* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Three-body dynamics in hydrogen ionization by fast highly charged
Survey
Document related concepts
Elementary particle wikipedia , lookup
Cross section (physics) wikipedia , lookup
Renormalization wikipedia , lookup
Wave–particle duality wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
X-ray photoelectron spectroscopy wikipedia , lookup
Tight binding wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Auger electron spectroscopy wikipedia , lookup
Atomic orbital wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Electron configuration wikipedia , lookup
Atomic theory wikipedia , lookup
Transcript
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS J. Phys. B: At. Mol. Opt. Phys. 35 (2002) 1759–1773 PII: S0953-4075(02)32735-4 Three-body dynamics in hydrogen ionization by fast highly charged particles J Fiol and R E Olson Physics Department, University of Missouri-Rolla, Rolla, MO 65401, USA E-mail: [email protected] Received 8 January 2002, in final form 15 February 2002 Published 26 March 2002 Online at stacks.iop.org/JPhysB/35/1759 Abstract Electron double and triple differential cross sections are calculated as a function of projectile momentum transfer for ionization of ground and excited state hydrogen by 3.6 MeV u−1 C6+ and Au53+ ions. These three-body Coulomb systems are investigated using the classical-trajectory Monte Carlo and continuum distorted wave methods that incorporate all interactions including the nuclear–nuclear potential. The calculations allow one to distinguish cross section features associated with three collision mechanisms. The first is due to distance collisions that lead primarily to a binary interaction between the projectile and electron. The second is a strong three-body interaction by the projectile with both the electron and the target ion when the projectile passes between the electron and its parent nucleus. These two mechanisms result in binary peak electrons located near the momentum transfer vector Q with active participation by the recoil ion along −Q in order to determine the overall momentum transfer magnitude. The third mechanism, which is present only for fast highly charged ion impact, yields electron spectra that here-to-fore have not been observed. Rather than the electrons being scattered near the angle θ associated with the momentum transfer vector Q, they are found at an angle of 360◦ − θ . Such electrons are due to a close collision of the projectile with the target nucleus with the electron being forced to swing by its parent. These electrons are not so-called recoil electrons associated with the angle 180◦ + θ. Moreover, in this case neither the recoil ion nor the electron spectra peak near Q. Calculations are also presented for collisions with excited hydrogen in order to assess the collision dynamics as a function of the radial dimensions of the target atom. 1. Introduction Single ionization of an atom induced by a collision with an ion provides a challenging field for the study of the three-body Coulomb problem. For many years the analysis of atomic 0953-4075/02/071759+15$30.00 © 2002 IOP Publishing Ltd Printed in the UK 1759 1760 J Fiol and R E Olson ionization produced by heavy ion impact was mainly conducted through the study of total cross sections, and differential cross sections in terms of the angle and energy of the ejected electron. Only recently has the development of experimental techniques permitted the coincidence measurement of both the ionized electron and recoil ion momentum vectors. In particular, momentum spectroscopy experiments now provide kinematically complete descriptions of heavy-particle three-body ionization collisions (Cocke and Olson 1991, Ullrich et al 1997). These data can be used to rigorously test different theoretical approaches. In electron–atom collisions the study of kinematically complete ionization cross sections has been carried out both experimentally and theoretically for more than two decades (see Ehrhardt et al (1985), Lahmann-Bennani (1991) and references therein). Research in this area has shown that for relatively high incident velocities the first Born approximation (FBA) gives reliable results even for highly differential cross sections. However, electron impact studies are limited because of the fixed, low charge of the projectile. In the traditional picture, the strength of the collision is estimated through the perturbation or Sommerfeld parameter νP = ZP /v, where ZP and v are the projectile’s charge and initial velocity, respectively. For electron impact the magnitude of the perturbation strength νP cannot exceed unity for ground state hydrogen or rare gas targets. The use of heavy ions provides an additional degree of freedom due to the possibility of varying the projectile charge. Increasing the projectile charge makes it possible to exceed perturbation strengths of unity. The description of ionization collisions in such high perturbation conditions is theoretically much more tenuous. This fact was recently illustrated by detailed studies of He single ionization. Double differential cross sections (DDCSs) were presented for the impact of 3.6 MeV u−1 Au53+ ions (νP = 4.42) as a function of the projectile momentum transfer (Moshammer et al 2001, Olson and Fiol 2001). Poor agreement between experimental data and theoretical predictions was observed. The study of DDCSs as a function of the momentum transfer in a collision gives valuable information on the dynamics of three interacting particles. While the ionized electron momentum distribution by itself is nearly independent of the nuclear–nuclear interaction, a precise inclusion of all interactions on the same basis is needed in order to properly describe cross sections related to the momentum transfer or projectile scattering angle. Previous research has shown that the inclusion of the internuclear interaction is necessary to describe ionization cross sections related to the projectile scattering angle for large values of this variable (Salin 1989, Rodrı́guez 1996). Moreover, it has recently been shown that the cross section for small deflections of the projectile is strongly affected by the nuclear–nuclear interaction in both classical and quantum theories (Olson and Fiol 2001). Previous work on this subject analysed the projectile deflection for He ionization by 100 MeV u−1 C6+ and 3.6 MeV u−1 Au53+ . Good agreement between the theoretical and experimental results was found for C6+ , but serious discrepancies arose for the high perturbation Au53+ case. The calculations show that, besides the inclusion of the internuclear interaction, the choice of the effective three-body model for the description of the four-body target is crucial in the quantum description of the experimental results. Furthermore, complications due to the double ionization of He have been addressed via classical trajectory Monte Carlo (CTMC) calculations. In order to avoid these two major difficulties, in this work we study the mechanisms of ionization for an atomic hydrogen target. This choice leads to a real three-body system with pure Coulomb interactions between all particles. A systematic study on atomic hydrogen ionization as a function of projectile momentum transfer allows one to elucidate threebody collision mechanisms and predict electron spectra signatures due to various dynamical processes. Three-body dynamics in hydrogen ionization by fast highly charged particles 1761 Single ionization of ground and n = 2, 3 excited state hydrogen by 3.6 MeV u−1 C6+ and Au are investigated. CTMC and quantum mechanical continuum distorted wave (CDW) theories are employed. Atomic units are used throughout this work (me , e, h̄ = 1). In section 2, the theoretical methods are briefly described. Next, the ionization of ground and excited states of hydrogen by C6+ are studied. In section 4, the ionization process in the non-perturbative regime (Au53+ + H) is investigated. In section 5, a detailed study of the collision dynamics and its interpretation in terms of classical collision mechanisms is given. 53+ 2. Theoretical approaches We have investigated hydrogen ionization by the impact of a charged particle. The projectiles, C6+ and Au53+ are considered point charges, neglecting the tightly bound electrons in the Au53+ case. The calculations were performed within two very different approximations, namely the CTMC and quantum mechanical CDW theories. In both theories the internuclear interaction is included along with the projectile–electron and target–electron interactions. As will be shown, the inclusion of all interactions is essential for the proper account of the momentum transferred to the target system during the collision. The hydrogen atom was chosen for the present study because it can be properly treated in both the quantum mechanical and classical theories since only Coulomb interactions are present. Moreover, it is well known that a classical microcanonical ensemble yields the exact quantum mechanical momentum density (Fock 1935, Abrines and Percival 1966) 8pn5 ϕ̃nm (p)2 = ρ̃n (p) = . (1) π 2 (p 2 + pn2 )4 ,m However, the classical spatial density is too compact compared to the quantum one and vanishes identically beyond r = 2 au. In the three-body CTMC method employed here the initial state of the H(1s) atom is described by means of a modified Wigner distribution (Hardie and Olson 1983, Wood et al 1997). This method utilizes a discrete superposition of microcanonical ensembles in order to provide a good approximation to the H(1s) initial radial distribution. This description of the initial state slightly degrades the momentum distribution. The choice of such a method over the single microcanonical ensemble approach is grounded on the fact that, as will be shown below, the results are strongly dependent on the spatial electron distribution. Furthermore, this method has been extensively tested for a wide range of energies and projectile incident charge states and provides excellent agreement with experiment (Illescas and Riera 1999). The CTMC method is well documented (Olson and Salop 1977, Reinhold and Olson 1989). The initial conditions are set such that the projectile impinges with a velocity v and a randomly chosen impact parameter b on the atomic target from a large distance, such that the interaction between the two fragments is negligible. The classical equations of motion are solved for the Hamiltonian ZP ZP 1 − − (2) H = H0 + rN rP rT where H0 is the kinetic energy, and the coordinates rN , rT are the positions of the projectile and electron relative to the target nucleus, and rP is the position of the electron relative to the projectile P of charge ZP . The CTMC method has successfully described atomic ionization collisions by heavy and light projectiles on a wide range of incidence energy for a large number of atomic and molecular targets (Schultz and Olson 1988, Olson 1996). In particular, it has recently been applied to 1762 J Fiol and R E Olson describe electron momentum spectra for helium ionization by impact of 3.6 MeV u−1 Au53+ (Schmidt et al 1998) and to the helium triple differential ionization cross section (TDCS) for 100 MeV u−1 C6+ collisions (Schulz et al 2001a). In this work we are interested in the evaluation of the TDCS and DDCS dσ dσ dσ and = de . (3) dQ⊥ dEe de dQ⊥ dEe dQ⊥ dEe de Here Ee and e are the ionized electron energy and solid angle, and Q⊥ is the perpendicular component of the momentum transferred to the atom by the projectile Q = MP v − K , where MP , v and K are the projectile’s mass, initial velocity and final momentum, respectively. Since the final projectile momentum is very close to the initial (Q K) the scattering angle is related to the perpendicular momentum transfer by Q⊥ ≈ MP vθ . Thus, the corresponding cross sections are trivially related by dσ dσ (MP v)2 ≈ . dEe de dP 2πQ⊥ dQ⊥ dEe de The quantum mechanical calculation of the TDCS is performed through (2π)5 dσ = kQ⊥ |tif |2 , dQ⊥ dEe de v2 tif = #f− |Vf |#i (4) where Q = (Ee + |εi |)/v and εi is the atomic binding energy. Here tif is the transition matrix element between the initial and final states. The final state is chosen to be the C3 or CDW wavefunction (Garibotti and Miraglia 1980) that is given by #f± (r, R) = ei (kT ·rT +KT ·RT ) ± D (νT , kT , rT )D ± (νN , kN , rN )D ± (νP , kP , rP ) (2π)3 (5) where D ± (να , kα , rα ) = )(1 ± iνα ) e−π να /2 1 F1 [∓iνα ; 1; −i(kα rα ∓ kα · rα )]. In (5) να is the Sommerfeld parameter for the Coulomb interaction Zα /rα between a given pair of particles, kα is their relative momentum, and mα is the corresponding two-body reduced mass. Rα , Kα are the position and the momentum of the remaining particle relative to the pair. The simplest description of the initial state is made through the unperturbed Born wavefunction (B1) #iB (rT , RT ) = ei KT ·RT φi (rT ) (2π)3/2 (6) that includes the motion of the projectile and the bound state of the target. In this work we present calculations with the Born initial state (CDW-B1) and with the eikonal initial state (CDW-EIS, see Crothers and McCann (1983), Fainstein et al (1991)) # EI S (rT , RT ) = #iB (rT , RT ) eiνP ln(kP rP −kP ·rP ) eiνN ln(kN rN −kN ·rN ) . (7) The difference between our CDW-EIS calculations and the standard CDW-EIS approximation is that, because we are interested in the scattering of the projectile, the internuclear interaction has not been neglected. We note that the internuclear interaction has previously been included for small scattering angle in both FBA and CDW-EIS theories in a semiclassical approximation by means of a phase factor (Salin 1989, Rodrı́guez 1996). In our calculations the three interactions have been taken into account on exactly the same footing in the wavefunctions and perturbation potential (Fiol et al 2001). Three-body dynamics in hydrogen ionization by fast highly charged particles 1763 -15 10 6+ 3.6 MeV/u C + H(1s) -16 Ee = 10 eV 2 dσ/dQ⊥dEe (cm /a. u.) 10 -17 10 50 eV -18 10 90 eV -19 10 0.1 1 2 3 4 5 Q⊥ (a. u.) Figure 1. DDCS as a function of the projectile transverse momentum transfer at fixed electron energies for ionization of hydrogen by 3.6 MeV u−1 C6+ . Curves: CDW-B1 calculations; and symbols: CTMC results, which are convoluted over the transverse momentum resolution of /Q⊥ = ±0.1 au and the electron energy /Ee = ±10 eV. The final distortion potential Vf in the transition matrix element (4) for the CDW wavefunction (5) can be written as ∇D −∗ (νj , kj , rj ) KP KN KT KN (8) Vf = − − KT KP Kj = MP MT D −∗ (νj , kj , rj ) where MP and MT are the masses of the projectile and target nucleus, respectively (Fiol and Olson (2002) and references therein). The above expression is valid for any relation of masses between the fragments. In the limit of infinitely heavy nucleus (MP , MT 1) only the last term contributes and the usual CDW result is recovered. 3. Perturbative collisions Very recently, single ionization of helium produced in collisions with 100 MeV u−1 C6+ and 3.6 MeV u−1 Au53+ was studied by investigating the DDCS dσ/dQ⊥ dEe as a function of the ionized electron energy and perpendicular momentum transfer (Moshammer et al 2001, Olson and Fiol 2001). Although the experimental data presented similar behaviour for both projectiles, the theoretical analysis showed marked differences. The calculations revealed that for the Au53+ + He system four-body effects are important. Moreover, the shape of the cross section was shown to strongly depend on the inclusion of the internuclear interaction and the choice of the model potential for the He+ core. Additionally, the CTMC calculations indicated that double ionization dominates over single ionization for large values of the momentum transfer. These problems are absent in calculations for the atomic hydrogen target. In this section we analyse the hydrogen ionization cross section (3) for 3.6 MeV u−1 6+ C impact (figure 1). The C6+ system is in the perturbative regime, νP = 0.5, which is characterized by close projectile–electron collisions. The projectile primarily transfers 1764 J Fiol and R E Olson 6+ 3.6 MeV/u C + H(n) -17 n=1 2 2 dσ/dQ⊥dEe (cm /a. u. ) 10 -18 10 n=2 n=3 -19 10 0. 2 3 45 Q⊥ (a. u.) Figure 2. Comparison of the DDCS for the ground, n = 2 and 3 excited states of hydrogen. The electron emission energy is 50 eV; the resolutions in the CTMC calculations are the same as in figure 1. momentum to the electron without a strong interaction with the nucleus target, leading to a readily apparent binary peak. The strong binary peak is the result of the collision distance between the projectile and the electron being smaller than the characteristic radius of the atom in the initial state. In two-body collisions with Coulomb interactions the perpendicular component of the momentum transferred by the projectile is given by 2νj j Q⊥ = (9) bj where bj is the two-body impact parameter and νj is the corresponding Sommerfeld parameter. For instance, √ with an electron energy Ee = 50 eV the corresponding momentum transfer is Qe = k = 2(Ee + |εi |) ≈ 2.16 au. The component parallel to the initial velocity is fixed by energy conservation Ee + |εi | Q = , (10) v which for this case is Q = 0.15 au. Thus, the projectile must approach to a distance of be ≈ 0.46 au with the electron. This characteristic close projectile–electron collision is even more pronounced for higher electron energy emission. For ionized electron energies Ee = 90 eV, the relative impact parameter decreases to be ≈ 0.36 au. The projectile–electron impact parameter be must be compared with the radial expectation value of the atom r0 . For the ground state this radius is r0 1s = 1.5 au, while the expectation values of the hydrogen atom in the n = 2 and 3 excited states are r0 n=2 = 5, 6 au for = 0, 1 and r0 n=3 = 10.5, 12.5, 13.5 au for = 0, 1, 2, 3, respectively. The distance of closest approach between the projectile and the target nucleus bR ≈ b ≈ r0 ± be increases with the quantum number n. Thus, according to (9) the momentum transferred to the recoil ion decreases and binary-encounter collisions dominate. Shown in figure 2 are the cross sections Three-body dynamics in hydrogen ionization by fast highly charged particles 1765 for ionization of the n = 2, 3 excited states. The initial n = 2 and 3 states were described by means of a microcanonical ensemble of energy En = −1/2n2 in the CTMC calculations. In order to perform a fair comparison between the two theories, the CDW results for n = 2 have been averaged over the 2s and 2p 0, ±1 states. Quantum mechanical calculations for the n = 3 initial state are not presented because their computing times are prohibitive within the present approximations. The cross sections for ionization of the excited states of hydrogen in figure 2 are strongly peaked around Q⊥ ≈ k accordingly with the simple projectile–electron binary collision picture. In fact, the binary peak width for the excited states is mainly determined by the electron’s Compton profile, since the target nucleus is far removed from the interaction region. For such perturbative collisions a laser excited target will present the opportunity to study collisions of ions with essentially a free-electron gas target. The study of the ionization of excited states of hydrogen does not only present theoretical interest. Although the experimental production of the atomic hydrogen target in either ground or excited states is difficult, the cross section for ionization of alkali atoms is closely related to the present calculations. These atoms, having only one active electron, are expected to present the same qualitative behaviour as atomic hydrogen. It has recently been possible to produce a cold target of alkali atoms by means of magneto-optical traps (MOT). This technique allows one to study collisional processes involving both ground and excited initial states (van der Poel et al 2001, Turkstra et al 2001, Flechard et al 2001). In order to study in more detail the mechanisms of ionization we have plotted in figure 3 the TDCS dσ/dQ⊥ dEe de in the plane defined by the initial velocity of the projectile and the perpendicular momentum transfer vector Q⊥ . We have chosen four values of Q⊥ = 0.5, 1, 2, 3 au. These values correspond to regions below, at and above the binary collision condition Q ≈ k for 50 eV electrons. For all cases the electrons are emitted along the direction of the momentum transfer (marked with the arrows in the plots). For Q⊥ = 0.5 and 1 au the recoil ion momentum is directed opposite to the direction of the electron emission. In these cases the electron and the recoil ion are emitted back-to-back, the recoil ion balancing a fraction of the momentum of the ionized electron since the total momentum transfer is fixed. For values of Q⊥ equal to, or greater than, the electron momentum, the recoil ion is constrained by momentum conservation KR = Q − k to the angular range θR− < θR < θR+ , where θR± = θQ ± sin−1 (k/Q) and θQ is the polar angle of the momentum transfer Q. For instance, for Q⊥ = 2 au the recoil must lie in the half of the plane given by the Q⊥ vector. For Q⊥ = 3 au such a restriction is more evident as the recoil momentum angle is limited to a narrow range, 46◦ < θR < 126◦ . This kinematic constraint gives rise to singularities in the recoil ion spectra. In figure 3 the CDW calculations clearly display ‘wings’ for Q⊥ = 2 and 3 au at the positions where several electron angles contribute to one recoil ion angle. The CTMC calculations also display the same tendency, but because of the finite resolution used for the electron energy, momentum transfer and collision plane azimuthal angle, the ‘wings’ are smoothed. The CDW predictions are clearly confirmed by the CTMC calculations. 4. Ionization by highly charged ion impact In figure 4 are shown the DDCSs for ionization of ground state hydrogen by 3.6 MeV u−1 Au53+ . The shapes of the DDCS differ from those observed for C6+ at the same electron energies and values of momentum transfer. The Au53+ system shows a dominance of low momentum transfer collisions. In contrast, binary-encounter electrons portrayed by the so-called Bethe ridge characterized the spectra in the C6+ case. However, neither the CTMC nor the CDW-B1 calculations display a strong contribution in the binary peak region for Au53+ impact. These 1766 J Fiol and R E Olson 90 3 120 90 60 120 2 30 150 0 180 0 1 0 180 0 3 210 Q⊥ = 0.5 a. u. 2 330 210 Q⊥ = 1.0 a. u. 330 6 2 2 TDCS (10 cm /a. u. sr) 30 150 3 1 240 3 300 240 -18 270 120 300 270 90 30 90 1.5 60 120 60 1.0 20 30 150 30 150 0.5 10 0 180 0 10 0.0 180 0 0.5 210 Q⊥ = 2.0 a. u. 20 30 60 6 240 300 270 330 210 Q⊥ = 3.0 a. u. 1.0 1.5 240 330 300 270 Figure 3. TDCS for ionization of H by 3.6 MeV u−1 C6+ in the plane defined by the initial velocity and the perpendicular momentum transfer Q⊥ . CTMC (symbols) results for fixed values of the projectile transverse momentum and electron energy are compared to CDW-B1 (lines) as a function of the electron (solid symbols and lines) and recoil (open symbols and dashed lines) angles. The CTMC resolutions were set to /Ee = ±10 eV, azimuthal angle /ϕ = ±20◦ and /Q⊥ = ±0.1 au. results for atomic hydrogen mirror those previously presented for 3.6 MeV u−1 Au53+ + He single ionization (Olson and Fiol 2001). The interpretation of the Au53+ + H(1s) behaviour, νP = 4.42, as compared to the perturbative C6+ case, νP = 0.5, follows the explanation given for the helium target in our previous paper. For coulomb collisions the Au53+ projectile–electron collision distance (equation (8)) for electrons emitted with 50 eV (Q⊥ ≈ 2.15 au) is be ≈ 4.1 au, which unlike in the C6+ case is considerably larger than the radius of the atom. The nuclear–nuclear impact parameter in such collisions is estimated to be about 4.1 ± 1.5 au. For these distances, the momentum transferred from the projectile to the target nucleus is also large and ranges from 1.5 to 3.5 au with the recoil ion momentum vector pointed opposite to that of the ionized electron. Vector addition of the electron and recoil ion momenta results in small values of overall momentum transfer. Thus, we find that Au53+ collisions strongly involve all three bodies and cannot simply be considered as a binary encounter collision between the projectile and electron as in the C6+ case. Three-body dynamics in hydrogen ionization by fast highly charged particles 1767 -13 10 3.6 MeV/u Au + H(1s) Ee = 10 eV -14 10 50 eV 2 dσ/dQ⊥dEe (cm /a. u. ) 53+ -15 2 10 90 eV -16 10 -17 10 -18 10 0. 2 3 4 5 Q⊥ (a. u.) Figure 4. The same as in figure 1 except that the projectile is now 3.6 MeV u−1 Au53+ . It is possible that collisions with highly charged projectiles can induce polarization of the initial state leading to projectile electron collisions at larger impact parameters than those given here (Moshammer et al 2001). However, one can argue that the collision time is too short for polarization of the target to be important. The collision time is approximately given by τ = 2b/v. For an impact parameter of 1.5 + 4.1 au, this leads to a collision time of 0.93 au. The orbital period of H(1s) is approximately 9.4 au. Thus, as in the He case we expect insignificant polarization of the target before the collision. Although the CDW quantum mechanical approach employed in this work may lack these polarization effects, they are included in the CTMC treatment (see, for example, Olson (1979)). The excellent agreement between theories observed in figure 4 indicates that polarization effects are not important in this system. From our discussion it is apparent that the underlying physics of these collisions is determined by the magnitude of the radial expectation value of the atom compared to the pairwise ranges of interaction. In figure 5 we show the average impact parameter b from our CTMC calculations for the two collision systems as a function of the perpendicular component of the momentum transfer at the three electron energies investigated, Ee = 10, 50 and 90 eV. The impact parameter of C6+ is observed to be comparable to r0 . Moreover, at 50 and 90 eV we observe a peak in the impact parameter distribution for momentum transfer corresponding to the binary peak position. Using equation (9), it is clear that the recoil ion has a minor contribution in the binary peak region for the C6+ system. On the other hand, the Au53+ impact parameters do not possess maxima and are considerably larger, confirming that both the ionized electron and recoil ion participate in the overall momentum balance. In order to assess the importance of the relative distances between particles as compared to the range of interactions, we display in figure 6 the ionization cross sections for the n = 1, 2 and 3 states of hydrogen. The shape of the DDCS for Au53+ impact on the excited states tends to that predicted for the C6+ perturbative case as the radius of the atom becomes larger than the range of the interaction. For instance, for 50 eV the projectile–electron distance be ≈ 4.1 au 1768 J Fiol and R E Olson 10 53+ < b > (a. u.) Au 1 6+ C 10 eV 50 eV 90 eV 0.1 0. 3 4 5 2 Q⊥ (a. u.) Figure 5. Average impact parameter for H(1s) ionization by 3.6 MeV u−1 Au53+ and C6+ . -14 10 3.6 MeV/u Au 53+ + H(n) -15 10 n=2 2 2 dσ/dEe dQ⊥ (cm /a. u. ) n=1 -16 10 n=3 -17 10 0. 2 3 4 5 Q⊥ (a. u.) Figure 6. Comparison of DDCS for the Au53+ + H(n) system with n = 1, 2 and 3 for 50 eV electron emission. Solid curves represent the CDW-B1 calculations; and the symbols represent the CTMC results. The CTMC resolutions are as in figure 1. is much smaller than the size of the atom in the n = 3 excited state r0 n=3 ≈ 12 au. Thus, the momentum transferred to the nucleus ranges from 0.5 to 1 au, similar to the C6+ + H(1s) case. For Au53+ impact the excited states show a small shoulder at Q⊥ ≈ 3 au, the origin of which will be discussed in the next section. Three-body dynamics in hydrogen ionization by fast highly charged particles 1769 5. Collision dynamics Our study of hydrogen ionization by C6+ and Au53+ impact indicates that the range of the interaction as compared to the atomic spatial distribution determines the overall shape of the DDCS. The marked differences in the results for ground and excited states of hydrogen support this idea. In this section the different collision dynamics that lead to the observed spectra are discussed. A detailed investigation of the three-particle dynamics can be performed through the study of kinematically complete cross sections (Schulz et al 2001a, 2001b). Figure 7 shows the Au53+ TDCS for electrons emitted at 50 eV in the plane defined by the v and Q vectors. For small values of Q⊥ the electrons are emitted along the direction of the momentum transfer, i.e. at an angle θQ associated with the Q vector. As expected, in these collisions the recoil ion is found along the −Q vector. Such angular distributions are in accord with a binary-encounter collision picture. As the perpendicular momentum transfer is increased, a second peak emerges at an angle 360◦ − θQ . With a further increase in momentum transfer this second peak dominates the spectra, as is shown in the right-most plots in figure 7 for Q⊥ = 1.6 au. This tendency is confirmed for higher perpendicular momentum transfer values. This behaviour is quite surprising since it has never been observed for other systems. In particular, the TDCS for C6+ impact, plotted in figure 3, does not present a similar trend. Moreover, the transition for electron scattering at angle θQ to the angle 360◦ − θQ occurs before one reaches momentum transfers associated with binary peak scattering, Q⊥ = 1.9 au. Our CDW and CTMC calculations indicate that the transition between the two different angular distributions occurs quite rapidly. Both theories show that the distribution switches between both regimes in a narrow interval /Q⊥ ≈ 0.2 au. However, there is a discrepancy in the value at which the transition takes place. For the CTMC results the value of the transition is Q⊥ ≈ 1.1 au, while in the CDW-B1 and CDW-EIS models it is slightly higher, Q⊥ ≈ 1.2 au. The angular distributions presented in figure 7 are related to different collision mechanisms. The peak observed near to the angle θQ of the momentum transfer Q is produced in a distant collision as shown in figure 8 (a). This kind of collision is similar to the binaryencounter mechanism in that the projectile–electron interaction is stronger than that between the two nuclei. This mechanism is the main one responsible for ionization in the C6+ case. Its signature is the electron spectra peaking near θQ with the recoil ions found scattered back-toback at θQ + 180◦ . A second collision mechanism is displayed in figure 8(b) with the projectile penetrating the electron cloud. These are related to small impact parameter collisions resulting in large momentum transfer, and contribute in the C6+ case for Q⊥ > k, but are not observed in the Au53+ + H(1s) system because the range of interaction exceeds the radial dimensions of the target atom. However, by increasing the atomic dimensions this kind of collision produces the peak observed in the DDCS (figure 6) near Q⊥ = 3 au in the Au53+ ionization of the n = 3 state of hydrogen. Here, the collision distance and the momentum transfer between the projectile and the electron are approximately be = 4 au and Qe⊥ ≈ 2.1 au. For collision 8(b) the distance between nuclei can be estimated to bR = r0 n=3 − be ≈ 8 au. Thus, the momentum transfer to the target ion is QR⊥ ≈ 1 au. In this case both the electron and the recoil scatter in the same direction so that the net momentum transfer is now Q⊥ ≈ 3 au. Figure 8(c) shows a third mechanism where the projectile suffers a stronger interaction with the recoil ion than with the electron. After the ionization takes place the electron is forced to swing by its parent nucleus. We observe that this mechanism only contributes when the range of the interaction is larger than the size of the atom. It is important to note that in the case of ‘swing by’ electrons, neither the electron nor the recoil ion is emitted along the direction of momentum transfer (see Q⊥ = 1.6 au in figure 7). 1770 J Fiol and R E Olson 90 -18 2 TDCS (10 cm /a. u.) 450 120 90 450 60 120 300 300 30 150 150 30 150 150 Q⊥= 0.8 ± 0.1 a. u. 0 180 Q⊥ = 0.8 a. u. 0 150 0 180 0 150 330 210 300 330 210 300 240 450 300 240 450 270 120 90 45 60 -18 120 60 30 30 150 15 0 180 30 150 15 Q⊥ = 1.1 ± 0.2 a. u. 2 TDCS (10 cm /a. u.) 30 Q⊥ = 1.2 a. u. 0 15 0 180 0 15 330 210 30 330 210 30 240 45 300 240 45 270 120 300 270 90 24 90 24 60 16 120 60 16 30 2 150 8 -18 TDCS (10 cm /a. u.) 300 270 90 45 0 180 30 150 8 Q⊥ = 1.6 ± 0.3 a. u. Q⊥ = 1.6 a. u. 0 8 0 180 0 8 330 210 16 24 60 330 210 16 240 300 270 24 240 300 270 Figure 7. Classical CTMC and quantum mechanical CDW TDCSs for ionization of ground state hydrogen by Au53+ impact to produce 50 eV electrons. In the left column the CTMC results for the electron (solid squares) and the recoil ion (open circles) were convolved over the energy /Ee = ±10 eV, and azimuthal angle /ϕ = ±20◦ resolutions. The resolutions in the perpendicular momentum transfer are specified in the plots. In the right panel the CDW-B1 (electron: full lines, recoil: dashed lines) and CDW-EIS (symbols as in CTMC) results are displayed. Three-body dynamics in hydrogen ionization by fast highly charged particles MP v + (a) (b) KR Q k 1771 + + - - k MP v Q + KR - k KR (c) + MP v + Q Figure 8. Schematic diagram of the three ionization mechanisms leading to (a) binary electrons, (b) three-body collisions and (c) swing by electrons. 6 5 x (a. u.) 4 t2 t1 t0 t2 3 "binary electrons" 2 t1 1 0 t0 -1 5 t2 4 "swing-by electrons" x (a. u.) 3 2 1 t1 t0 t2 0 -1 -2 t1 t0 -5 -4 -3 -2 -1 0 1 z (a. u.) 2 28 29 30 Figure 9. CTMC trajectories illustrating the mechanisms that produce binary electrons (upper plot) and swing by electrons (lower plot). The projectile follows an almost straight line trajectory and the recoil ion remains approximately at rest. The horizontal axis, displaying the component of the position of the particles parallel to the initial velocity is approximately proportional to the time. The projectile scattering plane defines the x component. The signature of this mechanism is observation of electrons emitted at 360◦ −θQ with the recoil ion being found peaked near 180◦ − θQ . Note, these are not the recoil electrons commonly found in electron impact ionization that peak at 180◦ + θQ . 1772 J Fiol and R E Olson To confirm our interpretation of the collision mechanisms, in figure 9 we show CTMC trajectories illustrating the ‘binary’ and ‘swing by’ electron dynamics for Au53+ + H(1s). In the upper plot, the binary electrons are produced in distant collisions (b ≈ 5 au). On the other hand, the swing-by electrons are produced in closer collisions, with impact parameter b ≈ 2 au. The projectile passes on one side of the target nucleus while the electron is on the opposite side. The direction of the perpendicular momentum transfer (not distinguished in the figure) is determined by the interaction with the recoil ion, which is opposite to the case for binary electrons. 6. Conclusions Ionization cross sections for fast ion–atom collisions have been calculated in the perturbative (C6+ , νP = 0.5) and non-perturbative (Au53+ , νP = 4.4) regimes. In general there is excellent agreement between calculations made using the CTMC and CDW-B1 and CDW-EIS theories. This agreement extends to doubly differential and kinematically complete TDCSs. Three collision mechanisms have been identified. The range of the projectile–electron interaction (collision distance) as compared to the size of the atom determines the relative importance of each of them. The first process is the well known binary encounter between the projectile and electron whose signature is the Bethe ridge. This mechanism is compromised for the case of highly charged ions impact because the long-range internuclear interaction allows the recoil ion to also actively participate in the collision. Because of the long range of interaction compared to the orbital dimensions of the target atom, no Bethe ridge is found for 3.6 MeV u−1 Au53+ + H(1s) collisions. In contrast, binary-like electrons are found for high incident velocities, even in nonperturbative collisions, when the size of the atom is larger than the collision distance. This was verified by studying the ionization of n = 2 and 3 excited states of hydrogen. Here, a second mechanism also occurs when the projectile penetrates the electron cloud interacting simultaneously with the recoil and the electron. Such collisions result in large momentum transfer and can give rise to structures on the DDCS. A laser-excited alkali atom target could provide the framework for the experimental study of these kind of processes. The third mechanism results in our prediction of ‘swing by’ electrons. For strong nonperturbative collisions where the range of interaction is greater than the target atom’s orbital dimensions, it is possible for the projectile to closely interact with the recoil ion while the electron is forced to swing by its parent ion. This mechanism leads to the ionized electron and recoil ion spectra not being found peaked parallel to the momentum transfer vector, but in a mirror position to it. To date these electrons have not been observed. Acknowledgment The authors acknowledge the support from the Office of Fusion Energy Sciences, DOE. References Abrines R and Percival I C 1966 Proc. Phys. Soc. London 88 861 Cocke C L and Olson R E 1991 Phys. Rep. 205 153 Crothers D S F and McCann J F 1983 J. Phys. B: At. Mol. Phys. 16 3229 Ehrhardt H, Knoth G, Schlemmer P and Jung K 1985 Phys. Lett. A 110 92 Fainstein P D, Ponce V H and Rivarola R D 1991 J. Phys. B: At. Mol. Opt. Phys. 24 3091 Fiol J, Rodrı́guez V D and Barrachina R O 2001 J. Phys. B: At. Mol. Opt. Phys. 34 933 Three-body dynamics in hydrogen ionization by fast highly charged particles 1773 Fiol J and Olson R E 2002 J. Phys. B: At. Mol. Opt. Phys. at press Flechard X, Nguyen H, Wells E, Ben-Itzhak I and DePaola B D 2001 Phys. Rev. Lett. 87 123203 Fock V 1935 Z. Phys. 98 145 Garibotti C R and Miraglia J E 1980 Phys. Rev. A 21 572 Hardie D J W and Olson R E 1983 J. Phys. B: At. Mol. Phys. 16 1983 Illescas C and Riera A 1999 Phys. Rev. A 60 4546 Lahmam-Bennani A 1991 J. Phys. B: At. Mol. Opt. Phys. 24 2401 Moshammer R, Perumal A, Schulz M, Kollmus H, Mann R, Hagmann S and Ullrich J 2001 Phys. Rev. Lett. 87 223201 Olson R E 1979 Phys. Rev. Lett. 43 126 Olson R E 1996 Atomic, Molecular & Optical Physics Handbook ed G W F Drake (New York: AIP) ch 56 Olson R E and Fiol J 2001 J. Phys. B: At. Mol. Opt. Phys. 34 L625 Olson R E and Salop A 1977 Phys. Rev. A 16 531 Reinhold C O and Olson R E 1989 Phys. Rev. A 39 3861 Rodrı́guez V D 1996 J. Phys. B: At. Mol. Opt. Phys. 26 275 Salin A 1989 J. Phys. B: At. Mol. Opt. Phys. 22 3901 Schmitt W, Moshammer R, O’Rourke F S C, Kollmus H, Sarkadi L, Mann R, Hagmann S, Olson R E and Ullrich J 1998 Phys. Rev. Lett. 81 4337 Schultz D R and Olson R E 1988 Phys. Rev. A 38 1866 Schulz M et al 2001a J. Phys. B: At. Mol. Opt. Phys. 34 L305 Schulz M, Moshammer R, Perumal A N and Ullrich J 2001b J. Phys. B: At. Mol. Opt. Phys. submitted Turskstra J W, Hoekstra R, Knoop S, Meyer D, Morgenstern R and Olson R E 2001 Phys. Rev. Lett. 87 123202 Ullrich J, Moshammer R, Dörner R, Jagutzki O, Mergel V, Schmidt-Böcking H and Spielberger L 1997 J. Phys. B: At. Mol. Opt. Phys. 30 2917 van der Poel M, Nielsen C V, Gearba M A and Andersen N 2001 Phys. Rev. Lett. 87 123201 Wood C J, Feeler C R and Olson R E 1997 Phys. Rev. A 56 3701