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Simultaneous ionization and excitation of helium by electron impact
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NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 266 (2008) 570–576 www.elsevier.com/locate/nimb
Simultaneous ionization and excitation of helium by electron impact C. Dal Cappello a,*, A.C. Roy b, X.G. Ren c,d, R. Dey e a
Universite´ Paul Verlaine-Metz, Laboratoire de Physique Mole´culaire et des Collisions, ICPMB (FR 2843), Institut de Physique, 1 rue Arago, 57078 Metz Cedex 3, France b Ramakrishna Mission Vivekananda University, Belur Math 711202, West Bengal, India c Department of Physics and Key Laboratory of Atomic and Molecular NanoSciences of MOE, Tsinghua University, Beijing 100084, People’s Republic of China d Max-Planck-Institute for Nuclear Physics, Saupfercheckweg 1, 68117 Heidelberg, Germany e Institute for Plasma Research, Bhat, Gandhinagar 382428, India Received 21 September 2007; received in revised form 16 November 2007 Available online 14 December 2007
Abstract We present numerical results for He (1s2) (e, 2e) He+ reaction process for transitions to the n = 1, 2 and 3 states of He+ for noncoplanar symmetric geometry at incident energies of 1000 and 1600 eV. The calculations are performed using the plane wave impulse approximation (PWIA) and the 3C method (also called the Brauner, Briggs and Klar (BBK) model) that includes post collision interaction and multiple scattering effects. In both the methods we have used the highly correlated configuration interaction wave function for the ground state of helium. A comparison of the present theoretical cross sections with the recent measured data of Ren et al. [X.G. Ren, C.G. Ning, J.K. Deng, G.L. Su, S.F. Zhang, Y.R. Huang, G.Q. Li, Phys. Rev. A 72 (2005) 042718] shows reasonably good agreement. Ó 2007 Elsevier B.V. All rights reserved. PACS: 34.50 Keywords: Triply differential cross sections; Ionization and excitation; 3C approximation
1. Introduction In the recent past, there has been an increasing interest in both the theoretical and experimental studies of simultaneous ionization-excitation phenomenon in few-body systems [1–6]. Watanabe et al. [2] have reported the (e, 2e) experiment on the above phenomenon for the transition of ground state of helium to the n = 2 excited state of He+ at large momentum transfer at the incident energy of 2080 eV in the symmetric non-coplanar geometry. They have also reported the momentum-dependent triply differential cross section (TDCS), also known as the momentum profile within the plane wave impulse approximation (PWIA) [7–9] and pointed out that the ionization-excitation *
Corresponding author. Tel.: +33 03 87 31 58 60; fax: +33 03 87 54 72
57. E-mail address: [email protected] (C. Dal Cappello). 0168-583X/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2007.12.003
event at large momentum transfer is very sensitive to the electron correlation in the target initial state, Although the shapes of cross sections were well reproduced by PWIA at small momentum on using highly correlated wave function for the ground state of helium, considerable discrepancies between experiment and theory remained in magnitude. Later on, in the same year Ren et al. [1] reported the results of their experimental and theoretical momentum profiles for the (e, 2e) reaction in helium for the transitions to the ground n = 1 state and the n = 2 and 3 excited states of He+ at impact energies of 1000 and 1600 eV. They also applied the PWIA method to study for the first time the impact energy dependence of the excited final ion states and confirmed that the PWIA description of ionizationexcitation event was inadequate for large recoil momentum p of the residual ion, especially for p greater than 1.5 a.u. Very recently, the simultaneous ionization and excitation of helium (up to n = 4) has also been investigated in
C. Dal Cappello et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 570–576
coplanar geometries with the asymmetric and symmetric energy-sharing cases [5] for low impact energy, following the pioneering works of Dupre´ et al. [10], Avaldi et al. [11] and Rouvellou et al. [12] for n = 2. Because of the energy degeneracy of the He+(2s) and He+(2p) states, the experimental data were the sum of their respective contributions. However, by also detecting the emitted 2p ? 1s photon in a triple coincidence measurement, Sakhelashvili et al. [4] were able to measure the 2p contribution individually. The theoretical calculations (DWB-RMPS and DWB2-RMPS models using a hybrid distorted wave and R matrix [13] and also the second Born approximation [3,14]) were able to give a reasonably good agreement for the above geometries [5,6]. Generally speaking, these models such as DWB, DWB2 and 3DW [15] describe well the experimental measurements when the momentum vectors of the incident projectile, the scattered projectile and the outgoing electron all lie in the same plane, but not for out-of-plane geometries (see, for instance, Schulz et al. [15] and Foster et al. [16]). In this work the non-coplanar symmetric (e, 2e) measurements on helium for transitions to the n = 1, 2 and 3 final states of He+ are studied at incident energies of 1000 eV and 1600 eV. The objective of this investigation is to apply a theoretical method to analyse and explain the experimental findings of Ren et al. [1] including the impact energy dependences of the momentum profiles of the excited states of He+. Since the PWIA is a one-step process that involves the interaction of the projectile with a target electron, it is unlikely to be adequate for the description of the ejection of a bound electron accompanied by the simultaneous excitation of another bound electron. Keeping that in mind we have applied the 3C method (also called the BBK method) [17,18] which has been extensively applied with success to (e, 2e) processes in atoms and molecules [19–23]. The BBK method contains both the post collision interaction (PCI) among the three Coulomb particles in the final state and multiple scattering effects. This model has a correct asymptotic Coulomb three-body wave function for the ejected and scattered electrons in the residual ion field. The effect of exchange between the scattered and ejected electrons has been incorporated in this model by the Pauli method. The present paper is organised as follows: Section 2 gives a brief description of the two theories, namely the plane wave impulse approximation and the BBK method which have been applied in the present investigation. In Section 3 we discuss and compare our results with the experimental data of Ren et al. [1]. Section 4 contains our conclusions. Atomic units will be used throughout, unless otherwise stated. 2. Theory 2.1. Plane-wave impulse approximation We consider an event in which a high energy electron knocks out a bound electron from the helium target in
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the ground state and the two escaping electrons are detected in coincidence. In this case the triply differential cross section (TDCS) for the transition to the nth state of He+ in the plane wave impulse approximation is given by X d3 rn kske 2 ¼2 rM j F n;l;m ð~ pÞj ; fi dXs dXe dEe ki l;m
ð1Þ
2pg and g ¼ j~k 1~k j where ~ ki, ~ k s and ~ k e denote the with rM ¼ e2pg 1 s e momentum of the incident electron, the scattered electron and the ejected electron, respectively. dXs and dXe are the elements of solid angles for the scattered electron and the ejected electron, respectively and dEe represents the energy interval of the ejected electron. In Eq. (1) a sum over the unresolved ionic excited states is assumed. Here ~ p stands for the recoil momentum of the residual ion and is governed by the conservation of linear momentum
~ ~ ~ p ¼~ ki ~ ks ~ ke ¼ K ke;
ð2Þ
~ is the momentum transfer. where K In the present investigation we have considered a kinematic scheme where the two outgoing electrons have equal energies (Ee = Es) and make polar angles he = hs = 45° with respect to the direction of the incident beam. The magnitude of the momentum p is given by pffiffiffi pffiffiffi p ¼ ½ðk i 2k s Þ2 þ ð 2k s sinðD/=2ÞÞ2 1=2 ; ð3Þ where D/ = /e/sp is the out-of-plane azimuthal angle difference between the two outgoing electrons. In Eq. (1) rM is the half-off-shell Mott scattering cross section [3,7]. F n;l;m ð~ pÞ is known as the overlap integral fi and written as Z 2 ei~p:~r2 n;l;m F n;l;m d~ r ð~ pÞ ¼ d~ r U ð~ r Þ Wð~ r2 ;~ r3 Þ; ð4Þ 2 3 3 fi f 3=2 K2 ð2pÞ where Un;l;m represents the wave function of the residual ion f in the nth state and Wð~ r2 ;~ r3 Þ is the ground state wave function of He. Since the energies of the two particles in the present investigation are not too high, it is not quite valid to describe them as plane waves. Besides, the post collision interaction between the two equi-speed Coulomb particles (although expected to be small) should not be totally ignored. In these circumstances, we have considered another method, called the BBK method which accounts for pairwise Coulomb distortions among the two escaping particles and the nucleus. In the following we give the outlines of this model. 2.2. The BBK approximation The TDCS for the (e, 2e) process is given by d3 rn k s k e X n;l;m 2 ¼2 j T fi j ; dXs dXe dEe k i l;m
ð5Þ
C. Dal Cappello et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 570–576
where the transition-matrix element T n;l;m is fi 1 hWn;l;m j V i j Ui i: 2p f
T n;l;m ¼ fi
ð6Þ
In Eq. (6) Wn;l;m and Ui represent the wave function describf ing the whole system in its final and initial states, respectively. The initial state Ui is a product of an incoming plane wave representing the incident particle with momentum ~ ki and the ground state of helium and can be represented by the asymptotic wave function ~
Ui ¼ eiki ~r1 Wi ð~ r2 ;~ r3 Þ;
ð7Þ
where ~ r1 , ~ r2 and ~ r3 are the positions, relative to the atomic centre, of the projectile and the two bound electrons, respectively. The perturbation Vi ¼
2 1 1 þ þ ; r1 r12 r13
ð8Þ
is the interaction of the projectile with the target atom, where ~ rij ¼ ~ ri ~ rj . The final state wave function Wn;l;m describes the system f consisting of the two outgoing electrons and the residual ion and is written as Wn;l;m ¼ W r1 ;~ r2 ÞUn;l;m ð~ r3 Þ; f f 3C ð~
ð9Þ
ð~ r3 Þ is the wave function of the residual hydrowhere Un;l;m f genic ion He+. In Eq. (9) W r1 ;~ r2 Þ is the BBK wave func3C ð~ tion for the two distinguishable emitted electrons and is given by W r1 ;~ r2 Þ ¼ 3C ð~
1 3=2
ð2pÞ
~ ~ eiks ~r1 eike ~r2 Cðas ; ~ k s ;~ r1 ÞCðae ; ~ k e ;~ r2 Þ
k se ;~ r12 Þ; Cðase ; ~
and a 3C function which consists of a product of three Coulomb distortion factors (one for each two-body Coulomb interaction) to describe the final double-continuum state. He+ ion states for n = 1, 2 and 3 are represented by hydrogenic wave functions which are known exactly. We have compared our (e, 2e) cross sections with the corresponding experimental momentum profiles of Ren et al. [1] for the above-mentioned transition processes. Calculations have been performed for non-coplanar symmetric geometry, i.e. h1 = h2 = 45° and Es = Ee. Since the experimental cross sections are relative they are normalised to the theoretical cross sections at the peak for each transition process at each incident energy. This will, of course, enable us to make a comparative study of the shapes of cross sections. Fig. 1 shows the present TDCS obtained in the BBK and PWIA methods at the incident energy of 1600 eV for the transition to the ground state of He+. Also included in this figure is the corresponding experimental momentum profile of Ren et al. [1]. We see that both the theoretical methods predict nearly identical cross sections in the momentum region below 1 a.u. The shapes of these cross sections are almost indistinguishable from the shape of the experimental profile. However, they differ from each other and also from experiment at larger momentum. The shape of BBK cross sections is seen to be conspicuously better than that of PWIA calculation. Fig. 2 displays the results of the same reaction process but at the lower incident energy of 1000 eV. We notice that with the decrease in impact energy both the BBK and measured cross sections increase for the same fixed momentum while the shapes remain more or less the same. The present BBK and PWIA cross sections for (e, 2e) reactions in helium for the transitions to n = 2 and n = 3
ð10Þ
k ~ rþ k;~ rÞ ¼ Cð1 iaÞ expð p2 aÞ1 F 1 ðia; 1; ið~ where Cða;~ Z krÞÞ, as ¼ ks , ae ¼ kZe , ase ¼ 2k1se and ~ k se ¼ 12 ð~ ks ~ k e Þ. Here Z is the charge of the residual ion and 1F1 represents the confluent hypergeometric function. In the present work, we have considered the spatially symmetric part of the above 3C wave function W r1 ;~ r2 Þ; thus exchange between the two emitted elec3C ð~ trons is included in accordance with the Pauli principle.
1E-3
TDCS (a.u.)
572
1E-4
1E-5
3. Results and discussion We have evaluated the TDCS for electron impact ionization of helium at incident energies of 1000 and 1600 eV for the transitions to the He+ n = 1, 2 and 3 states. The present calculations are performed using the BBK model and PWIA methods. In both methods we have used the highly correlated CI function of Nesbet and Watson [24] for the ground state of helium. The energy of the ground state is E = 2.90276 a.u. differing from the experimental value by 0.001 a.u. Whereas in the PWIA both the incident and the scattered particles are represented by plane waves, the BBK model uses a plane wave for the incident projectile
1E-6 0
1
2
3
Momentum (a.u.)
Fig. 1. Comparison of present triply differential cross section for (e, 2e) reaction in He as a function of the momentum of the residual ion for the transition to the ground state of He+ with the measurements of Ren et al. [1] at the incident energy of 1600 eV. The relative data of Ren et al. [1] have been normalised at the peak to the present theoretical value. The solid line represents the present calculation using the BBK model. The dotted curve represents the present plane wave impulse approximation (PWIA) calculation. The solid squares are the experimental data.
C. Dal Cappello et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 570–576
573
1E-6
TDCS (a.u.)
TDCS (a.u.)
1E-3
1E-4
1E-7
1E-8
1E-5 0
0
2
1
Momentum (a.u.)
TDCS (a.u.)
1E-5
1E-6
1E-7 1
3
Fig. 5. Same as in Fig. 3, but for the transition to the n = 3 excited state of He+.
Fig. 2. Same as in Fig. 1, but with an incident energy of 1000 eV.
0
2
Momentum (a.u.)
2
3
Momentum (a.u.)
Fig. 3. Same as in Fig. 1, but for the transition to the n = 2 excited state of He+.
states of residual He+ are displayed in Figs. 3–6. Both the present BBK and PWIA calculations show that the contributions from the sublevels other than the s levels are too low and are at least of one order smaller than those of the corresponding s level. This confirms the calculations of Cook et al. [25]. Since these sublevels are energetically unresolved, we have compared the summed cross sections with the experimental momentum profiles. Figs. 3 and 4 compare the present TDCS with the experimental momentum profiles for n = 2 at the impact energies of 1600 and 1000 eV, respectively. In contrast to the n = 1 case, we notice that there is a discrepancy between theory and experiment. Both the BBK and the PWIA cross sections are found to rapidly decrease with every increase in momentum for p greater than 0.5 a.u., unlike the experimental data. BBK and the PWIA curves are nearly the same with the former showing a slightly better agreement with experiment than the latter. Figs. 5 and 6 exhibit analogous comparisons of n = 3 momentum profiles at impact
1E-6
TDCS (a.u.)
TDCS (a.u.)
1E-5
1E-6
1E-7
1E-8 0
1
2
Momentum (a.u.)
Fig. 4. Same as in Fig. 3, but with an incident energy of 1000 eV.
3
0
1
2
Momentum (a.u.)
Fig. 6. Same as in Fig. 5, but with an incident energy of 1000 eV.
3
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C. Dal Cappello et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 570–576
energies of 1600 and 1000 eV, respectively. The two theoretical profiles are almost indistinguishable and are in reasonably good agreement with the experimental data.
a 0.10
a
0.08
0.08
0.06 0.06
0.04
n=2/n=1
n=2/n=1
Ren et al. [1] have also given the cross section ratios. The cross sections corresponding to n = 1, 2 and 3 have been measured at the same out of plane azimuthal angles /. Figs. 7 and 8 display a comparison between the experimental values of Ren et al. [1], the theoretical calculations of McCarthy and Mitroy [26] and the results of the BBK
0.02
0.00 0
1
2
3
0.04
0.02
4
Momentum (a.u.)
b
0.030
0.00 0
1
2
3
2
3
2
3
Momentum (a.u.)
0.025
b
0.020
0.018
n=3/n=1
0.016
0.015
0.014 0.012
n=3/n=1
0.010
0.005
0.010 0.008 0.006
0.000 0
2
0.004
Momentum (a.u.) 0.002
c
0.5
0.000 0
1
Momentum (a.u.) 0.4
c
0.3
0.2
0.2
n=3/n=2
n=3/n=2
0.3
0.1
0.1
0.0 0
1
2
3
Momentum (a.u.)
Fig. 7. Cross section ratios of helium (a) n = 2/n = 1, (b) n = 3/n = 1 and (c) n = 3/n = 2 at the incident energy of 1600 eV. The solid line represents the present calculation using the BBK model whereas the solid squares are the experimental data of Ren et al. [1]. The DWIA calculation of McCarthy and Mitroy [26] (dashed line) is also included for the case (a).
0.0 0
1
Momentum (a.u.)
Fig. 8. Same as in Fig. 7, but with an incident energy of 1000 eV.
C. Dal Cappello et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 570–576
model at the impact energy of 1600 eV and 1000 eV, respectively. For the ratio n = 2/n = 1 at 1600 eV the agreement between experiments and the BBK model is good although this theoretical model underestimates the experiments for p greater more than 2.2 a.u. We notice that the BBK model gives a better agreement than the DWIA model of McCarthy and Mitroy [26]. For the ratio n = 2/ n = 1 at 1000 eV the agreement between the BBK model and experiments begins to deteriorate from p = 1 a.u. In fact the BBK model underestimates the experiments. The DWIA model underestimates the experiments too. For the other ratios, n = 3/n = 1 and n = 3/n = 2, respectively, the agreement between BBK and experiments is reasonably good although the BBK model slightly underestimates the experiments. Generally speaking, the agreement between the BBK model and experiments is always better at 1600 eV than at 1000 eV. We also notice a relatively good agreement for n = 1 and n = 3 but, some disagreement for n = 2 for values of p greater more than 1 a.u. It is true that in the case n = 3 the statistical errors are larger than in the case n = 2 and the agreement could be a little bit fortuitous. The present model is also found to be in better agreement with experiment than the PWIA model. However the improvement of the present BBK model over the PWIA method is seen to be marginal, since the incident energy considered in this investigation is pretty high. It may be noted that whereas in the BBK model, the final state wave function contains pairwise Coulomb interactions among all the three particles in the final state, this model uses a plane wave for the incident electron and fails to account for distortion in the entrance channel. Probably, this is the reason for the discrepancy observed between our results and experiment. 4. Conclusions We have evaluated the TDCS for the transition of He to the ground and the n = 2 and 3 excited states of He+ at impact energies of 1000 and 1600 eV using the BBK and plane wave impulse approximation methods. We have found that the (e, 2e) reaction process for the excitation of He+ to n = 2 and 3 excited states are mainly dominated by ns sublevels belonging to the level n, even when the highly correlated CI wave function is used to describe the ground state of He. We have also compared the present calculations with the experimental momentum profiles of Ren et al. [1]. In the case of n = 1 momentum profile, both the BBK and PWIA are able to reproduce the shape of the experimental cross section in the momentum range below about 1.5 a.u. There is, however, a substantial discrepancy in the range 1.5–3.1 a.u. with the BBK showing better agreement with experiment than the PWIA. The shapes of TDCS predicted by the BBK are in reasonably good agreement with the experimental momentum profiles for n = 2 and n = 3 excited states. The discrepancy between theory and experiment still
575
persists, especially in the high momentum region. Furthermore, we notice that the present BBK, unlike the PWIA, exhibits the impact energy dependence of n = 2 and 3 momentum profiles in agreement with the experimental findings. The n = 2/n = 1 ratio cross sections are reasonably well described by the BBK model and are better than the DWIA model. Except for large values of p the other ratios (n = 3/n = 1 and n = 3/n = 2) are well described by the BBK model. From the comparisons between theory and experiment, we can conclude that the BBK method is feasible to describe the processes of simultaneous ionization-excitation better than the PWIA or DWIA model. But some discrepancies still persist especially for the n = 2 case. We must remember that the BBK wave function which consists of the product of three coulomb waves having the final state electron–electron interaction is the leading term of the exact scattering wave function in the entire coordinate space only when any two particles have large relative speed [27]. Maybe an elaborate close-coupling calculation is needed to explain the discrepancy for the transition from the ground state of helium to the n = 2 excited state. It is also necessary to get new (and more accurate) data at higher energies in order to ascertain where the PWIA model can be successfully applied for the description of ionization-excitation of helium. Acknowledgements We are very grateful to Prof. Takahashi M and Prof. Popov Y.V. for many fruitful discussions and comments about this work. The authors would like to thank the CINES (Centre Informatique National de l’Enseignement Supe´rieur) of Montpellier (France) for large free computer time. References [1] X.G. Ren, C.G. Ning, J.K. Deng, G.L. Su, S.F. Zhang, Y.R. Huang, G.Q. Li, Phys. Rev. A 72 (2005) 042718. [2] N. Watanabe, Y. Khajuria, M. Takahashi, Y. Udagawa, P.S. Vinitsky, Yu.V. Popov, O. Chuluunbaatar, K.A. Kouzakov, Phys. Rev. A 72 (2005) 032705. [3] N. Watanabe, M. Takahashi, Y. Udagawa, K.A. Kouzakov, Yu. V. Popov, Phys. Rev. A 75 (2007) 052701. [4] G. Sakhelashvili, A. Dorn, C. Hohr, J. Ullrich, A.S. Kheifets, L. Lower, K. Bartschat, Phys. Rev. Lett 95 (2005) 033201. [5] S. Bellm, J. Lower, K. Bartschat, Phys. Rev. Lett. 96 (2006) 223201. [6] S. Bellm, J. Lower, K. Bartschat, X. Guan, D. Weflen, M. Foster, A.L. Harris, D.H. Madison, Phys. Rev. A 75 (2007) 042704. [7] I.E. McCarthy, E. Weigold, Phys. Rep. Phys. Lett. 27 (1976) 275. [8] C.E. Brion, Int. J. Quantum Chem. 29 (1986) 1397. [9] M.A. Coplan, J.H. Moore, J.P. Doering, Rev. Mod. Phys. 66 (1994) 985. [10] C. Dupre´, A. Lahmam-Bennani, A. Duguet, F. Mota-Furtado, P. O’Mahony, C. Dal Cappello, J. Phys. B 25 (1992) 259. [11] L. Avaldi, R. Camilloni, R. Multari, G. Stefani, J. Langlois, O. Robaux, R.J. Tweed, N. Nguyen Vien, J. Phys. B 31 (1998) 2981. [12] B. Rouvellou, S. Rioual, A. Pochat, R.J. Tweed, J. Langlois, N. Nguyen Vien, J. Phys. B 33 (2000) L599.
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[13] K. Bartschat, P.G. Burke, J. Phys. B 20 (1987) 3191. [14] R.H. Reid, K. Bartschat, A. Raeker, J. Phys. B 31 (1998) 563. [15] M. Schulz, R. Moshammer, D. Fischer, H. Kollmus, D.H. Madison, S. Jones, J. Ullrich, Nature 48 (2003) 422. [16] M. Foster, J.L. Peacher, M. Schulz, D.H. Madison, Z. Chen, H.R.J. Walters, Phys. Rev. Lett. 97 (2006) 093202. [17] M. Brauner, J.S. Briggs, H. Klar, J. Phys. B 22 (1989) 2265. [18] B. Joulakian, C. Dal Cappello, M. Brauner, J. Phys. B 25 (1992) 2863. [19] M. Brauner, J.S. Briggs, H. Klar, T.J. Broad, T. Rosel, K. Jung, H. Ehrhardt, J. Phys. B 24 (1991) 657. [20] J. Berakdar, J.S. Briggs, Phys. Rev. Lett. 72 (1994) 3799.
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NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 266 (2008) 570–576 www.elsevier.com/locate/nimb
Simultaneous ionization and excitation of helium by electron impact C. Dal Cappello a,*, A.C. Roy b, X.G. Ren c,d, R. Dey e a
Universite´ Paul Verlaine-Metz, Laboratoire de Physique Mole´culaire et des Collisions, ICPMB (FR 2843), Institut de Physique, 1 rue Arago, 57078 Metz Cedex 3, France b Ramakrishna Mission Vivekananda University, Belur Math 711202, West Bengal, India c Department of Physics and Key Laboratory of Atomic and Molecular NanoSciences of MOE, Tsinghua University, Beijing 100084, People’s Republic of China d Max-Planck-Institute for Nuclear Physics, Saupfercheckweg 1, 68117 Heidelberg, Germany e Institute for Plasma Research, Bhat, Gandhinagar 382428, India Received 21 September 2007; received in revised form 16 November 2007 Available online 14 December 2007
Abstract We present numerical results for He (1s2) (e, 2e) He+ reaction process for transitions to the n = 1, 2 and 3 states of He+ for noncoplanar symmetric geometry at incident energies of 1000 and 1600 eV. The calculations are performed using the plane wave impulse approximation (PWIA) and the 3C method (also called the Brauner, Briggs and Klar (BBK) model) that includes post collision interaction and multiple scattering effects. In both the methods we have used the highly correlated configuration interaction wave function for the ground state of helium. A comparison of the present theoretical cross sections with the recent measured data of Ren et al. [X.G. Ren, C.G. Ning, J.K. Deng, G.L. Su, S.F. Zhang, Y.R. Huang, G.Q. Li, Phys. Rev. A 72 (2005) 042718] shows reasonably good agreement. Ó 2007 Elsevier B.V. All rights reserved. PACS: 34.50 Keywords: Triply differential cross sections; Ionization and excitation; 3C approximation
1. Introduction In the recent past, there has been an increasing interest in both the theoretical and experimental studies of simultaneous ionization-excitation phenomenon in few-body systems [1–6]. Watanabe et al. [2] have reported the (e, 2e) experiment on the above phenomenon for the transition of ground state of helium to the n = 2 excited state of He+ at large momentum transfer at the incident energy of 2080 eV in the symmetric non-coplanar geometry. They have also reported the momentum-dependent triply differential cross section (TDCS), also known as the momentum profile within the plane wave impulse approximation (PWIA) [7–9] and pointed out that the ionization-excitation *
Corresponding author. Tel.: +33 03 87 31 58 60; fax: +33 03 87 54 72
57. E-mail address: [email protected] (C. Dal Cappello). 0168-583X/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2007.12.003
event at large momentum transfer is very sensitive to the electron correlation in the target initial state, Although the shapes of cross sections were well reproduced by PWIA at small momentum on using highly correlated wave function for the ground state of helium, considerable discrepancies between experiment and theory remained in magnitude. Later on, in the same year Ren et al. [1] reported the results of their experimental and theoretical momentum profiles for the (e, 2e) reaction in helium for the transitions to the ground n = 1 state and the n = 2 and 3 excited states of He+ at impact energies of 1000 and 1600 eV. They also applied the PWIA method to study for the first time the impact energy dependence of the excited final ion states and confirmed that the PWIA description of ionizationexcitation event was inadequate for large recoil momentum p of the residual ion, especially for p greater than 1.5 a.u. Very recently, the simultaneous ionization and excitation of helium (up to n = 4) has also been investigated in
C. Dal Cappello et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 570–576
coplanar geometries with the asymmetric and symmetric energy-sharing cases [5] for low impact energy, following the pioneering works of Dupre´ et al. [10], Avaldi et al. [11] and Rouvellou et al. [12] for n = 2. Because of the energy degeneracy of the He+(2s) and He+(2p) states, the experimental data were the sum of their respective contributions. However, by also detecting the emitted 2p ? 1s photon in a triple coincidence measurement, Sakhelashvili et al. [4] were able to measure the 2p contribution individually. The theoretical calculations (DWB-RMPS and DWB2-RMPS models using a hybrid distorted wave and R matrix [13] and also the second Born approximation [3,14]) were able to give a reasonably good agreement for the above geometries [5,6]. Generally speaking, these models such as DWB, DWB2 and 3DW [15] describe well the experimental measurements when the momentum vectors of the incident projectile, the scattered projectile and the outgoing electron all lie in the same plane, but not for out-of-plane geometries (see, for instance, Schulz et al. [15] and Foster et al. [16]). In this work the non-coplanar symmetric (e, 2e) measurements on helium for transitions to the n = 1, 2 and 3 final states of He+ are studied at incident energies of 1000 eV and 1600 eV. The objective of this investigation is to apply a theoretical method to analyse and explain the experimental findings of Ren et al. [1] including the impact energy dependences of the momentum profiles of the excited states of He+. Since the PWIA is a one-step process that involves the interaction of the projectile with a target electron, it is unlikely to be adequate for the description of the ejection of a bound electron accompanied by the simultaneous excitation of another bound electron. Keeping that in mind we have applied the 3C method (also called the BBK method) [17,18] which has been extensively applied with success to (e, 2e) processes in atoms and molecules [19–23]. The BBK method contains both the post collision interaction (PCI) among the three Coulomb particles in the final state and multiple scattering effects. This model has a correct asymptotic Coulomb three-body wave function for the ejected and scattered electrons in the residual ion field. The effect of exchange between the scattered and ejected electrons has been incorporated in this model by the Pauli method. The present paper is organised as follows: Section 2 gives a brief description of the two theories, namely the plane wave impulse approximation and the BBK method which have been applied in the present investigation. In Section 3 we discuss and compare our results with the experimental data of Ren et al. [1]. Section 4 contains our conclusions. Atomic units will be used throughout, unless otherwise stated. 2. Theory 2.1. Plane-wave impulse approximation We consider an event in which a high energy electron knocks out a bound electron from the helium target in
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the ground state and the two escaping electrons are detected in coincidence. In this case the triply differential cross section (TDCS) for the transition to the nth state of He+ in the plane wave impulse approximation is given by X d3 rn kske 2 ¼2 rM j F n;l;m ð~ pÞj ; fi dXs dXe dEe ki l;m
ð1Þ
2pg and g ¼ j~k 1~k j where ~ ki, ~ k s and ~ k e denote the with rM ¼ e2pg 1 s e momentum of the incident electron, the scattered electron and the ejected electron, respectively. dXs and dXe are the elements of solid angles for the scattered electron and the ejected electron, respectively and dEe represents the energy interval of the ejected electron. In Eq. (1) a sum over the unresolved ionic excited states is assumed. Here ~ p stands for the recoil momentum of the residual ion and is governed by the conservation of linear momentum
~ ~ ~ p ¼~ ki ~ ks ~ ke ¼ K ke;
ð2Þ
~ is the momentum transfer. where K In the present investigation we have considered a kinematic scheme where the two outgoing electrons have equal energies (Ee = Es) and make polar angles he = hs = 45° with respect to the direction of the incident beam. The magnitude of the momentum p is given by pffiffiffi pffiffiffi p ¼ ½ðk i 2k s Þ2 þ ð 2k s sinðD/=2ÞÞ2 1=2 ; ð3Þ where D/ = /e/sp is the out-of-plane azimuthal angle difference between the two outgoing electrons. In Eq. (1) rM is the half-off-shell Mott scattering cross section [3,7]. F n;l;m ð~ pÞ is known as the overlap integral fi and written as Z 2 ei~p:~r2 n;l;m F n;l;m d~ r ð~ pÞ ¼ d~ r U ð~ r Þ Wð~ r2 ;~ r3 Þ; ð4Þ 2 3 3 fi f 3=2 K2 ð2pÞ where Un;l;m represents the wave function of the residual ion f in the nth state and Wð~ r2 ;~ r3 Þ is the ground state wave function of He. Since the energies of the two particles in the present investigation are not too high, it is not quite valid to describe them as plane waves. Besides, the post collision interaction between the two equi-speed Coulomb particles (although expected to be small) should not be totally ignored. In these circumstances, we have considered another method, called the BBK method which accounts for pairwise Coulomb distortions among the two escaping particles and the nucleus. In the following we give the outlines of this model. 2.2. The BBK approximation The TDCS for the (e, 2e) process is given by d3 rn k s k e X n;l;m 2 ¼2 j T fi j ; dXs dXe dEe k i l;m
ð5Þ
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where the transition-matrix element T n;l;m is fi 1 hWn;l;m j V i j Ui i: 2p f
T n;l;m ¼ fi
ð6Þ
In Eq. (6) Wn;l;m and Ui represent the wave function describf ing the whole system in its final and initial states, respectively. The initial state Ui is a product of an incoming plane wave representing the incident particle with momentum ~ ki and the ground state of helium and can be represented by the asymptotic wave function ~
Ui ¼ eiki ~r1 Wi ð~ r2 ;~ r3 Þ;
ð7Þ
where ~ r1 , ~ r2 and ~ r3 are the positions, relative to the atomic centre, of the projectile and the two bound electrons, respectively. The perturbation Vi ¼
2 1 1 þ þ ; r1 r12 r13
ð8Þ
is the interaction of the projectile with the target atom, where ~ rij ¼ ~ ri ~ rj . The final state wave function Wn;l;m describes the system f consisting of the two outgoing electrons and the residual ion and is written as Wn;l;m ¼ W r1 ;~ r2 ÞUn;l;m ð~ r3 Þ; f f 3C ð~
ð9Þ
ð~ r3 Þ is the wave function of the residual hydrowhere Un;l;m f genic ion He+. In Eq. (9) W r1 ;~ r2 Þ is the BBK wave func3C ð~ tion for the two distinguishable emitted electrons and is given by W r1 ;~ r2 Þ ¼ 3C ð~
1 3=2
ð2pÞ
~ ~ eiks ~r1 eike ~r2 Cðas ; ~ k s ;~ r1 ÞCðae ; ~ k e ;~ r2 Þ
k se ;~ r12 Þ; Cðase ; ~
and a 3C function which consists of a product of three Coulomb distortion factors (one for each two-body Coulomb interaction) to describe the final double-continuum state. He+ ion states for n = 1, 2 and 3 are represented by hydrogenic wave functions which are known exactly. We have compared our (e, 2e) cross sections with the corresponding experimental momentum profiles of Ren et al. [1] for the above-mentioned transition processes. Calculations have been performed for non-coplanar symmetric geometry, i.e. h1 = h2 = 45° and Es = Ee. Since the experimental cross sections are relative they are normalised to the theoretical cross sections at the peak for each transition process at each incident energy. This will, of course, enable us to make a comparative study of the shapes of cross sections. Fig. 1 shows the present TDCS obtained in the BBK and PWIA methods at the incident energy of 1600 eV for the transition to the ground state of He+. Also included in this figure is the corresponding experimental momentum profile of Ren et al. [1]. We see that both the theoretical methods predict nearly identical cross sections in the momentum region below 1 a.u. The shapes of these cross sections are almost indistinguishable from the shape of the experimental profile. However, they differ from each other and also from experiment at larger momentum. The shape of BBK cross sections is seen to be conspicuously better than that of PWIA calculation. Fig. 2 displays the results of the same reaction process but at the lower incident energy of 1000 eV. We notice that with the decrease in impact energy both the BBK and measured cross sections increase for the same fixed momentum while the shapes remain more or less the same. The present BBK and PWIA cross sections for (e, 2e) reactions in helium for the transitions to n = 2 and n = 3
ð10Þ
k ~ rþ k;~ rÞ ¼ Cð1 iaÞ expð p2 aÞ1 F 1 ðia; 1; ið~ where Cða;~ Z krÞÞ, as ¼ ks , ae ¼ kZe , ase ¼ 2k1se and ~ k se ¼ 12 ð~ ks ~ k e Þ. Here Z is the charge of the residual ion and 1F1 represents the confluent hypergeometric function. In the present work, we have considered the spatially symmetric part of the above 3C wave function W r1 ;~ r2 Þ; thus exchange between the two emitted elec3C ð~ trons is included in accordance with the Pauli principle.
1E-3
TDCS (a.u.)
572
1E-4
1E-5
3. Results and discussion We have evaluated the TDCS for electron impact ionization of helium at incident energies of 1000 and 1600 eV for the transitions to the He+ n = 1, 2 and 3 states. The present calculations are performed using the BBK model and PWIA methods. In both methods we have used the highly correlated CI function of Nesbet and Watson [24] for the ground state of helium. The energy of the ground state is E = 2.90276 a.u. differing from the experimental value by 0.001 a.u. Whereas in the PWIA both the incident and the scattered particles are represented by plane waves, the BBK model uses a plane wave for the incident projectile
1E-6 0
1
2
3
Momentum (a.u.)
Fig. 1. Comparison of present triply differential cross section for (e, 2e) reaction in He as a function of the momentum of the residual ion for the transition to the ground state of He+ with the measurements of Ren et al. [1] at the incident energy of 1600 eV. The relative data of Ren et al. [1] have been normalised at the peak to the present theoretical value. The solid line represents the present calculation using the BBK model. The dotted curve represents the present plane wave impulse approximation (PWIA) calculation. The solid squares are the experimental data.
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573
1E-6
TDCS (a.u.)
TDCS (a.u.)
1E-3
1E-4
1E-7
1E-8
1E-5 0
0
2
1
Momentum (a.u.)
TDCS (a.u.)
1E-5
1E-6
1E-7 1
3
Fig. 5. Same as in Fig. 3, but for the transition to the n = 3 excited state of He+.
Fig. 2. Same as in Fig. 1, but with an incident energy of 1000 eV.
0
2
Momentum (a.u.)
2
3
Momentum (a.u.)
Fig. 3. Same as in Fig. 1, but for the transition to the n = 2 excited state of He+.
states of residual He+ are displayed in Figs. 3–6. Both the present BBK and PWIA calculations show that the contributions from the sublevels other than the s levels are too low and are at least of one order smaller than those of the corresponding s level. This confirms the calculations of Cook et al. [25]. Since these sublevels are energetically unresolved, we have compared the summed cross sections with the experimental momentum profiles. Figs. 3 and 4 compare the present TDCS with the experimental momentum profiles for n = 2 at the impact energies of 1600 and 1000 eV, respectively. In contrast to the n = 1 case, we notice that there is a discrepancy between theory and experiment. Both the BBK and the PWIA cross sections are found to rapidly decrease with every increase in momentum for p greater than 0.5 a.u., unlike the experimental data. BBK and the PWIA curves are nearly the same with the former showing a slightly better agreement with experiment than the latter. Figs. 5 and 6 exhibit analogous comparisons of n = 3 momentum profiles at impact
1E-6
TDCS (a.u.)
TDCS (a.u.)
1E-5
1E-6
1E-7
1E-8 0
1
2
Momentum (a.u.)
Fig. 4. Same as in Fig. 3, but with an incident energy of 1000 eV.
3
0
1
2
Momentum (a.u.)
Fig. 6. Same as in Fig. 5, but with an incident energy of 1000 eV.
3
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energies of 1600 and 1000 eV, respectively. The two theoretical profiles are almost indistinguishable and are in reasonably good agreement with the experimental data.
a 0.10
a
0.08
0.08
0.06 0.06
0.04
n=2/n=1
n=2/n=1
Ren et al. [1] have also given the cross section ratios. The cross sections corresponding to n = 1, 2 and 3 have been measured at the same out of plane azimuthal angles /. Figs. 7 and 8 display a comparison between the experimental values of Ren et al. [1], the theoretical calculations of McCarthy and Mitroy [26] and the results of the BBK
0.02
0.00 0
1
2
3
0.04
0.02
4
Momentum (a.u.)
b
0.030
0.00 0
1
2
3
2
3
2
3
Momentum (a.u.)
0.025
b
0.020
0.018
n=3/n=1
0.016
0.015
0.014 0.012
n=3/n=1
0.010
0.005
0.010 0.008 0.006
0.000 0
2
0.004
Momentum (a.u.) 0.002
c
0.5
0.000 0
1
Momentum (a.u.) 0.4
c
0.3
0.2
0.2
n=3/n=2
n=3/n=2
0.3
0.1
0.1
0.0 0
1
2
3
Momentum (a.u.)
Fig. 7. Cross section ratios of helium (a) n = 2/n = 1, (b) n = 3/n = 1 and (c) n = 3/n = 2 at the incident energy of 1600 eV. The solid line represents the present calculation using the BBK model whereas the solid squares are the experimental data of Ren et al. [1]. The DWIA calculation of McCarthy and Mitroy [26] (dashed line) is also included for the case (a).
0.0 0
1
Momentum (a.u.)
Fig. 8. Same as in Fig. 7, but with an incident energy of 1000 eV.
C. Dal Cappello et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 570–576
model at the impact energy of 1600 eV and 1000 eV, respectively. For the ratio n = 2/n = 1 at 1600 eV the agreement between experiments and the BBK model is good although this theoretical model underestimates the experiments for p greater more than 2.2 a.u. We notice that the BBK model gives a better agreement than the DWIA model of McCarthy and Mitroy [26]. For the ratio n = 2/ n = 1 at 1000 eV the agreement between the BBK model and experiments begins to deteriorate from p = 1 a.u. In fact the BBK model underestimates the experiments. The DWIA model underestimates the experiments too. For the other ratios, n = 3/n = 1 and n = 3/n = 2, respectively, the agreement between BBK and experiments is reasonably good although the BBK model slightly underestimates the experiments. Generally speaking, the agreement between the BBK model and experiments is always better at 1600 eV than at 1000 eV. We also notice a relatively good agreement for n = 1 and n = 3 but, some disagreement for n = 2 for values of p greater more than 1 a.u. It is true that in the case n = 3 the statistical errors are larger than in the case n = 2 and the agreement could be a little bit fortuitous. The present model is also found to be in better agreement with experiment than the PWIA model. However the improvement of the present BBK model over the PWIA method is seen to be marginal, since the incident energy considered in this investigation is pretty high. It may be noted that whereas in the BBK model, the final state wave function contains pairwise Coulomb interactions among all the three particles in the final state, this model uses a plane wave for the incident electron and fails to account for distortion in the entrance channel. Probably, this is the reason for the discrepancy observed between our results and experiment. 4. Conclusions We have evaluated the TDCS for the transition of He to the ground and the n = 2 and 3 excited states of He+ at impact energies of 1000 and 1600 eV using the BBK and plane wave impulse approximation methods. We have found that the (e, 2e) reaction process for the excitation of He+ to n = 2 and 3 excited states are mainly dominated by ns sublevels belonging to the level n, even when the highly correlated CI wave function is used to describe the ground state of He. We have also compared the present calculations with the experimental momentum profiles of Ren et al. [1]. In the case of n = 1 momentum profile, both the BBK and PWIA are able to reproduce the shape of the experimental cross section in the momentum range below about 1.5 a.u. There is, however, a substantial discrepancy in the range 1.5–3.1 a.u. with the BBK showing better agreement with experiment than the PWIA. The shapes of TDCS predicted by the BBK are in reasonably good agreement with the experimental momentum profiles for n = 2 and n = 3 excited states. The discrepancy between theory and experiment still
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persists, especially in the high momentum region. Furthermore, we notice that the present BBK, unlike the PWIA, exhibits the impact energy dependence of n = 2 and 3 momentum profiles in agreement with the experimental findings. The n = 2/n = 1 ratio cross sections are reasonably well described by the BBK model and are better than the DWIA model. Except for large values of p the other ratios (n = 3/n = 1 and n = 3/n = 2) are well described by the BBK model. From the comparisons between theory and experiment, we can conclude that the BBK method is feasible to describe the processes of simultaneous ionization-excitation better than the PWIA or DWIA model. But some discrepancies still persist especially for the n = 2 case. We must remember that the BBK wave function which consists of the product of three coulomb waves having the final state electron–electron interaction is the leading term of the exact scattering wave function in the entire coordinate space only when any two particles have large relative speed [27]. Maybe an elaborate close-coupling calculation is needed to explain the discrepancy for the transition from the ground state of helium to the n = 2 excited state. It is also necessary to get new (and more accurate) data at higher energies in order to ascertain where the PWIA model can be successfully applied for the description of ionization-excitation of helium. Acknowledgements We are very grateful to Prof. Takahashi M and Prof. Popov Y.V. for many fruitful discussions and comments about this work. The authors would like to thank the CINES (Centre Informatique National de l’Enseignement Supe´rieur) of Montpellier (France) for large free computer time. References [1] X.G. Ren, C.G. Ning, J.K. Deng, G.L. Su, S.F. Zhang, Y.R. Huang, G.Q. Li, Phys. Rev. A 72 (2005) 042718. [2] N. Watanabe, Y. Khajuria, M. Takahashi, Y. Udagawa, P.S. Vinitsky, Yu.V. Popov, O. Chuluunbaatar, K.A. Kouzakov, Phys. Rev. A 72 (2005) 032705. [3] N. Watanabe, M. Takahashi, Y. Udagawa, K.A. Kouzakov, Yu. V. Popov, Phys. Rev. A 75 (2007) 052701. [4] G. Sakhelashvili, A. Dorn, C. Hohr, J. Ullrich, A.S. Kheifets, L. Lower, K. Bartschat, Phys. Rev. Lett 95 (2005) 033201. [5] S. Bellm, J. Lower, K. Bartschat, Phys. Rev. Lett. 96 (2006) 223201. [6] S. Bellm, J. Lower, K. Bartschat, X. Guan, D. Weflen, M. Foster, A.L. Harris, D.H. Madison, Phys. Rev. A 75 (2007) 042704. [7] I.E. McCarthy, E. Weigold, Phys. Rep. Phys. Lett. 27 (1976) 275. [8] C.E. Brion, Int. J. Quantum Chem. 29 (1986) 1397. [9] M.A. Coplan, J.H. Moore, J.P. Doering, Rev. Mod. Phys. 66 (1994) 985. [10] C. Dupre´, A. Lahmam-Bennani, A. Duguet, F. Mota-Furtado, P. O’Mahony, C. Dal Cappello, J. Phys. B 25 (1992) 259. [11] L. Avaldi, R. Camilloni, R. Multari, G. Stefani, J. Langlois, O. Robaux, R.J. Tweed, N. Nguyen Vien, J. Phys. B 31 (1998) 2981. [12] B. Rouvellou, S. Rioual, A. Pochat, R.J. Tweed, J. Langlois, N. Nguyen Vien, J. Phys. B 33 (2000) L599.
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