The dynamics of He double ionization by 2 keV electron impact is studied experimentally for a momentum
transfer of 0.6 a.u. at excess energies of 10 and 40 eV. Complete sets of fivefold differential
cross sections are presented for all electron emission angles in coplanar geometry. Contributions beyond
the first Born approximation are identified comparing experimental data with first order convergent
close-coupling calculations which are in considerably better agreement with the present experiment than
with the earlier measurement of Kheifets et al. [J. Phys. B 32, 5047 (1999)]

Dorn, Alexander

Hohr, C

Kheifets, Anatoli

Moshammer, R

Najjari, B

Schroter, C D

Ullrich, Joachim

Physical Review Letters

The Australian National University

VOLUME 86, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 23 APRIL 2001Double Ionization of Helium by Electron-Impact: Complete Picturesof the Four-Body Breakup DynamicsA. Dorn,1 A. Kheifets,2 C. D. Schröter,1 B. Najjari,1 C. Höhr,1 R. Moshammer,1 and J. Ullrich1,31Universität Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg, Germany2Research School of Physical Sciences and Engineering, Australian National University, Canberra, ACT 0200, Australia3Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany(Received 14 November 2000)The dynamics of He double ionization by 2 keV electron impact is studied experimentally for a mo-mentum transfer of 0.6 a.u. at excess energies of 10 and 40 eV. Complete sets of fivefold differentialcross sections are presented for all electron emission angles in coplanar geometry. Contributions be-yond the first Born approximation are identified comparing experimental data with first order convergentclose-coupling calculations which are in considerably better agreement with the present experiment thanwith the earlier measurement of Kheifetset al. [J. Phys. B32, 5047 (1999)].DOI: 10.1103/PhysRevLett.86.3755 PACS numbers: 34.80.Dpnet-hivnrvsdeirr-uf-s.thedl asmi-ryallfittedrerstoner-wndthevelasasu-dc-p-ticneleseas-A full understanding of the correlated fragmentatiodynamics of an atomic system upon impact of a chargparticle or a photon is one of the central aims of atomcollision physics. While for the most basic three-bodbreakup processes such as electron impact single ionizaof hydrogen or photo double ionization (PDI) of heliuma rather profound understanding is emerging (see [1–and numerous references therein), the transition to trmany-body systems still remains a challenge, both expementally and theoretically. With the availability of new experimental multicoincidence techniques, we now have topportunity to obtain complete momentum space picturof complex many-body processes such as double ionizatof helium by electron impact, a so-callede, 3e reactionleading to four unbound particles [4,5].On the theoretical side, fairly sophisticated models habeen employed [6–8] to describe the firste, 3e experi-ment on helium [9]. Because of a large disparity in the eergies of the fast projectile (5.6 keV) and the slow ejecteelectrons (10 or 4 eV), it was natural to limit the projectiletarget interaction to the first order while treating the inteaction of the two slow ejected electrons nonperturbative[6,7]. However, these first Born calculations produced cofusing results. Agreement with the experiment was pooboth in magnitude and in shape, especially at the lowe4 eV ejected electron energy. Some of the experimenfeatures, such as strong peaks in the cross section for smrelative electron emission angles, could not be reproducat all. Agreement with the experiment seemed to improwhen a full four-body model was employed [8]. In themeantime, the authors of Ref. [7] did not rule out the posibility that some “hidden” artifacts had actually affectethe experiment. Therefore, the matter remains controvsial, and clarifying experimental and theoretical studies ahighly desirable.With the aim to resolve this controversy, we presentthis Letter the data from a kinematically complete experment for 2 keV electron impact on helium in which, fo0031-90070186(17)3755(4)$15.00dicyion3]ueri-eesone-d--lyn-r,sttalallede-r-reni-the first time, a large part of the 12-dimensional, fouparticle final-state momentum space is mapped with sficient statistics to extract fully differential cross sectionThus, theory can be tested for kinematics ranging fromoptical limit to high momentum transfer up to 5 a.u. anover a large range of relative emission angles as welenergy partitions between the two ejected electrons. Silarly to the previous experiments [7,9] performed veclose to the optical limit, we select here a relatively smmomentum transfer ofjqj  0.6 a.u., equal energies othe ejected electrons ofEb  Ec  5 and 20 eV, and thecoplanar geometry where the ejected electrons are emwithin the scattering plane of the projectile. We compathese data with calculations performed within the same fiBorn model as employed in [7]. The slow ejected electrinteraction was treated nonperturbatively using the convgent close coupling (CCC) method. The method is knoto yield very reliable quantitative results for the relatePDI process when the two electrons are ejected fromhelium atom by photon impact [10,11].The present experiment is performed applying a nomultielectron–recoil-ion coincidence technique which hbeen demonstrated to enablee, 3e experiments on he-lium [4]. Details of the working principle and the datanalysis are described elsewhere [5,12]. In short, apersonic jet provided a well localized (2 mm diam) andense1011 atomscm3 helium target at the intersectionpoint with a pulsed electron beam. Ions and slow eletrons produced in ionizing collisions are extracted to oposite directions by means of static electric and magnefields and detected by two position sensitive multichanplate detectors. From the measured positions and timof flight, the momentum vectors of two slow electronskbandkc Eb,c , 30 eV and the momentum vector of threcoiling ionkHe11 emitted in ane, 3e reaction are de-termined. The kinematics of the fast scattered electronwell as the momentumq transferred by the scattered projectile follow from momentum conservation:© 2001 The American Physical Society 3755VOLUME 86, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 23 APRIL 2001k0 2 ka  q  kb 1 kc 1 kHe11. (1)Here, k0 and ka are the momenta of the incoming and thescattered projectile, respectively.The electron detector is equipped with a fast delay-linereadout and a multihit time-to-digital converter. There-fore, position as well as arrival time of the second elec-tron emitted in a double ionization event are determinedif the flight-time difference between two electrons exceeds15 ns. Whereas the complete final-state momentum spaceis mapped for all ions with kHe11 , 5 a.u., this detectordead time results in a small loss of the total momentumspace for the second electron hitting the detector.In Fig. 1(a), the fivefold differential cross section(FDCS) is given as a function of the ejected electronsemission angles ub and uc with respect to the forwardprojectile beam direction. The momentum transfer isjqj  0.6 6 0.2 a.u.; the energies of the ejected electronsare Eb  Ec  5 6 2.5 eV. We have chosen a densityplot representation to visualize a large range of angles uband uc being particularly well suited to study the overallstructure of the cross section and to reveal symmetriesas well as nodal lines. The angular range which isnot affected by reduced detection efficiency due to theelectron detectors dead time is inside the circular solidlines in the diagrams.The cross section pattern consists of a four peak struc-ture. Both peaks in the upper left are equivalent to thepeaks in the lower right of the diagram (marked A and B)since, for symmetric energy sharing Eb  Ec, both ejectedelectrons are interchangeable.In Fig. 1 we also present the theoretical CCC result[1(b)] as well as the corresponding photo double ionizationcross section [1(c)] obtained by using the phenomenologi-cal parametrization from Huetz et al. [13], which is com-monly accepted to yield good agreement with experimentaldata. This parametrization nicely reveals the origin of theobserved structures in factorizing the cross section in anangular part arising from photoabsorption and a correla-tion part describing the electron-electron repulsion in thefinal state [3]. For equal energy sharing, the angular dis-tribution of the photoelectrons is given bysPDI  cosubE 1 cosucE2 ? Cb, c .Here, ubE and ucE are the emission angles relative tothe electric field vector E. The angular factor mirrors the1P0 symmetry of the final state and, hence, it gives riseto zeros in the cross section arising from dipole selectionrules: There is a first node (i) for back to back emis-sion of both ejected electrons jub 2 ucj  180± [dashedlines in Figs. 1(a)–1(c)] and a second node (ii) if the summomentum of the ejected electrons is perpendicular tothe electric field vector kb 1 kc  E or, equivalently,ubE 1 ucE  180± (dotted lines in Fig. 1). The correla-tion factor Cb, c is zero for parallel emission ub  ucand gives rise to a broad minimum of the cross sectionin the vicinity of small relative angles. Since electron3756FIG. 1. Fivefold differential cross section in coplanar scatter-ing geometry as a function of the ejected electron emissionangles ub and uc relative to the primary beam forward direction.For all diagrams, the ejected electron energies are Eb,c  5 eV.(a) Experimental cross section for E0  2 keV, jqj  0.6 a.u.The direction of the momentum transfer q is marked by arrowsin the diagram. The angular ranges which are not affected by thedetector dead time are encircled by solid lines. (b) CCC calcu-lation for jqj  0.5 a.u. (c) Cross section for photo double ion-ization obtained with the parametrization from Huetz et al. (seetext). The electric field vector E has been oriented to coincidewith the corresponding quantization axis q for the e, 3e pro-cess in (b) uE  uq  55±.VOLUME 86, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 23 APRIL 2001repulsion acts similarly for the e, 3e process, a largepart of the experimentally nonaccessible angular range, foremission of both electrons into the same direction ub  ucat equal energies, is not of major importance as the crosssection is close to zero in this region. A further prop-erty of the PDI cross section is the reflection symmetrywith respect to a plane perpendicular to the electric fieldaxis since both directions along the axis are equivalent andindistinguishable. Therefore the cross section shows re-flection symmetry with respect to the lines (i) and (ii).Examining the experimental and theoretical e, 3e crosssections, it immediately becomes obvious that both aregoverned by the same nodal lines showing close similari-ties to PDI, despite the fairly large momentum transfer ofjqj  0.6 a.u. This has been discussed previously for con-ditions even closer to the optical limit by Lahmam-Bennaniet al. [6]. The experimental uncertainty in the amount ofthe momentum transfer gives rise to an uncertainty in thedirection of the q indicated by the length of the thick solidline in Fig. 1(b) which marks angles where both electronsare emitted parallel to q. Therefore the peak widths shouldbe smeared out accordingly in this direction. The experi-mental uncertainty in the electron energies Eb,c can causethe observed partial filling of the node for back to backemission since the corresponding selection rule is strictonly for symmetric energy sharing between both electrons.The remaining differences compared to the PDI crosssection can be classified as being, first, due to nondipolecontributions and, second, due to higher order interac-tions with the projectile. First, contributions beyond thedipole approximation are the result of momentum transfersjqj larger than the minimum transfer for photoabsorptionjqgj  Egc and, consequently, do not obey the dipoleselection rules. As a result, the cross section in the CCCcalculation is finite for back to back emission with oneelectron going into the momentum transfer direction, i.e.,a configuration which is strictly forbidden by both dipoleselection rules (i) and (ii). As a further consequence, anyq . qg breaks the reflection symmetry of the cross sectionwith respect to a plane perpendicular to the quantizationaxis since the direction along q now is distinguishable fromthe one opposite to q. Thus, in Figs. 1(a) and 1(b), thecross section maximum A is stronger and less elongatedcompared to the maximum B for which both electronsenclose angles larger than 90± with q. These features be-come even more pronounced if the energies of the ejectedelectrons are higher. For Eb  Ec  20 eV [Figs. 2(a)and 2(b)] peak A becomes the dominant structure in thecross section which can be understood as follows: Dipoletransitions dominate in the transition matrix element notonly in the limit of small momentum transfer, where thetransition operator reduces to the dipole operator, but alsoif the energies of the ejected electrons are small. With de-creasing energies Eb,c, the low angular momentum termsin a partial wave expansion of the two-electron final-statewave function dominate more and more at small distancesfrom the ion, i.e., in the volume which largely determinesFIG. 2. (a) Experimental FDCS for Eb,c  20 6 4 eV andjqj  0.6 a.u. (b) CCC calculation for Eb,c  20 eV and jqj 0.5 a.u.the transition matrix element. Therefore, dipole transitionsare emphasized for Eb,c  5 eV despite the relatively largemomentum transfer. For Eb,c  20 eV, peak A is domi-nant since for the corresponding emission angles so-calledBethe kinematics is closely fulfilled. Here, the sum mo-mentum of the ejected electrons is close to the momentumtransfer kb 1 kc  q both in magnitude and direction,i.e., the ion momentum is small. This condition has beenshown by Berakdar and Klar [14] to be the preferred col-lision kinematics for the e, 3e process. Correspondingly,to the nomenclature used for the e, 2e cross sections, wename maximum A the “binary peak” and maximum Bthe “ recoil peak” since, for this maximum, the electronssum momentum is directed opposite q and the residual ioncarries considerable momentum.In all features discussed thus far, experiment and theCCC theory agree remarkably well. The most strikingremaining disagreement of experiment and CCC theory,namely, the position of peak B, can be assigned tohigher order projectile-target interactions by the following3757VOLUME 86, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 23 APRIL 2001symmetry arguments: A collision process of the firstorder reveals axial symmetry of the cross section withrespect to the momentum transfer direction. Thus, thetheoretical cross sections [Figs. 1(b) and 2(b)], which areobtained in first Born approximation for the projectile-target interaction, are invariant for inversion at the pointwhere both electron’s emission angles are equal to theangle of the momentum transfer or, equivalently, forreflection of both momentum vectors at the direction ofq. Taking into account the symmetry with respect to theline ub  uc mentioned previously, it follows that there isalso reflection symmetry with respect to the straight solidline in Fig. 1(b) corresponding to electron emission intoequal but opposite angles with respect to the momentumtransfer direction: ub 1 uc2  uq. While in theexperimental data this symmetry is closely fulfilled forthe binary peak A, the recoil peak B shows a distinctshift to larger angles for both electrons. Therefore, higherorder projectile-target interactions are of importanceand explain the failure of first order theories for certainparts of the final-state momentum space. For the higherenergies of the ejected electrons, the recoil peak Bdecreases strongly in magnitude becoming very weak forEb,c  20 eV (see Fig. 2). Since peak B is the onlyfeature showing signatures of higher order projectile-targetinteraction, we can clearly prove experimentally previousassumptions made by Kheifets et al. [7], namely, thatsecond order effects in the projectile-target interactionare more important for smaller energies of the ejectedelectrons Eb,c.Finally, the cross section for Eb,c  5 eV as functionof the mutual angle ubc  ub 2 uc integrated over thewhole range of the individual angles ub and uc is pre-sented in Fig. 3. The agreement with theory, which in thiscase is integrated over the experimental uncertainty of Eb,c,is quite satisfactory, in particular, if the reduced acces-sible angular range of the experiment is taken into account(full line in the figure). The width of the cross sectionminimum for small mutual angles is consistent with ob-servations from PDI at similar excess energies [15]. Thus,within the angular range accessible in the present experi-ment, we cannot confirm the somewhat surprising experi-mental observations from Kheifets et al. [7], who reporteda four-lobe structure for Eb,c  4 eV with two strongmaxima at relative angles of ubc  60± and 300±. It mustbe emphasized that these extra lobes where observed forboth electrons going to small forward angles, a configu-ration which is not part of the present accessible angularrange [see Fig. 1(a)]. On the other hand, we can confirma filling of the node at ubc  180± which cannot be at-tributed to be due only to instrumental angular resolutionbut also indicates nonfirst Born processes.In conclusion, the present experimental data allow fora large scale view of the final-state momentum space and,therefore, enables the identification of general symmetriesand structures of the cross section. In contrast to earlierstudies, remarkably good agreement with the CCC cal-3758FIG. 3. Cross section as a function of the mutual angle ubc jub 2 ucj integrated over all values of the individual ub and ucfor Eb,c  5 eV. The CCC calculation for the full angular rangeand for the angular range accessible experimentally is given asa dashed line and a continuous line, respectively.culations is found concerning all features in the shape ofthe cross sections which can be described within a firstorder approach, i.e., relative peak heights and widths aswell as the position of nodal lines for both cases Eb,c  5and 20 eV. We also observe a break of symmetry of thecross section with respect to the momentum transfer direc-tion which is a clear signature of higher order interactionswhich are not included in theory. For the partially inte-grated cross section as a function of the mutual angle ubc,good agreement is obtained which is in contrast to recentexperiments closer to the optical limit.Support from the Deutsche Forschungsgemeinschaftwithin the SFB 276, project B7, and within the Leibnizprogram is gratefully acknowledged.[1] T. N. Rescigno et al., Science 286, 2474 (1999).[2] C. T. Whelan, Science 286, 2457 (1999).[3] J. S. Briggs and V. Schmidt, J. Phys. B 33, R1 (2000).[4] A. Dorn et al., Phys. Rev. Lett. 82, 2496 (1999).[5] R. Moshammer et al., Nucl. Instrum. Methods Phys. Res.,Sect. B 108, 425 (1996); H. Kollmus et al., Nucl. Instrum.Methods Phys. Res., Sect. B 124, 377 (1997).[6] A. Lahmam-Bennani, I. Taouil, A. Duguet, M. Lecas,L. Avaldi, and J. Berakdar, Phys. Rev. A 59, 3548 (1999).[7] A. Kheifets, I. Bray, A. Lahmam-Bennani, A. Duguet, andI. Taouil, J. Phys. B 32, 5047 (1999).[8] J. Berakdar, Phys. Rev. Lett. 85, 4036 (2000).[9] I. Taouil, A. Lahmam-Bennani, A. Duguet, M. Lecas, andL. Avaldi, Phys. Rev. Lett. 81, 4600 (1998).[10] A. S. Kheifets and I. Bray, J. Phys. B 31, 5149 (1998).[11] A. S. Kheifets and I. Bray, Phys. Rev. Lett. 81, 4588 (1998).[12] J. Ullrich et al., J. Phys. B 30, 2917 (1997).[13] A. Huetz et al., J. Phys. B 24, 1917 (1991).[14] J. Berakdar and H. Klar, J. Phys. B 26, 4219 (1993).[15] O. Schwarzkopf, B. Krässig, and V. Schmidt, J. Phys.(Paris), Colloq. 3, C6–169 (1993).

English

Double ionization of helium by electron-impact: complete pictures of the four-body breakup dynamics

Double ionization of helium by electron-impact: complete pictures of the four-body breakup dynamics

Abstract

Similar works

Full text

Available Versions

The Australian National University