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(e,2e) ionization of helium leading to the n=2 final state under asymmetric Bethe-ridge conditions
Journal of Electron Spectroscopy and Related Phenomena, 58 (1992 ) 17-22 Elsevier Science Publishers B.V., Amsterdam
17
(e,Ze) ionization of helium leading to the n- 2 final state under asymmetric Bethe-ridge conditions A. Lahmam-Bennani”, A. Duguet*, C. Dupre” and C. Dal Cappellob !‘Laboratoire des CoZlisionsAtomiques et Mokkuluires (URADO281). B&t. 351, Uniuersitk Paris XI, 91405 Orsay C&2x (France) bLaboratoire de Physique Mole’culaire et de Collisions, Institut de Physique, Rue Arago, Technopole 2000,57070 Metz (France) (First received 25 March 1991; in final form 11 September 1991)
Abstract The (e,2e) triple differential cross-sections for ionization leading to the ground ( n = 1) and first excited (n = 2 ) states of the ion are measured under high energy asymmetric Bethe-ridge kinematics, at scattered and ejected electron energies of 5506 and 75 eV respectively. With respect to older experiments, the present data are free from electron distortion effects. Hence, when compared with calculations using different correlated wavefunctions, it is found that the best agreement for the n= 1 to n=2 ratio is obtained with the variationally best wavefunction in energy space. This is in contrast with the experimental findings of Cook et al. [J.P.D. Cook, I.E. McCarthy, A-T. Stelbovics and E. Weigold, J. Phys. B, 17 (1984) 2339 J and supports the theoretical arguments given by Smith et al. [A.D. Smith, M.A. Coplan, D.J. Chomay, J.H. Moore, J.A. Tossell, J. Mroxek, V.H. Smith, Jr. and S. Chant, J. Phys. B, 19 (1986) 9691.
INTRODUCTION
(e,2e ) studies have mostly been confined to processes leaving the residual ion in its ground state. However, a new interest has recently developed in considering cases where the ion is left in an excited final state. Such processes are much more sensitive to correlation effects present both in the initial and the final state of the collision and hence are of prime importance towards a better understanding of these effects. Also, these processes which involve two active “atomic” electrons, one being ionized and the second one being promoted to some empty orbital, bear a strong resemblance to (e,3e) processes where two electrons are brought into the ionization continuum. Therefore, they should constitute a very valuable bridge between the widely investigated (e,2e) domain and the almost unexplored, experimentally and theoretically challenging (e,3e ) field. Recently, Stefani et al. [ 1 ] and Duprt5et al. [2] used intermediate and high
03682048/92/$05.06
0 1992 Elsevier Science Publishers B-V. All rights reserved.
18
impact energy and an asymmetric geometry to study the process He +e + He+ (n = 2 ) + 2e, under dipolar kinematics. They reported new effects in the triple differential cross-section (TDCS ) angular distributions, not present in the corresponding n = 1 distributions. These effects were ascribed to final state interactions between the residual excited ion and the slow ejected electron, mostly due to the larger spatial extent of the n = 2 ionic potential. They were shown to strongly depend on the velocity of the ejected electron, and to vanish at high enough velocities. Earlier studies by McCarthy et al. [3], Dixon et al. [ 41 and Cook et al. [ 51 used intermediate energy (about 1 keV) and a non-coplanar symmetric arrangement (so-called electron momentum spectroscopy, EMS) to investigate the target initial state electron correlations, and to discriminate between various helium atom wavefunctions of different sophistication levels. Larkins [ 61 and Larkins and Richards [ 71 computed within the framework of the plane wave impulse approximation (PWIA) the ratio of (e,2e) TDCSs leading to the ground and first excited states of the helium ion. They assumed that the distortion effects were small at the impact energy of 1200 eV, and furthermore that if they did exist they were the same for the n= 1 and n = 2 final states, and hence they would cancel in the cross-section ratio, Based on this assumption, measurements of this ratio were compared by Cook et al. [ 5 ] with these PWIA calculations. Good agreement was reported with the values obtained using the correlated wavefunction of Joachain and Vanderpoorten [8] which accounts for 98% of the correlation energy. Next best was the simple five-term CI wavefunction of Taylor and Parr [9] which accounts for 85% of the correlation energy. Surprisingly, a far inferior fit to the data was provided by the elaborate 20-term CI wavefunction of Nesbet and Watson [lo] which yields 97.7% of the correlation energy. However, this point was criticized by Smith et al. [ 111 who showed that the apparent failure of this complicated wavefunction is an artefact of the method of analysis of the (e,2e) data. Performing distorted waves impulse approximation (DWIA) calculations, they found that electron distortion effects are larger on the n= 2 cross-sections than on the corresponding n= 1 cross-sections at high momentum, and hence do not cancel when taking the 2/l ratio. The differences observed by Cook et al. [ 51 between the various correlated wavefunctions are of the same order as the differences between the DWIA and the PWIA calculations. On the grounds of these arguments, Smith et al. [ll] corrected for the distortion of Cook et al’s data. By doing so, they found better agreement with the Nesbet and Watson [ 10 ] wavefunction. Hence, they concluded that, “the criterion that the variationally best wavefunction is the one most likely to be best as a function of distance probably remains intact”. It is the aim of this paper to provide new insight into this important question. For this purpose, advantage is taken of the high energy asymmetric Betheridge kinematics where distortion effects are expected to be small [ 12 ], and the measured TDCS angular distribution reduces to an electron momentum
19
density within the framework of the PWIA. Hence, direct comparison can be made with the various theoretical wavefunctions, without the need to introduce more or less reliable corrections to the data. EXPERIMENTAL
The electron-electron coincidence spectrometer used for the present experiments has been extensively described [ 13-15 1. Briefly, a well-collimated beam of monochromatic electrons crosses an effusive gaseous beam at right angles. The 127” cylindrical, high energy electron analyzer is kept fixed, while the electron gun and the MOohemispherical, low energy electron analyzer rotate independently around the scattering centre, thus defining the scattering angle 0, and the ejection angle &,. After angle and energy selection, only those pairs of electrons coincident in time are registered. Owing to the large difference in cross-sections between the n = 1 and n = 2 cases (a ratio of about 200) and the subsequent low coincidence rates for n = 2, it was necessary to operate at modest energy and angular resolutions. The acceptance solid angles were AS& N 2 X 10m5 sr, A&, N 2 x low2 sr, and the effective coincidence energy resolution [ 141 was A& N 4.6 eV, resulting in a momentum resolution &x 0.2 a.u. This value is reasonably low so as not to seriously distort the measured electron momentum distribution (see below). Besides, both the n = 1 and the n= 2 distributions are affected in a similar (though not identical} manner, the effect of a finite momentum resolution being an apparent decrease in intensity at p= 0 and an increase at large p values. However, we are here mostly interested in the n = 2 to n= 1 ratio and hence, to first order, these effects essentially cancel, and only second-order differences may be expected to remain. Another influence of the modest energy resolution is twofold. Firstly, due to the high energy tail of the resolution function, a small fraction, about O.l%, of the n= 1 ground state intensity is present with the observed n = 2 intensity. This effect was discussed in detail [ 2 ] and was corrected for by measuring a ground state angular distribution under angles and energies identical to those for n= 2 and subtracting the corresponding fraction from the n = 2 measurements. Such correction amounts to lo-20% of the recorded n= 2 intensity. Similarly, the observed n = 2 signal also includes unresolved contributions from n = 3, . . . up to 00 (double ionization limit ) final states. From a deconvolution procedure [ 21, these contributions were estimated to amount to less than 5% of the measured intensity and were therefore neglected. For both final states n= 1 and n= 2, an angular distribution of the ejected electron was mapped, keeping the scattering angle 9, constant. The ejected electron energy was fixed at Eb - 75 eV, and the momentum transfer value was fixed at K=2.35 a.u. which corresponds to the bound-electron Bethe-ridge condition [ 121, K= (2&) t. These choices were dictated, (i) by the requirement of a large enough Eb value so that the final state interactions between
the ejected electron and the residual ion have practically vanished [ 2 1, but also (ii) by a small enough I$, value so that the n = 2 cross-section is not unreasonably small, and (iii) by the need for the TDCS to be described within the framework of the PWIA so that it can be reduced to an electron momentum density [M-18]. Though the ejected and scattered electron energies are the same for both ground and excited He+ final states experiments (75 and 5500 eV respectively ) , the amount of energy transferred to the target upon collision is different in both cases. This induces a slight difference in the incident energy, E$dL;J$j99.6 and Eg=’ =5640.3 eV, and in the scattering angle OEE1= 6.61” a =6.64”. RESULTS AND DISCUSSION
Figure 1 shows the measured n= 1 and n= 2 TDCS angular distributions. They are compared with first-Born OCW calculations where the ejected electron is described using a Coulomb wave, with an effective charge Z= 1 (full screening of the remaining bound electron) [ 16,191. The data are measured only on a relative scale; hence they have been normalized to the maximum of the n = 1 theoretical results. This simultaneously brings onto an absolute scale both experimental n-- 1 and n = 2 distributions since they are obtained on the same relative scale. Very good agreement is observed between theory and experiment. This is a well-established conclusion for the ionization to the ground state. As a matter of fact, the OCW/Z = 1 model has been extensively shown [ 16,18,20] to reproduce very well the He (n= 1) TDCS at the binary lobe, in the energy regime considered here. Therefore, the observed good agreement in Fig. 1, (i) rules out major systematic experiment& &facts which might distort the measured angular distributions, and (ii) confirms that the effect of the finite angular
0
45 Ejection
90 angle, eb
135
180
(den)
Fig. 1. Experimental and first-Born OCW/Z= 1 theoretical triple differential cross-sections for helium leading to the n = 1 (open circles and full line ) and n= 2 (filled circles and broken line ) ion states plotted as a function of ejection angle 0,,. Kinematic parameters are: Em=5500 eV, I$,=75 eV and Kz2.35 a.u. (Bethe-ridge condition).
21
resolution on the measured distributions is negligible. Moreover, an n = I PWIA calculation was also performed (not shown in Fig. 1) and it was found to be indistinguishable from the OCW one when renormalized at the maximum, thus confirming our hypothesis of neglect of electron distortion effects, especially in the tails of the distribution which correspond to large p values. In contrast Dupr6 et al, [ 21 showed that at low E,.,values the shape of the G= 2 distribution is severely distorted by the electron-ion interactions in the final state, and that the distortions tend to vanish with increasing Eb values, and the binary lobe of the n=2 distribution tends to be well described by an OCW/Z= 1 model just as in the n= 1 case. Therefore, the good agreement observed in Fig. 1 between n = 2 experiments and theory suggests that the final state interactions have become negligible under the present kinematics, and that both n= 1 and n= 2 ionization processes are essentially governed by the same dynamics. The n = 2 to n = 1 cross-section ratio is plotted in Fig. 2 against initial momentum p. The data of Cook et al. [ 51 at 1200 eV are substantially higher than ours in the large p range, which was attributed by Smith et al. [ 111 to electron distortion effects present in these experiments. Our data are free from, or at least much less contaminated by, these effects. The horizontal dotted line shows the cross-section ratio obtained using the Hartree-Fock (HF) helium ground state wavefunction. As stated by previous authors, this ratio is independent of p. Using correlated wavefunctions the ratio needs no longer to be con&ant. Four correlated wavefunctions of different quality (in terms of energy) have been considered. The two best ones, due to Joachain and Vanderpoorten [8] (JV) and Nesbet and Watson [lo] (NW), which account for 97.7% and 98% of the correlation energy respectively, give excellent fits tq the data over the entire investigated p range. Conversely, the simplest wavefunction, proposed
i; 0.06 -5 :: .z
0.04
f
i
0.02
1 6 1
P
I--)
2
3
Fig. 2. Cross-section ratio for the n = 2 to n = 1 ionization of helium plotted as a function of the momentum p. Full circles, present data, open circles, Cook et al.% data [ 5 1. film shown for comparison are the theoretical ratios obtained using the Hartree-Fock helium ground-state wavefunction (HF, *.....), and accurate correlated wavefunctions: TL ( ), TP (- m-e-), (----) (seetext). JV (---)andNW
22
by Taylor and Parr [ 9 ] (TP, 85% of the correlation energy) substantially overestimates the ratio at all p values. The multiconfiguration variational wavefunction of Tweed and Langlois [ 211 (TL, 94% of the correlation energy) significantly overestimates the ratio in the large momentum region, but gives better agreement with the data than the JV and NW functions in the low momentum region. In conclusion, our present measurements support the arguments theoretically invoked by Smith et al. [111todemonstrate that the best wavefunction in terms of energy is still the best one when electron correlations are investigated in the momentum space. Previous (e,2e) experiments on helium at 1200 eV impact energy are definitely affected by electron distortions at large p values, while the present ones are not. This again confirms the superiority of the high energy asymmetric geometry under Bethe-ridge kinematics for doing (e,2e) spectroscopy.
REFERENCES 1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20 21
E. Stefani, L. AvaIdi and R. CamiIloni, J. Phys. B, 23 (1990) L227. C. Dupre, A. Lahmam-Bennani, A. Duguet, F&Mota-Furtado, P.F. O’Mahony and C. Dal CappeIlo, J. Phys. B, 24 (1991) in press. I.E. McCarthy, A. Ugbabe, E. Weigold and P.J.O. Teubner, Phys. Rev. Lett., 33 (1974) 459. A.J. Dixon, I.E. McCarthyand E. Weigoid, J. Phys. B, 9 (1976) L195. J.P.D. Cook, I.E. McCarthy, A.T. Stelbovics and E. Weigold, J. Phys. B, 17 (1984) 2339. F.P. Larkins, J. Phys. B, 14 (1981) 1477. F.P. Larkins and J.A. Richards, Chem. Phys., 81 (1983) 329. C.J. Joachain and R. Vanderpoorten, Physica (Utrecht), 46 (1970) 333. G.R. Taylor and R.G. Parr, Proc. Natl. Acad. Sci. U.S.A., 38 (1952) 154. R.K. Nesbet and R.E. Watson, Phys. Rev., 110 (1958) 1073: A.D. Smith, M.A. Coplan, D.J. Chornay, J.H. Moore, J.A. Tossell, J. Mrozek, V.H. Smith, Jr. and S. Chant, J. Phys. B, 19 (1986) 969, A. Lahmam-Bennani, A. Duguet and C. Dal Cappeho, J. Electron Spectrosc. Relat. Phenom., 40 (1986) 141. M. Ch&id, A. Lahmam-Bennani, A. Duguet, R.W. Zurales, R.R. Lucchese, M.C. Dal Cappello and C. Dal CappelIo, J. Phys. B, 22 (1989) 3483. A. Duguet, M. Ch&id, A. Lahmam-Bennani, A. Franz and H. Klar, J. Phys. B, 20 (1987) 6145. A. Lahmam-Bennani, H.F. Wellenstein, A. Duguet and M. Lecas, Rev. Sci. Instrum., 56 (1985) 43. A. Lahmam-Bennani, H.F. Wellenstein, C. Dal Cappello andA. Duguet, J. Phys. B, 17 ( 1984) 3159. L. Avaldi, R. Camilioni, E. Fainelli, G. Stefani, A. Franz, H. Klar and I.E. McCarthy, J. Phys. B, 20 (1987) 5827. A. Lahmam-Bennani, L. Avaldi, E. Fainelli and G. Stefani, J. Phys. B, 21 ( 1988) 2145. M.J. Brothers and R.A. Bonham, J. Phys. B, 17 (1984) 4235. A. Lahmam-Bennani, H.F. Wellenstein, C. Dal Cappello, M. Rouault and A. Duguet, J. Phys. B, 16 (1983) 2219. R.J. Tweed and J. Langlois, J. Phys. B, 20 (1987) 5213.
17
(e,Ze) ionization of helium leading to the n- 2 final state under asymmetric Bethe-ridge conditions A. Lahmam-Bennani”, A. Duguet*, C. Dupre” and C. Dal Cappellob !‘Laboratoire des CoZlisionsAtomiques et Mokkuluires (URADO281). B&t. 351, Uniuersitk Paris XI, 91405 Orsay C&2x (France) bLaboratoire de Physique Mole’culaire et de Collisions, Institut de Physique, Rue Arago, Technopole 2000,57070 Metz (France) (First received 25 March 1991; in final form 11 September 1991)
Abstract The (e,2e) triple differential cross-sections for ionization leading to the ground ( n = 1) and first excited (n = 2 ) states of the ion are measured under high energy asymmetric Bethe-ridge kinematics, at scattered and ejected electron energies of 5506 and 75 eV respectively. With respect to older experiments, the present data are free from electron distortion effects. Hence, when compared with calculations using different correlated wavefunctions, it is found that the best agreement for the n= 1 to n=2 ratio is obtained with the variationally best wavefunction in energy space. This is in contrast with the experimental findings of Cook et al. [J.P.D. Cook, I.E. McCarthy, A-T. Stelbovics and E. Weigold, J. Phys. B, 17 (1984) 2339 J and supports the theoretical arguments given by Smith et al. [A.D. Smith, M.A. Coplan, D.J. Chomay, J.H. Moore, J.A. Tossell, J. Mroxek, V.H. Smith, Jr. and S. Chant, J. Phys. B, 19 (1986) 9691.
INTRODUCTION
(e,2e ) studies have mostly been confined to processes leaving the residual ion in its ground state. However, a new interest has recently developed in considering cases where the ion is left in an excited final state. Such processes are much more sensitive to correlation effects present both in the initial and the final state of the collision and hence are of prime importance towards a better understanding of these effects. Also, these processes which involve two active “atomic” electrons, one being ionized and the second one being promoted to some empty orbital, bear a strong resemblance to (e,3e) processes where two electrons are brought into the ionization continuum. Therefore, they should constitute a very valuable bridge between the widely investigated (e,2e) domain and the almost unexplored, experimentally and theoretically challenging (e,3e ) field. Recently, Stefani et al. [ 1 ] and Duprt5et al. [2] used intermediate and high
03682048/92/$05.06
0 1992 Elsevier Science Publishers B-V. All rights reserved.
18
impact energy and an asymmetric geometry to study the process He +e + He+ (n = 2 ) + 2e, under dipolar kinematics. They reported new effects in the triple differential cross-section (TDCS ) angular distributions, not present in the corresponding n = 1 distributions. These effects were ascribed to final state interactions between the residual excited ion and the slow ejected electron, mostly due to the larger spatial extent of the n = 2 ionic potential. They were shown to strongly depend on the velocity of the ejected electron, and to vanish at high enough velocities. Earlier studies by McCarthy et al. [3], Dixon et al. [ 41 and Cook et al. [ 51 used intermediate energy (about 1 keV) and a non-coplanar symmetric arrangement (so-called electron momentum spectroscopy, EMS) to investigate the target initial state electron correlations, and to discriminate between various helium atom wavefunctions of different sophistication levels. Larkins [ 61 and Larkins and Richards [ 71 computed within the framework of the plane wave impulse approximation (PWIA) the ratio of (e,2e) TDCSs leading to the ground and first excited states of the helium ion. They assumed that the distortion effects were small at the impact energy of 1200 eV, and furthermore that if they did exist they were the same for the n= 1 and n = 2 final states, and hence they would cancel in the cross-section ratio, Based on this assumption, measurements of this ratio were compared by Cook et al. [ 5 ] with these PWIA calculations. Good agreement was reported with the values obtained using the correlated wavefunction of Joachain and Vanderpoorten [8] which accounts for 98% of the correlation energy. Next best was the simple five-term CI wavefunction of Taylor and Parr [9] which accounts for 85% of the correlation energy. Surprisingly, a far inferior fit to the data was provided by the elaborate 20-term CI wavefunction of Nesbet and Watson [lo] which yields 97.7% of the correlation energy. However, this point was criticized by Smith et al. [ 111 who showed that the apparent failure of this complicated wavefunction is an artefact of the method of analysis of the (e,2e) data. Performing distorted waves impulse approximation (DWIA) calculations, they found that electron distortion effects are larger on the n= 2 cross-sections than on the corresponding n= 1 cross-sections at high momentum, and hence do not cancel when taking the 2/l ratio. The differences observed by Cook et al. [ 51 between the various correlated wavefunctions are of the same order as the differences between the DWIA and the PWIA calculations. On the grounds of these arguments, Smith et al. [ll] corrected for the distortion of Cook et al’s data. By doing so, they found better agreement with the Nesbet and Watson [ 10 ] wavefunction. Hence, they concluded that, “the criterion that the variationally best wavefunction is the one most likely to be best as a function of distance probably remains intact”. It is the aim of this paper to provide new insight into this important question. For this purpose, advantage is taken of the high energy asymmetric Betheridge kinematics where distortion effects are expected to be small [ 12 ], and the measured TDCS angular distribution reduces to an electron momentum
19
density within the framework of the PWIA. Hence, direct comparison can be made with the various theoretical wavefunctions, without the need to introduce more or less reliable corrections to the data. EXPERIMENTAL
The electron-electron coincidence spectrometer used for the present experiments has been extensively described [ 13-15 1. Briefly, a well-collimated beam of monochromatic electrons crosses an effusive gaseous beam at right angles. The 127” cylindrical, high energy electron analyzer is kept fixed, while the electron gun and the MOohemispherical, low energy electron analyzer rotate independently around the scattering centre, thus defining the scattering angle 0, and the ejection angle &,. After angle and energy selection, only those pairs of electrons coincident in time are registered. Owing to the large difference in cross-sections between the n = 1 and n = 2 cases (a ratio of about 200) and the subsequent low coincidence rates for n = 2, it was necessary to operate at modest energy and angular resolutions. The acceptance solid angles were AS& N 2 X 10m5 sr, A&, N 2 x low2 sr, and the effective coincidence energy resolution [ 141 was A& N 4.6 eV, resulting in a momentum resolution &x 0.2 a.u. This value is reasonably low so as not to seriously distort the measured electron momentum distribution (see below). Besides, both the n = 1 and the n= 2 distributions are affected in a similar (though not identical} manner, the effect of a finite momentum resolution being an apparent decrease in intensity at p= 0 and an increase at large p values. However, we are here mostly interested in the n = 2 to n= 1 ratio and hence, to first order, these effects essentially cancel, and only second-order differences may be expected to remain. Another influence of the modest energy resolution is twofold. Firstly, due to the high energy tail of the resolution function, a small fraction, about O.l%, of the n= 1 ground state intensity is present with the observed n = 2 intensity. This effect was discussed in detail [ 2 ] and was corrected for by measuring a ground state angular distribution under angles and energies identical to those for n= 2 and subtracting the corresponding fraction from the n = 2 measurements. Such correction amounts to lo-20% of the recorded n= 2 intensity. Similarly, the observed n = 2 signal also includes unresolved contributions from n = 3, . . . up to 00 (double ionization limit ) final states. From a deconvolution procedure [ 21, these contributions were estimated to amount to less than 5% of the measured intensity and were therefore neglected. For both final states n= 1 and n= 2, an angular distribution of the ejected electron was mapped, keeping the scattering angle 9, constant. The ejected electron energy was fixed at Eb - 75 eV, and the momentum transfer value was fixed at K=2.35 a.u. which corresponds to the bound-electron Bethe-ridge condition [ 121, K= (2&) t. These choices were dictated, (i) by the requirement of a large enough Eb value so that the final state interactions between
the ejected electron and the residual ion have practically vanished [ 2 1, but also (ii) by a small enough I$, value so that the n = 2 cross-section is not unreasonably small, and (iii) by the need for the TDCS to be described within the framework of the PWIA so that it can be reduced to an electron momentum density [M-18]. Though the ejected and scattered electron energies are the same for both ground and excited He+ final states experiments (75 and 5500 eV respectively ) , the amount of energy transferred to the target upon collision is different in both cases. This induces a slight difference in the incident energy, E$dL;J$j99.6 and Eg=’ =5640.3 eV, and in the scattering angle OEE1= 6.61” a =6.64”. RESULTS AND DISCUSSION
Figure 1 shows the measured n= 1 and n= 2 TDCS angular distributions. They are compared with first-Born OCW calculations where the ejected electron is described using a Coulomb wave, with an effective charge Z= 1 (full screening of the remaining bound electron) [ 16,191. The data are measured only on a relative scale; hence they have been normalized to the maximum of the n = 1 theoretical results. This simultaneously brings onto an absolute scale both experimental n-- 1 and n = 2 distributions since they are obtained on the same relative scale. Very good agreement is observed between theory and experiment. This is a well-established conclusion for the ionization to the ground state. As a matter of fact, the OCW/Z = 1 model has been extensively shown [ 16,18,20] to reproduce very well the He (n= 1) TDCS at the binary lobe, in the energy regime considered here. Therefore, the observed good agreement in Fig. 1, (i) rules out major systematic experiment& &facts which might distort the measured angular distributions, and (ii) confirms that the effect of the finite angular
0
45 Ejection
90 angle, eb
135
180
(den)
Fig. 1. Experimental and first-Born OCW/Z= 1 theoretical triple differential cross-sections for helium leading to the n = 1 (open circles and full line ) and n= 2 (filled circles and broken line ) ion states plotted as a function of ejection angle 0,,. Kinematic parameters are: Em=5500 eV, I$,=75 eV and Kz2.35 a.u. (Bethe-ridge condition).
21
resolution on the measured distributions is negligible. Moreover, an n = I PWIA calculation was also performed (not shown in Fig. 1) and it was found to be indistinguishable from the OCW one when renormalized at the maximum, thus confirming our hypothesis of neglect of electron distortion effects, especially in the tails of the distribution which correspond to large p values. In contrast Dupr6 et al, [ 21 showed that at low E,.,values the shape of the G= 2 distribution is severely distorted by the electron-ion interactions in the final state, and that the distortions tend to vanish with increasing Eb values, and the binary lobe of the n=2 distribution tends to be well described by an OCW/Z= 1 model just as in the n= 1 case. Therefore, the good agreement observed in Fig. 1 between n = 2 experiments and theory suggests that the final state interactions have become negligible under the present kinematics, and that both n= 1 and n= 2 ionization processes are essentially governed by the same dynamics. The n = 2 to n = 1 cross-section ratio is plotted in Fig. 2 against initial momentum p. The data of Cook et al. [ 51 at 1200 eV are substantially higher than ours in the large p range, which was attributed by Smith et al. [ 111 to electron distortion effects present in these experiments. Our data are free from, or at least much less contaminated by, these effects. The horizontal dotted line shows the cross-section ratio obtained using the Hartree-Fock (HF) helium ground state wavefunction. As stated by previous authors, this ratio is independent of p. Using correlated wavefunctions the ratio needs no longer to be con&ant. Four correlated wavefunctions of different quality (in terms of energy) have been considered. The two best ones, due to Joachain and Vanderpoorten [8] (JV) and Nesbet and Watson [lo] (NW), which account for 97.7% and 98% of the correlation energy respectively, give excellent fits tq the data over the entire investigated p range. Conversely, the simplest wavefunction, proposed
i; 0.06 -5 :: .z
0.04
f
i
0.02
1 6 1
P
I--)
2
3
Fig. 2. Cross-section ratio for the n = 2 to n = 1 ionization of helium plotted as a function of the momentum p. Full circles, present data, open circles, Cook et al.% data [ 5 1. film shown for comparison are the theoretical ratios obtained using the Hartree-Fock helium ground-state wavefunction (HF, *.....), and accurate correlated wavefunctions: TL ( ), TP (- m-e-), (----) (seetext). JV (---)andNW
22
by Taylor and Parr [ 9 ] (TP, 85% of the correlation energy) substantially overestimates the ratio at all p values. The multiconfiguration variational wavefunction of Tweed and Langlois [ 211 (TL, 94% of the correlation energy) significantly overestimates the ratio in the large momentum region, but gives better agreement with the data than the JV and NW functions in the low momentum region. In conclusion, our present measurements support the arguments theoretically invoked by Smith et al. [111todemonstrate that the best wavefunction in terms of energy is still the best one when electron correlations are investigated in the momentum space. Previous (e,2e) experiments on helium at 1200 eV impact energy are definitely affected by electron distortions at large p values, while the present ones are not. This again confirms the superiority of the high energy asymmetric geometry under Bethe-ridge kinematics for doing (e,2e) spectroscopy.
REFERENCES 1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20 21
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