Analysis of final-state momentum distributions of ionization products in ion-atom collisions

Transcription

1 PHYSICAL REVIEW A VOLUME 53, NUMBER 5 MAY 1996 Analyi of final-tate momentum ditribution of ionization product in ion-atom colliion Y. D. Wang, 1, * V. D. Rodríguez, 2, C. D. Lin, 3, C. L. Cocke, 1 S. Kravi, 1 M. Abdallah, 1 and R. Dörner 1,4 1 J. R. Macdonald Laboratory, Department of Phyic, Kana State Univerity, Manhattan, Kana Departamento de Fíica, Univeridad de Bueno Aire, 1428 Bueno Aire, Argentina 3 The Joint Intitute for Laboratory Atrophyic, Univerity of Colorado, Boulder, Colorado Intitut für Kernphyik, Univerität Frankfurt, D60486 Frankfurt, Germany Received 2 November 1995 A general formulation utilizing three-body kinematic wa developed to analyze the final-tate momentum ditribution of the electron, the recoil, and the projectile ion, for the ionization proce in ion-atom colliion. Information on ionization dynamic can be identified and analyzed from the perpective of momentum ditribution. The mechanim of electron capture into the projectile continuum wa found to contribute a finite value at the kinematical threhold in the longitudinal recoil-ion momentum ditribution. Detailed calculation uing the continuum ditorted wave eikonal initial tate approximation are compared with two recent meaurement on momentum ditribution of the recoiling ion and the ionized electron in ingle ionization of He by proton and by highly charged projectile. PACS number : x, Fa I. INTRODUCTION The ionization proce in ion-atom colliion provide fertile ground to tet our undertanding and our ability to decribe the breakup of baic Coulombic three-body ytem. While much of our knowledge of ionization dynamic originate from the tudy of the ejected electron pectra and ome from the projectile angular ditribution, recent development in recoil-ion momentum pectrocopy have added new dimenion for the detailed tudy of ion-atom colliion dynamic. Momentum ditribution of the recoiling ion and the electron have been carried out in the lat few year 1 7. Together with the meaurement of ejected electron pectra and the projectile angular ditribution, thee meaurement offer a wealth of information on ionization dynamic and can erve a a tringent tet for theory. Mot of the exiting theoretical analyi on recoil-ion momentum ditribution have been carried out uing the n-body claical trajectory Monte Carlo method (nctmc 1,2,4 6. The momentum ditribution of the recoil ion, however, are not independent of the momentum ditribution of the cattered projectile and/or of the ejected electron. In a recent paper, Rodríguez, Wang, and Lin 8 analyzed the longitudinal recoil-ion momentum ditribution in ion-atom ionization by conidering the three-body kinematic. Uing the continuum ditorted wave eikonal initial tate CDW- EIS approximation 9, it wa hown how the mot important ionization mechanim in fat ion-atom colliion can be identified from the recoil-ion momentum ditribution. Thee include: electron capture into the projectile continuum ECC 10 12, the emiion of oft electron SE 13, and projectile-electron binary colliion 14. The analyi wa demontrated for ingle ionization of He by proton 8 and * Electronic addre: ydwang@phy.ku.edu Electronic addre: vladimir@chico.df.uba.ar Permanent addre: Department of Phyic, Kana State Univerity, Manhattan, KS by highly charged Ni 24 ion 15. The only other publihed non-ctmc calculation were done in 1991 by Fukuda et al. 16 who ued the firt Born and the eikonal ditortedwave approximation to calculate the recoil-ion momentum ditribution. Recently the CDW-EIS method wa alo ued by O Rourke, Shinamura, and Crother to analyze the tranvere recoil-ion energy ditribution 17. However, none of thee calculation addreed the important conequence of three-body kinematic on longitudinal recoil-ion momentum ditribution that were detailed in 8. While the major feature of ionization dynamic at high velocitie are relatively well undertood 18,19, mechanim of ionization at intermediate to low energie and by highly charged ion are till a ubject of great controvery. In the low-energy region, ionization i a rather weak proce compared with the dominant charge tranfer. Thi i alo the region where the Coulomb interaction among the three charged particle in the final tate are expected to play an important role. In term of total ionization cro ection, ionization of one-electron target by proton and by highly charged ion ha been addreed uing the adiabatic electron uperpromotion model and by the extenive twocenter cloe-coupling method of Wang et al. 23. Experimental data on the momentum ditribution of the ionized electron will undoubtedly be ueful to help better clarify the validity of the theoretical model and to provide inight into the importance of final-tate interaction among the three charged particle. In thi paper, the analyi of recoil-ion momentum ditribution reported in 8 i further developed. Our goal i to give a complete determination of the final-tate momentum ditribution for the electron, the projectile, and the target recoiling ion. From thee ditribution, we identify and decribe the important feature of the ionization dynamic. The method i ued to analyze the recent meaurement by Dörner et al. 6 and by Kravi et al. 7 for momentum ditribution of the recoil ion and electron in the ingle ionization of helium atom. Comparion with other theoretical approache are made wherever available. Throughout thi /96/53 5 / /$ The American Phyical Society

2 53 ANALYSIS OF FINAL-STATE MOMENTUM DISTRIBUTIONS paper, atomic unit are ued unle otherwie tated. II. THEORY A. The quintuply differential cro ection The mot detailed information about ingle ionization of atom by heavy-ion impact can be obtained experimentally from the meaurement of five of the nine momentum component for the three particle in the final tate. The other four component can be deduced from energy and momentum conervation of the three particle. Naturally there are numerou way to define the quintuply differential cro ection. One choice that ha been tudied extenively in ionatom ionizing colliion i the quintuply differential cro ection in projectile cattering angle ( P ) and the ejected electron energy ( e ) and angle ( e ): d 5 P e e T if 2, where T if i the tranition matrix, and i the reduced ma. Equation 1 i given in the center-of-ma frame. For heavyion colliion, we may alo introduce d 5 dp e v 2 T if 2, the quintuply differential cro ection relating the tranvere momentum tranfer ( ) and the electron momentum (p e ). Here v i the velocity of the projectile and the momentum tranfer i P P K i K f P P v ˆ, where K i (K f ) i the initial final momentum of the projectile. Integrating over P ( ), we can obtain the doubly differential cro ection in electron energy momentum and angle. In thi paper, our goal i to tudy the momentum ditribution of the recoiling ion and the electron. It i thu deirable to conider a quintuply differential cro ection relating the recoil-ion momentum p R and the electron momentum p e. Below we obtain the quintuply differential cro ection d 5 /dp e dp R dp R from d 5 / dp e by mean of energymomentum conervation. The longitudinal momentum balance for the three particle along the incident beam direction, p P p R p e Q/v e i /v, where p P i the longitudinal momentum tranfer of the projectile, i i the binding energy of the target atom in the initial tate, and e i the ejected electron energy. Thi equation i correct to O(1/M P ) and O(1/M T ), where M P (M T )i the ma of the heavy projectile target. The fact that the longitudinal projectile momentum tranfer i related to the Q value of the ytem lead to the imple tranformation among longitudinal momentum ditribution of the projectile, the electron and the recoiling ion. It alo connect the longitudinal recoil-ion momentum ditribution with the electron pectrocopy DDCS 8. On the other hand, the tranvere momentum conervation p e p R doe not lead to any imple tranformation. It would then be more convenient to obtain tranvere momentum ditribution from the T matrix. Detail will be given later. Uing Eq. 3 and 4, we can how that d 5,p e p R,p R d5, 5 dp e dp e dp R dp R J where the Jacobian J i given by,p J e v. 6 p R,p R v i 2p R v p e Thi expreion yield differential cro ection relating all three particle. The variou form of the quintuply differential cro ection introduced above are related to each other through the fundamental law of energy and momentum conervation. They are directly related to the tranition T matrix and can be ued to derive a variety of differential cro ection of fewer dimenion. Complete experiment, where the momentum of each of the three particle in the final tate i determined, are a tandard technique in electron impact ionization tudie (e- 2e). For ion-atom colliion they only recently became feaible 24 and no fivefold differential cro ection have been publihed o far to our knowledge. However, much information on ionization dynamic can be obtained by tudying differential cro ection concerning one or two of the three particle. In the following we derive ome differential cro ection uitable for the decription of final-tate momentum ditribution in ion-atom ionizing colliion. B. Recoil-ion momentum ditribution Let u firt conider the longitudinal recoil-ion momentum ditribution, /dp R. From the energy-momentum conervation relation Eq. 3, we can how that the recoil-ion momentum ditribution i related to the doubly differential cro ection in electron energy and angle DDCS 8, dp R e e 1 p e d 2 e co e e, where the lower and upper integration limit are implicitly given by alo from Eq. 3 7 p e v co e v 2 co 2 e 2 p R v i 8 and e 1 2 (p e ) 2. The baic kinematic relation given in Eq. 3 impoe a evere contraint on the longitudinal momentum ditribution in ion-atom ionization. In 8, Rodríguez, Wang, and Lin firt pointed out that there i a kinematic threhold in the longitudinal recoil-ion momentum ditribution given by p min R v 2 i v. At p R p min R electron are emitted at zero degree with the ame velocity v a the projectile. Thi correpond to electron capture into the projectile continuum ECC. The ECC 9

3 3280 Y. D. WANG et al. 53 electron are characterized a a cup Coulomb divergence in the zero-degree DDCS pectrocopy. In the longitudinal recoil-ion momentum ditribution, however, thee electron contribute to a finite cro ection. Thi can be undertood ince the upper and lower integration limit in Eq. 7 approach the ame value at ECC i.e., p e p e v). Meaurement on the longitudinal recoil-ion momentum ditribution can therefore provide unambiguou evidence for the ECC ionization mechanim. We now turn to the tranvere recoil-ion momentum ditribution /dp R. Uing the quintuply differential cro ection introduced in Eq. 5 we can ee that the tranvere recoil-ion momentum ditribution i expreed a 2 p dp Rdp R e dpr d 5 dp e dp dp. 10 R R The kinematic relation of Eq. 4 will be ued to carry out the integration. It hould be pointed out that projectile-target internuclear interaction i known to make an important contribution to the projectile angular ditribution. It alo affect the tranvere recoil-ion momentum ditribution. C. Electron momentum ditribution Experiment on electron momentum ditribution in coincidence with recoil-ion momentum ditribution have only been done very recently. Kravi et al. 7 meaured electron momentum ditribution in ingle ionization of He by proton and by C 6 ion at low to intermediate energie. Mohammer et al. 5 meaured longitudinal electron momentum ditribution in ingle ionization of He by highly charged Ni 24 ion. Theoretically, both the longitudinal and the tranvere electron momentum ditribution can be obtained directly from the DDCS or from the tranition T matrix. For a given p e, the longitudinal electron momentum ditribution /dp e can be obtained by integrating the DDCS over the electron energy, dp e 2 pe/2 1 p e d 2 e co e e. 11 The integral i regular everywhere except at the ECC where p e v. In the neighborhood of p e v, the derivative of /dp e i dicontinuou. The change of lope acro p e v arie from the behavior of the DDCS at the ECC. The tranvere electron momentum ditribution i not influenced by the internuclear interaction. One can obtain /dp e from the DDCS, 1 dp e 2 pe /2 p e p e p e d 2 e co e e, 12 where p 2 e p 2 e 2 e. In the experiment of Kravi et al. 7, the momentum projection along the y axi wa alo meaured. The y-direction electron momentum ditribution can be obtained by where dp ey dpex dp e dp e dp ez, d5 dp e The perpendicular ditribution /dp ey i ymmetric about p ey 0. D. Projectile momentum tranfer ditribution The longitudinal projectile momentum tranfer ditribution /dp P i imply related to the ingly differential cro ection of the ejected electron: dp P v e. 15 From Eq. 3, the longitudinal projectile momentum ditribution tart at p P i /v, correponding to zero-energy electron emiion. For fat colliion, the longitudinal projectile momentum tranfer ditribution decreae with increaing p P. The ditribution reemble the ingle differential cro ection in electron energy. Since p P i related to the colliion Q value (p P Q/v), the longitudinal projectile momentum tranfer i a meaure of the overall inelaticity of the colliion proce. The tranvere projectile momentum tranfer ditribution /dp P follow from Eq. 2, dp P 2 p P d5 dp e, 16 where p P. Colliion dynamic on the tranvere projectile momentum ditribution may be obtained from the conventional meaurement of the projectile cattering angle P ince v P, for mall P. However, meaurement of the projectile angular ditribution in fat ion-atom colliion i quite difficult becaue of the extremely mall deflection of the projectile On the other hand, it i much eaier to meaure the tranvere recoil-ion and ejected electron momenta. The conervation of tranvere momentum in Eq. 4 can be ued to extract information on projectile tranvere momentum. III. RESULTS AND DISCUSSION The formulation outlined in Sec. II i quite general and independent of theoretical model ued for evaluating the tranition T matrix. However, no exact olution for threebody breakup i available. In thi paper we employed the CDW-EIS approximation of Crother and McCann 9 to evaluate T if. The CDW-EIS model ha the alient feature that the ionized electron ee the Coulomb field from both the target and the projectile ion. The wave function employed by the CDW-EIS atify the correct aymptotic boundary condition of the Coulombic three-body problem. Thi model include effect due to the long-range nature of the target and the projectile interaction in the entrance and

4 53 ANALYSIS OF FINAL-STATE MOMENTUM DISTRIBUTIONS exit channel. It ha been proven to be quite ucceful in decribing the ionization of atom by proton, antiproton and by highly charged ion. Concerning the detail of thi method and it application to the tudy of ejected electron pectrocopy, a review ha been given by Faintein, Ponce, and Rivarola 19. The tandard CDW-EIS approximation of Crother and McCann 9 wa developed for ionization of a hydrogenic atom by a bare ion. Following the work of Faintein, Ponce, and Rivarola 29, we ued an independent electron model to treat the two-electron helium target. The initial target atomic tate i decribed by the Hartree-Fock wave function and the final tate i given by the hydrogenic wave function with an effective charge. We could have alo adopted the recent approach of Gulyá, Faintein, and Salin 30 who replaced the hydrogenic final Coulomb function with numerical continuum wave function obtained from the Hartree-Fock potential. The advantage would be that the continuum function are orthogonal to the bound one. However, the ue of numerical wave function will make the evaluation of multiple integral in the momentum ditribution more complicated and the main feature of the reult are not expected to change due to thi improvement. The independent electron model adopted here i baically identical to what ha been in ue for treating ingle-electron procee in colliion with multielectron target cf. 31. The model i expected to be le applicable for colliion at lower energie. For the dominant colliion proce thi i expected to be valid to the firt order. The preent model adopt the ame approximation, with the emphai on the momentum ditribution of the colliion product. In thi paper we alo evaluated T if uing the firt Born approximation 32. Comparion between the CDW-EIS and the firt Born approximation i ued to demontrate the importance of including the long-range projectile ion-target electron interaction in the final tate. In the following we preent detailed reult for final-tate momentum ditribution in the ionization of helium by proton and by ome highly charged ion for which the meaurement have been done. In carrying out the CDW-EIS and the firt Born calculation, we ued two different effective charge Z T. The firt i Z T 1.344, ariing from the ionization potential of the He atom ( i a.u.. The econd i the variational charge Z T Magnitude of the cro ection obtained from the two effective charge generally differ by no more than 20%. For clarity of the preentation, we preent reult obtained with Z T A. Longitudinal recoil-ion momentum ditribution At intermediate to high colliion energie, Dörner et al. 6 meaured the longitudinal recoil-ion momentum ditribution in ingle ionization of He by fat proton. In Fig. 1, we compare their meaurement with the preent CDW-EIS and firt Born calculation for ionization of He by proton at 0.25, 0.5, and 1 MeV. The overall agreement between the calculation and meaurement i excellent. The longitudinal recoil-ion momentum ditribution at the three energie hown in Fig. 1 have imilar hape. Each ditribution how a ingle peak centered around p R 0. Cro ection drop rapidly on both ide of the peak. It i FIG. 1. Longitudinal recoil-ion momentum ditribution for ingle ionization of He by proton at a 0.25, b 0.5, and c 1 MeV. Experimental data are from Dörner et al. 6. Solid line: preent CDW-EIS calculation; dahed line: preent firt Born calculation. Arrow indicate the poition of p R and p min R ee text. a p R 0.29 a.u., p min min R 1.30 a.u.; b p R 0.20 a.u., p R 2.03 a.u.; c p R 0.14 a.u., p min R 3.02 a.u.. clear that the peak in the longitudinal recoil-ion momentum ditribution correpond to the emiion of low-energy electron. Thee o-called SE are mot important in the total ionization cro ection. Reult hown in Fig. 1 reflect the importance of oft-electron emiion from the perpective of recoil-ion momentum ditribution. The peak poition in the longitudinal recoil-ion momentum ditribution need more detailed analyi. Since we have identified the peak a being due to the emiion of oft electron, we would expect the peak poition at p R p R i /v, which correpond to the extreme ituation in Eq. 3 where electron are emitted with zero energie. If thi were true, we would expect that the peak poition for ionization of He by 0.25-, 0.5-, and 1-MeV proton would appear at 0.29, 0.20, and 0.14 a.u., repectively. However, a careful obervation of Fig. 1 how that the peak poition in the longitudinal recoil-ion momentum ditribution i generally hifted to a lower value of p R. The ditribution i thu

5 3282 Y. D. WANG et al. 53 FIG. 2. Tranvere recoil-ion momentum ditribution for ingle ionization of He by proton at 0.5 MeV. Experimental data are from Dörner et al. 6. Solid line: preent CDW-EIS calculation. backward hifted. In fact, the momentum ditribution at the three colliion energie are all peaked near p R 0. The enhancement of the longitudinal recoil-ion momentum ditribution in the backward direction i due to the enhancement of low-energy electron emitted in the forward direction. The enhancement of forward low-energy electron i a wellknown phenomenon in electron pectrocopy and ha been tudied in both theory and experiment 13,33. In Fig. 1, the peak poition predicted by the CDW-EIS theory i in excellent agreement with the experimental obervation. We note that the imple firt Born approximation alo predicted a backward hift in the longitudinal recoil-ion momentum ditribution although the magnitude of the hift i too mall. Thi i not urpriing. The firt Born approximation i known to partially account for the enhancement of electron emitted in the forward direction 33. With the increae of projectile energy, the backward enhancement of the longitudinal recoil-ion momentum ditribution decreae and the peak poition move to p R. Alo, the difference between the CDW-EIS and firt Born approximation diminihe with increaing projectile velocity. In Sec. II, we pointed out that there i a kinematic threhold characterizing the electron capture into projectile continuum in the longitudinal recoil-ion momentum ditribution. In Fig. 1, we can identify the kinematic threhold according to Eq. 3. At 0.25 MeV, we can ee a clear threhold at p min R from the CDW-EIS calculation. The meaured data below thi value could be acribed to background noie and hould be dicarded. At higher energie, the cro ection near the threhold become much maller. There i not enough tatitic in the data to indicate the threhold. Since the ECC threhold i a pure kinematic effect, it cannot depend on the projectile charge. In the ame paper, Dörner et al. 6 alo reported meaurement for ionization of He by He 2 ion at 0.25 MeV/amu. The threhold occur at the ame p R a in the ionization by 0.25-MeV proton. Dörner et al. reported calculation uing the claical trajectory Monte Carlo (nctmc method 6. The peak poition for the longitudinal momentum ditribution predicted by the nctmc i conitently hifted to maller p R a compared with the experiment. In other word, the nctmc predicted an even larger backward hift in the longitudinal momentum ditribution than the CDW-EIS theory and the experiment. Furthermore, the agreement between nctmc and the experiment doe not eem to improve a the projectile energy i increaed. In fact, the nctmc how the wort agreement with the experiment at 1 MeV, which i the highet colliion energy meaured. In our calculation, the CDW- EIS approache the firt Born approximation with increaing projectile velocity. The agreement with the experiment i alo improved at higher energie. Our integrated total ionization cro ection at the three colliion energie are in good agreement with thoe reported by Shah and Gilbody 37. Dörner et al. 6 alo reported the longitudinal recoil-ion momentum ditribution at a given tranvere recoil-ion momentum. We will dicu thi meaurement in Sec. III B. B. Tranvere recoil-ion momentum ditribution The tranvere momentum ditribution in ion-atom ionization repreent a delicate energy-momentum balance among all colliion particle ee Eq. 4. The meaurement of the tranvere recoil-ion momentum ditribution can probe the impact parameter dependence of the colliion proce. Previouly, our undertanding of tranvere momentum balance wa largely baed on meaurement of projectile angular ditribution cf Such meaurement are undoubtedly quite difficult at high velocitie. With recent progre in recoil-ion momentum pectrocopy, it ha become poible to meaure the tranvere recoil-ion momentum ditribution. In Fig. 2, we compare our calculation with the recent meaurement of Dörner et al. 6 for ingle ionization of He by 0.5-MeV proton impact. The tranvere recoil-ion momentum ditribution i preented a a function of the ratio between the tranvere-recoil ion momentum (p R ) and the initial projectile momentum (p 0 M P v). There i a relatively large difference between the theory and the meaurement at large p R. Thi i expected becaue the tandard CDW-EIS formulation 9 doe not include the internuclear interaction between the projectile and the target nucleu. In the previou calculation on the projectile angular ditribution, it wa hown that projectile-target internuclear interaction make an important contribution at large cattering

6 53 ANALYSIS OF FINAL-STATE MOMENTUM DISTRIBUTIONS angle 16,35,36. The projectile-target internuclear interaction account for projectile angular ditribution beyond the critical angle c P 0.55 mrad repreenting the maximum cattering angle for a proton being deflected by an electron at ret in a binary projectile-electron colliion. In the cae of the tranvere recoil-ion momentum ditribution, however, the role of projectile-target internuclear interaction i le clear becaue of the delicate momentum balance Eq. 4 among the three particle in the final tate. In Fig. 2, it doe not eem to be obviou to identify a region where the internuclear interaction i more important although dicrepancie between the preent CDW-EIS theory and the meaurement increae with increaing p R. In Fig. 3, we compare the calculated longitudinal recoilion momentum ditribution for variou tranvere momenta ranging from p R 0 to 7 a.u. with the meaurement of Dörner et al. 6 for p He ionization at 0.5 MeV. Our calculation how imilar dependence on tranvere momentum a oberved in the experiment. Cro ection decreae rapidly with increaing tranvere momentum. The oberved momentum ditribution are well decribed by the preent theory for p R between 0 and 3 a.u. The agreement between theory and experiment become wore with increaing p R, indicating again the increaing importance of the internuclear interaction. Experimental uncertaintie alo increae with p R. In Fig. 3, the ECC mechanim i hown in the calculation a a finite value at the kinematic threhold p min R a.u. We point out that at the ECC, the tranvere recoil-ion momentum p R exactly balance off the tranvere projectile momentum tranfer ince p e 0. The broadening in the momentum ditribution hown by the experimental reult i alo preent in the theoretical calculation. It follow from the increaing importance of the finite value at threhold. Dörner et al. reported the nctmc calculation for their meaurement 6. The nctmc reult alo how large dicrepancy with experiment on the hape of the longitudinal momentum ditribution at large p R. It doe, however, reproduce the ingle differential cro ection a a function of recoil tranvere momenum quite well, ince it include the internuclear interaction claically. C. Electron momentum ditribution At high velocitie, Dörner et al. 6 extracted the tranvere electron momentum ditribution /dp e in ingle ionization of He by 0.5-MeV proton from the doubly differential electron pectra meaured by Rudd, Toburen, and Stolterfoht 37. In Fig. 4, both the CDW-EIS and the firt Born calculation agree with the data quite well ince the projectile energy i ufficiently high. It hould be pointed out that thi ditribution i calculated from the DDCS ee Eq. 12, and therefore doe not depend on the incluion of the internuclear interaction in the calculation. Thi i true ince the DDCS are obtained by integrating over the projectile cattering angle. In the figure we can ee there i a change of lope at p e /p Thi i alo related to the critical cattering angle c P 0.55 mrad. In thi limit cae the ejected electron carrie out a tranvere momentum p e v. Beyond that point the projectile-electron binary colliion mechanim cannot contribute and the ditribution decreae quickly. At low to intermediate energie, Kravi 7 meaured both longitudinal and tranvere electron momentum ditribution in ingle ionization of He by proton and C 6 at projectile velocitie between 1 and 2 a.u. Thee meaurement provide FIG. 3. Longitudinal recoil-ion momentum ditribution for ingle ionization of He by 0.5-MeV proton for variou tranvere recoil-ion momenta. Experimental data are from Dörner et al. 6. min Solid line: preent CDW-EIS calculation. Arrow indicate the p R threhold at 2.03 a.u.

7 3284 Y. D. WANG et al. 53 FIG. 4. Tranvere electron momentum ditribution for ingle ionization of He by proton at 0.5 MeV. Experimental data are from Rudd, Toburen, and Stolterfoht 37 a quoted in Dörner et al. 6. Solid line: preent CDW-EIS calculation. P c 0.55 mrad i the critical cattering angle ee text. information on the momentum pace ditribution of the ionized electron in the preence of two Coulomb potential. Below we focu on the longitudinal electron momentum ditribution. In Fig. 5, we compare the CDW-EIS and the firt Born calculation for the longitudinal electron momentum ditribution /dp e in ionization of He by proton at v 2.39, 1.71, and 1.15 a.u. or E 143, 73, and 33 kev. Though the Born approximation i not uppoed to be valid at uch low energie, it i intereting to ee how it doe in predicting the momentum ditribution. For ionization of He by proton, the CDW-EIS theory i hown to be able to give accurate total cro ection 9,19 at energie a low a 30 kev. The preent CDW-EIS calculation agree very well with the meaurement. Mot importantly, the CDW-EIS theory correctly predict the peak poition in the electron momentum ditribution at the three energie. The peak poition predicted by the firt Born approximation i almot independent of the projectile velocity, indicating that it doe not include the final-tate interaction. We hould point out that even the Born approximation doe not give the peak poition at p e 0. That would be the cae only if the oft electron are emitted iotropically. Thi fact i alo reflected in the decription of the longitudinal recoil-ion momentum ditribution where the firt Born approximation reulted in a backward hift ee Sec. III B. The contribution of the o-called ECC electron i centered preciely at p e /v 1 in Fig. 5 in the CDW-EIS calculation. Thee electron are ejected with the ame velocity a the projectile. They reult in a kink change of lope in the momentum ditribution at v. The kink in the theoretical calculation i quite clear at v 2.39 and 1.71 a.u. At v 1.15 a.u., the projectile velocity i rather mall and the ECC electron contribution i mixed with that of the oft electron. Figure 5 how that the parallel component of the electron momentum fall mot likely between the projectile and the target. Thi latter obervation i more obviou at low velocitie e.g., v 1.71 and 1.15 a.u.. The reaonably good agreement between the CDW-EIS and the meaurement regarding the hape and location of the electron momentum ditribution how that the CDW-EIS can provide a qualitative decription of the main feature of three-body ionization dynamic. We now turn to the ionization of He by highly charged ion at comparable velocitie. In 7, longitudinal electron momentum ditribution were meaured for ionization of He by C 6 ion at v 1.63, 1.38, and 1.16 a.u. Thi i the lowenergy region where the cro ection for electron tranfer i larger than that of ionization by two order of magnitude 38,39. The CDW-EIS model i not expected to work in thi energy region and for thee colliion ytem ince the effect of charge tranfer on the ionization i not explicitly included in the perturbative treatment. Below the CDW-EIS model wa ued to check how much the o-called two-center effect i reflected in uch colliion ytem. In Fig. 6, we how the CDW-EIS cro ection normalized to the peak of the meaured ditribution. The hape of the momentum ditribution are only moderately repreented by the CDW-EIS except at the highet velocity v In all cae, the CDW-EIS tend to highlight the importance of projectile center or the ECC mechanim while the firt Born theory predict that mot electron hould be emitted around the target center. The three ditribution hown in Fig. 6 for ionization by C 6 eem to have quite different velocity dependence from the correponding ditribution for proton impact ionization. The pulling of electron toward the highly charged projectile may indicate a trong pot-colliion effect. In the cae of proton impact, uch effect are weaker becaue of the lower projectile charge. In a recent calculation for total ionization of He by C 6 uing the two-center cloe-coupling method, Wang et al. 23 found that the projectile center play an important role. Within the ame two-center bai et, projectile continuum tate become more important with increaing velocity. The increaing importance of the projectile center wa ued to explain the oberved trong onet in the ionization cro ection oberved by Wu et al. 38. With decreaing velocity, the target center will eventually become important. We mut keep in mind that thi concluion i obtained for low-energy ionization by highly charged ion, where the ionization probability i extremely mall. When compared with the cloecoupling calculation 23, the CDW-EIS often overetimate the total ionization cro ection by about a factor of 2 in Fig. 6 a and 6 b. At the lowet colliion velocity hown in Fig. 6 c, however, the CDW-EIS underetimate the total

8 53 ANALYSIS OF FINAL-STATE MOMENTUM DISTRIBUTIONS FIG. 5. Longitudinal electron momentum ditribution a function of p e /v for ionization of He by proton at intermediate to low velocity: a v 2.39 a.u.; b v 1.71 a.u.; and c v 1.15 a.u. Experimental data are from Kravi et al. 7. Solid line: preent CDW-EIS calculation; dahed line: preent firt Born calculation. cro ection by about a factor of 2, reflecting the rapid change of the total cro ection at thee low velocitie. Finally, we note that the dicrepancy between the CDW- EIS and the meaurement on ionization of He by highly charged ion i not becaue of the ue of independent electron model for treating the two-electron target. A demontrated by Wang et al. 23 in their cloe-coupling calculation, the independent electron approximation work rather well in predicting ingle ionization and ingle charge tranfer cro ection in the C 6 He colliion ytem in the preent velocity region. IV. CONCLUSIONS In thi paper we have formulated a quantum mechanical theory for decribing the momentum ditribution in ionatom ingle ionization. Our formulation i quite general and i independent of the colliion model ued for calculating the tranition matrix. The energy-momentum balance among the three particle i ued to extract a variety of differential cro FIG. 6. Longitudinal electron momentum ditribution a function of p e /v for ionization of He by C 6 highly charged ion at low velocity: a v 1.63 a.u.; b v 1.38 a.u.; and c v 1.16 a.u. Experimental data are from Kravi et al. 7. Solid line, preent CDW-EIS calculation; dahed line, preent firt Born calculation FBA. The calculation are normalized to experimental peak value. ection. The three prominent feature in ion-atom ionization proce i.e., oft electron emiion, electron capture into the projectile continuum, and projectile-electron binary colliion can all have ignature in the final-tate momentum ditribution of the projectile, the electron, and the recoiling ion. Colliion at both high and low velocitie and by both low and highly charged projectile were conidered in thi paper. The tandard CDW-EIS theory wa applied to calculate the cro ection. Thi theory account for the longrange interaction of the projectile and target Coulomb field. A a comparion, the firt Born approximation, which doe not take into account thee refinement, wa alo applied. The comparion between the CDW-EIS and the Born approximation how the importance of two-center effect. In general, the CDW-EIS theory i able to identify and decribe the main feature of ionization dynamic. It give both qualitative and quantitative decription for final-tate momentum ditribution in the ingle ionization of He by proton at intermediate to low energie.

9 3286 Y. D. WANG et al. 53 ACKNOWLEDGMENTS We are grateful to D.S.F. Crother and O Rourke for ueful dicuion. We alo acknowledge dicuion with P. Richard, L. Tribedi, and J. Ullrich. Thi work i upported by the Diviion of Chemical Science, Office of Baic Energy Science, Office of Energy Reearch, U.S. Department of Energy. V.D.R. ha been partially upported by the Conejo Nacional de Invetigacione Científica y Técnica under PID 3357/92-CONICET ARGENTINA. R.D. wa upported by the Alexander von Humboldt Stiftung. 1 J. Ullrich, R.E. Olon, R. Dörner, V. Dangerdorf, S. Kelbch, H. Berg, and H. Schmidt-Böking, J. Phy. B 22, R. Dörner, R.E. Olon, and H. Schmidt-Böking, Phy. Rev. Lett. 63, R. Ali, V. Frohne, C.L. Cocke, M. Stöckli, and M. Raphaeliam, Phy. Rev. Lett. 69, V. Frohne, S. Cheng, R. Ali, M. Raphaeliam, C.L. Cocke, and R.E. Olon, Phy. Rev. Lett. 71, R. Mohammer, J. Ullrich, M. Unverzagt, W. Schmidt, P. Jardin, R.E. Olon, R. Mann, R. Dörner, V. Mergel, U. Buck, and H. Schmidt-Böcking, Phy. Rev. Lett. 73, R. Dörner, V. Mergel, L. Zaoyuan, J. Ullrich, L. Spielberger, R.E. Olon, and H. Schmidt-Böcking, J. Phy. B 28, S. Kravi et al. unpublihe. 8 V.D. Rodríguez, Y.D. Wang, and C.D. Lin, Phy. Rev. A 52, R D.S.F. Crother and J.F. McCann, J. Phy. B 16, G.B. Crook and M.E. Rudd, Phy. Rev. Lett. 25, K.G. Harrion and M. Luca, Phy. Lett. 33A, M.W. Luca and W. Steckelmacker, in Proceeding of the Third Workhop on High-Energy Ion-Atom Colliion, Debrecen, Hungary, 1987, edited by D. Berenyi and G. Hock, Lecture Note in Phyic Vol. 294 Springer-Verlag, Berlin, 1987, p S. Suárez, C. Garibotti, W. Meckbach, and G. Bernardi, Phy. Rev. Lett. 70, D.H. Lee, P. Richard, T.J.M. Zouru, J.M. Sander, J.L. Sinpaugh, and H. Hidmi, Phy. Rev. A 41, V.D. Rodríguez, Y.D. Wang, and C.D. Lin, J. Phy. B 28, L H. Fukuda, I. Shimamura, L. Végh, and T. Watanabe, Phy. Rev. A 44, S.F. O Rourke, I. Shimamura, and D.S.F. Crother, Proc. R. Soc. London A 452, J.E. Miraglia and J. Macek, Phy. Rev. A 43, P.D. Faintein, V.H. Ponce, and R.D. Rivarola, J. Phy. B 24, E.A. Solov ev, Zh. Ekp. Teor. Fiz. 81, Sov. Phy. JETP 54, J.H. Macek and S.Y. Ovchinnikov, Phy. Rev. A 50, R.K. Janev, G. Ivanovki, and E.A. Solov ev, Phy. Rev. A 49, R Y.D. Wang, C.D. Lin, N. Tohima, and Z. Chen, Phy. Rev. A 52, R. Mohammer, M. Unverzagt, W. Schmitt, J. Ullrich, and H. Schmidt-Böcking, Rev. Sci. Intrum. to be publihe. 25 E.Y. Kamber, C.L. Cocke, S. Cheng, J.H. McGuire, and S.L. Varghee, J. Phy. B 21, L E.Y. Kamber, C.L. Cocke, S. Cheng, and S.L. Varghee, Phy. Rev. Lett. 60, F.G. Kritenen and E. Hordal-Pederen, J. Phy. B 23, G. Schiwietz, P. Grande, C. Auth, H. Winter, and A. Salin, Phy. Rev. Lett. 72, P.D. Faintein, V.H. Ponce, and R.D. Rivarola, J. Phy. B 23, L. Gulyá, P.D. Faintein, and A. Salin, J. Phy. B 28, J.H. McGuire and O.L. Weaver, Phy. Rev. A 16, D.R. Bate and G. Griffing, Proc. Phy. Soc. A 66, D.H. Madion, Phy. Rev. A 8, M.B. Shah and H.B. Gilbody, J. Phy. B 18, A. Salin, J. Phy. B 22, V.D. Rodríguez, J. Phy. B 29, M.E. Rudd, L.H. Toburen, and N. Stolterfoht, Nucl. Data Table 18, W. Wu, C.L. Cocke, J.P. Giee, F. Melchert, M.L.A. Raphaelian, and M. Stöckli, Phy. Rev. Lett. 75, W. Wu, Ph.D. diertation, Kana State Univerity, 1994.