includes an advanced technique, described in details elsewhere [ 3 1, which speeds up simulations to the point of rendering calculations of this type quite affordable even for a small personal computer. For the sake of completeness, we briefly mention that our MC follows the motion of the two nuclei in the H! molecule as one “reduced” particle subjected to the Coulomb repulsion, energy loss straggling, multiple scattering with target atom and wake forces. Energy loss is purposely excluded since it cannot modify the relative motion of the nuclei. Similarly, one assumes that right upon penetration the nuclei become dissociated, i.e., the binding electron is lost. Whereas capture of an electron into a binding state can occur only at the exit of the foil. The wake forces used in our calculations are obtained from an approximate expression given in ref. [2]. Although such forces slightly differ from exact calculations [ 41, they seem acceptable at this first attempt to include wake forces in transmission yield calculations. In order to cancel the probability of the electronic capture out of the transmission yield, it is customary to introduce the relative transmission yield, i.e., Y/2&, where Y is the transmission yield and, #0 is the neutral fraction for protons at same velocity. The relative transmission yield, therefore, can be seen as the transmission yield if the electron capture into the binding state occur with a probability of one. In Fig. 1 we show the results of our MC calculations both with- and without wake forces. As one can clearly see, by including the wake forces it removes the apparent insensitivity with the bombarding energy
showed by the transmission yield in the cases where wake forces are not present. For 0.1 MeV/amu bombarding energy, the yields obtained including wake forces become larger than those where wake forces are not. Whereas for 1.O and 0.4 MeV/amu the yields go well below the no-wake counte~a~. An interesting result however appears in the case of 0.4 Mev/amu. Here the transmission yield begins to increase around 10 fs, reaching the values of no-wake
Dwell
Time
(fs)
Fig. 1. Relative transmission yield for Hz -ions as a function of dwell-time in carbon foils. Experiments (crosses); Monte Carlo calculations: Without wake forces (full symbols); with wake forces (open symbols). Bombarding energies are in MeV/amu.
0168-583X/94/$07.00 @ 1994 - Elsevier Science B.V. All rights reserved SSDIO168-583X(94)00189-3
M.M. Jakas / Nucl. Instr. and Meth. in Phys. Res. B 93 (1994) 11-13
12
3
0
(a)
E = 0.1
(b)
E = 1 .O MeV/amu
2
Internuclear
MeV/amu
4
distance
J 8
6
(a.u.)
Fig. 2. Interatomic potential for proton fragments during passage through the foil, i.e. wake plus screened Coulomb potential. The internuclear axis is parallel-to- (continuous line), and perpendicular-to-the-beam (long-dashed line), respectively. Initial distribution of internuclear distance (short-dashed line). cases at approximately 20 fs. Although the origin of such an increase has not been investigated in detail, it seems related to the fact that at large dwell-times the multiple scattering gets more developed and so the occurrence of wake forces becomes less relevant. In this regard, such a behavior might also appear in the case of 1.O MeV/amu. However, as multiple scattering results smaller with increasing energy such an effect cannot be seen within dwell-times smaller than 30 fs. The strong influence of the wake forces upon transmission yields can be readily understood by looking at the interatomic potential of the nuclei. In Figs. 2a and 2b we plot the potential energy of the two nuclei as a function of the internuclear position. For a comparison, the initial distribution of internuclear separations is also plotted in the same figures. We can see that for molecules oriented perpendicular-to-the-beam the wake forces do not appreciably change the Coulomb repulsion. However for those nuclei which have their axis aligned with the beam direction the wake potential exhibits a strong dependence with bombarding energy. In fact, for 1.0 MeV/amu the wake potential translates into a much stronger re-
pulsion between the nuclei, as the nearest minimum appears far apart from the initial distribution of the molecules. For 0.1 MeV/amu, however, a significant part of the incoming nuclei find themselves into the potential well. A well which, on the other hand, also occurs along lateral displacements starting from wellaligned positions. Accordingly, one can expect that for 0.1 MeV/amu molecules the transmission yields should benefit from the wake forces. Whereas at 1.0 MeV/amu the wake potential, as mentioned before, actually adds up a repulsive force, thus making more difftcult for the nuclei to reach the region in the configuration space where recombination is possible. One can be tempted to think that as number of molecules which are subjected to the wake “attraction” is pretty small, i.e. lo%, then the wake-effect cannot be expected to be greater than 10%. This reasoning however is in error. First, although it is true that only a small fraction of the incoming molecules result within the “well”, it does not imply that nuclei sitting off the well will be less influenced by wake forces. For a nucleus initially off the well it will be accelerated towards regions of smaller potential. In this way its kinetic energy will get increased and consequently, the possibility of recombination reduced. Second, one must recall that the transmission of molecules relates to an exceedingly particular sub-set of trajectories, i.e. those leading to bound clusters. Therefore, any mechanism which can modify such histories may have a dramatic consequences upon recombination, even though it amounts to a small perturbation upon the whole set of trajectories. For example, the fact that at 0.1 MeV/amu the wake-well becomes both deeper and closer to the origin compared to those of higher energies appears to make a substantial difference here. It is evident that at 0.1 MeV/amu the particles can be more easily “confined” within the walls of the wake-well. Then, the population of low relative-velocity increases due to particles which are either approaching to or bouncing off the potential wall. In addition these nuclei become nearly “trapped” within distances where the binding potential becomes larger (in absolute sense) and therefore recombination is highly favored. Notice that this is not possible at higher energies, because most of the nuclei initiate penetration sitting far away from the bottom of the wake-well. Then, as mentioned above they will be subjected to an additional force where otherwise, i.e. with no wake, there was only the nearly vanishing force in the tail of the Coulomb repulsion. It is rather striking however, that the experiments appear closer to calculations obtained without wake forces rather than to those where the wake is “on”. Moreover, the sensitivity with energy resulting from
MM. Jakas I Nucl. Instr. and Meth. in Phys. Res. B 93 (1994) 11-13
our calculation is not observed in the experiments, which, as mentioned elsewhere [ 51 appear to follow a “universal curve”. It should be mentioned that the expe~ments shown in Fig. 1 cover a range of bombarding energy between 0.4 and 1.0 MeV/amu, while the ones at much larger dwell times are for 0.5 , 0.75 and 1.0 MeV/amu, respectively. Moreover, the wake model used in our MC is the same as the one previously used to calculate the energy-angle spectra of dissociated fragments 161, for which calculations compare remarkably well with experiments. The discrepancy between our calculations and the experiments pointed out above does not necessarily imply a contradiction. In fact, observe that at large dwell-times only high-energy data are available and, deviations between these high energy data and our calculations are not conclusive. This is particularly so, after considering the difficulties associated with measuring such low transmission yields. Second, the results for 0.4 MeV/amu do not reach large dwelltimes where wake effects become apparent. In the same manner as the low energy data in ref. [ 71 are all for dwell-times smaller or equal to 4 fs where, again, calculations show nothing but a small sensitivity upon the occurrence of wake forces. One can start speculating about the validity of several assumptions entering our calculations. This however, would not be worth doing in the light of current experimental data, It is evident that new measurements of transmission yields at large dwell-times in carbon foils, i.e., 10 fs or more, and bombarding energies within the range of 0.1 to 0.5 MeV/amu appear as crucial in order to determine whether or not our current understanding of wake-forces and transmission of molecules are both adequate.
13
Acknowledgements
The author would like to thank the A. von Humboldt Foundation for providing financial support during earlier stage of this work. Discussions with Dr. N.E. Capuj are grateful acknowledged.
References
[l] D. Zajfman, Phys. Rev. A 42 (1990) 5374. 121Z. Vager and D. Gemmel, Phys. Rev. Lett. 37 (1976) 1352. [ 31 M.M. Jakas and N.E. Capuj, Nucl. Instr. and Meth. B 93 (this issue) ( 1994) 14. [4] P. Echenique, R. Ritchie and W. Brandt, Phys. Rev. B 20 (1979) 2567. [51 N. Cue, N.V. de Castro-Faria, M.J. Gaillard, J.-C. Poizat, J. Remillieux, D.S. Gemmel and I. Plesser, Phys. Rev. Lett. 45 (1980) 613. [ 61 A. Faibis, R. Kaim, I. Plesser and Z. Vager, Nucl. Instr. and Meth. 170 ( 1980) 99. [7] T.R. Fox, K. Lamb and R.Levi-Seti, NucLInstr. and Meth. 194 (1982) 285. See also R. Levi-Seti, K. Lamb and T.R. Fox, ibid. p. 281.