dichroism in the electron-impact ionization of excited and

ering,

PHYSICAL REVIEW A, VOLUME 62, 012706

Dichroism in the electron-impact ionization of excited and oriented sodium atoms

J. Lower,1 A. Elliott,1 E. Weigold,1 S. Mazevet,1,* and J. Berakdar2

1Atomic and Molecular Physics Laboratories, Institute of Advance Studies, Research School of Physical Sciences and EngineAustralian National University, Canberra ACT 0200, Australia

2Max-Planck Institute for Microstructure Physics, Weinberg 2, 06120 Halle, Germany~Received 9 December 1999; published 9 June 2000!

Measurements are reported for kinematically complete electron-impact ionization collisions with oriented3P sodium atoms excited from the ground state by right- and left-hand circularly polarized light. The mea-surements reveal a strong dependence of the ionization cross section on the helicity inversion of the excitingphoton. This dichroic effect and the magnetic state resolved cross sections are described using the distorted-wave Born approximation and the dynamically screened Coulomb wave method. Within the context of thesemethods we investigate the role of different short- and long-range interactions involved in the process. Formoderate values of incident energy we conclude that the shape and magnitude of the dichroism is predomi-nantly determined by the Coulomb interactions of the outgoing electrons with the residual ion. For slowescaping electrons the dichroism is also influenced by the interelectronic correlations.

PACS number~s!: 34.80.Dp, 34.80.2i, 34.80.Pa, 34.80.Qb

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I. INTRODUCTION

The need for a detailed understanding of the procesionization by electron impact for various fields of physicsuch as atmospheric physics, plasma physics, and gascharge physics, has led to extensive studies both experimtal and theoretical of this particular process. The most coplete information on the electron-impact ionization procesobtained from so-called (e,2e) experiments, where both thscattered and the ejected electrons from the target aretected in coincidence and their momenta are resolved. Tkind of measurement has proved indispensible when addring the details of collision dynamics. Moreover, the (e,2e)technique can be employed for the spectroscopy of electrstructure of atoms, molecules, and solids@1,2#. These differ-ent aspects of the (e,2e) reaction can be highlighted by judicious choice of the kinematic arrangement.

In recent years, a new generation of (e,2e) experimentshas emerged that uses polarized electron beams anatomic beams in which the constituent atoms are aligand/or oriented. The first (e,2e) experiment of this type usea polarized electron beam to ionize lithium atoms preparea particular quantum state by use of inhomogeneous mnetic fields, and was primarily concerned with the studythe exchange process@3#. These experiments were followeby experimental arrangements using a polarized elecbeam at relativistic energy for the inner-shell ionizationvarious atomic targets such as silver@4# and at intermediateenergy for the outer-shell ionization of xenon, where the fistructure energy splitting of the residual ion can be resolexperimentally@5,6#. High-energy experiments on the innshells of heavy target atoms were primarily concerned wthe study of the influence of relativistic interactions durithe ionization process@7#. In the xenon experiments, expermental verification of the so-called fine structure effect

*Permanent address: Theoretical Division, Los Alamos NatioLaboratory, Los Alamos, NM 87545.

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electron-impact ionization@8–10# was established and comparison of the results with theory revealed detailed informtion on the influence of the short-range interactions betwthe two outgoing electrons and the residual ion@6,9#.

In this paper we focus on the electron-impact ionizatiof sodium atoms laser pumped into a well defined angumomentum magnetic substate. In this situation the (e,2e)cross section reveals a dependence on the inversion ohelicity of the exciting photon@11,12#. This dichroic effectresults from the transfer of the initial-state orientation to toutgoing correlated electron pair and hence has been terorientational dichroism. Experimental evidence for the existence of this effect has been provided recently by Dornet al.@12#. Here, we present further experimental and theoretresults and investigate in more detail the underlying physprocesses.

Experimentally, the sodium atoms are prepared in(32P3/2, F53, mF563) hyperfine states and ionized uing an initially unpolarized electron beam. Initial theoreticand experimental studies@11–14# indicate a strong dependence of the cross section on the direction of the initial-statomic orientation. This dependence of the cross sectionmost easily analyzed within the theoretical framework of tfirst Born approximation~FBA!, i.e., when we assume thathe ejected electron moves in the field of the residualwhereas the scattered one is free. In this case the geomeproperties of the orientational dichroism are given by ttriple product of the quantization axis of the atom, the mmentum transfer direction, and the vector momentum ofejected electron@11,12#. According to the FBA model, theorigin of the dichroism lies in the interaction of the sloelectron with the residual ion; the dichroism vanishes whthis interaction is neglected@15#. Furthermore, the FBA predicts that the cross sections for the electron-pair emissfrom left- or right-hand circularly laser-pumped Na atomare related to each other via reflection symmetry aboutdirection of momentum transfer@15#.

Recent experimental measurements@12–14# reveal a con-siderable break of this symmetry property which indicatel

©2000 The American Physical Society06-1

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LOWER, ELLIOTT, WEIGOLD, MAZEVET, AND BERAKDAR PHYSICAL REVIEW A 62 012706

FIG. 1. Schematic of the electron source acoincidence spectrometer.

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dynamical transfer of the initial atomic orientation to the twoutgoing electrons through mechanisms beyond those slated by the FBA. On the grounds of an approximate mofor the correlated many-body dynamics, it was proposedthis symmetry breaking has its origin in the final-state eltron correlation@12–14#. As the theoretical interpretation isubject to the approximation adopted for the treatment ofreaction dynamics, we contrast in this work the predictioof various approximative scattering models with experimtal findings. We first employ the distorted-wave Born aproximation@16,17#, which is particularly convenient for isolating the influence of the various interactions betweencontinuum and target electrons, and then the more refidynamically screened Coulomb wave method@18# to quan-tify the influence of the final-state electron correlations.

The paper is organized as follows. In the next sectiongive an outline of the experimental procedure. We thendress the theoretical framework necessary to describeparticular experimental situation. In the fourth section,outline the main features of the scattering methods thatparticularly relevant to the present study. In the last sectwe investigate the influence of the short- and long-rangeteractions and the electron-electron correlations on the calated cross sections and compare them with our experimedata.

II. EXPERIMENTAL ARRANGEMENT

A. Apparatus

In Fig. 1 the apparatus is shown schematically. It candivided into three components comprising the sourcedifferential pumping chambers in which generation atransport of the primary electron beam is achieved, the stering chamber in which the atom beam is formed, the insection of the laser, electron, and atom beams occurs, an(e,2e) ionization processes are measured, and finallyMott chamber in which the degree of the incident electrbeam polarization can be determined. The Mott chambernot utilized in the present measurements and will notdescribed here. Each of the other stages is discussed iquence below.

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The first stage of the apparatus consists of two ultrahivacuum~UHV! chambers connected in series with one aother. In the first of these chambers, the so-called souchamber, electrons are produced by irradiation of a negataffinity p-doped GaAs crystal surface by laser light. The dtails of the crystal composition, cleaning, and coating produre have been described previously@19# and will not berepeated here. This first chamber is pumped by a 30 l/spump which maintains a background pressure in the 10211

torr range. The combination of an UHV environmentminimize contamination of the GaAs photocathode andcontinuous deposition of cesium onto the photocathodeface enables stable emission currents to be maintained operiod of many months. Linearly polarized light fromGaAlAs diode laser of around 810 nm wavelength is usedgenerate the photoelectrons. The laser beam is first pathrough a focusing lens and a 1/4 wave plate to prodcircularly polarized radiation before impinging upon thcrystal surface. Photon fluxes of a few milliwatts are sucient to produce microamperes of current.

Using extraction by an electrostatic field, the photoeletrons are formed into an electron beam. For excitationcircularly polarized light incident perpendicular to the suface, the beam is polarized either parallel or antiparallethe direction of electron motion, depending upon the helicof the radiation used in the photoexcitation process.GaAs the polarization of the electron beam is theoreticalimited to a maximum value of 50% due to the selectirules governing the transitions excited in the photoemissprocess. We obtain a polarization value routinely of arou24–30 % in our system, the difference between this vaand the theoretical maximum being attributed to depolarizeffects acting on the photoelectrons as they are transpofrom within the bulk to the surface of the photocathode@20#.Inversion of the electron beam polarization can be achieby reversing the helicity of the laser light through rotationthe 1/4 wave plate by 90°. As an applied electrostatic fiwill act on the trajectory of an electron beam but not onangular momentum, the initially longitudinally polarizeelectron beam is converted to one of transverse polarizaby deflecting it through a 90° electrostatic deflector. Th

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DICHROISM IN THE ELECTRON-IMPACT IONIZATION . . . PHYSICAL REVIEW A62 012706

action is desirable as for many of our experiments a traversely polarized electron beam is required. However, forpresent measurements unpolarized electrons were used.was achieved by rotating the quarter wave plate to an angposition producing linearly polarized radiation. After extration of the photoelectrons and their transmission through90° deflector, a system of electrostatic cylindrical tube lenand deflectors is used to focus and accelerate the elecbeam to 1000 eV and steer it through the 3-mm-diameaperture separating the source chamber from the differepumping chamber.

The purpose of the differential pumping chamber isprovide a differential pumping stage between the souchamber and the non-UHV scattering chamber to minimcontamination of the GaAs photocathode. The differenpumping chamber is pumped by a 180 l/s turbo molecupump and maintains a base pressure of around 5310210 torr.It is separated from the scattering chamber by a second 3aperture at its exit. Within the differential pumping chambare two sets of quadrupole deflectors to steer the 1000electron beam through the exit aperture into the main stering chamber, compensating for perturbations to the betrajectory from spurious magnetic fields and surface chargeffects within the electron optics. Transporting the befrom the source to the scattering chamber at high energytwo advantages. First, it renders the beam less sensitivthe effects of deflection and defocusing by stray magnfields. Secondly, the narrower beam profile achieved bying higher beam energies enables the use of smaller aperat the entrance and exit of the differential pumping chambincreasing the pressure gradient that can be maintainedtween the source and scattering chambers.

In the scattering chamber the electron passes througfurther series of cylindrical tube lenses which are usedboth collimate and decelerate the beam to the requiredpact energy. A target beam of sodium atoms is producedeffusion of sodium vapor through a 1 mmaperture in theoutput stage of a recirculating metal vapor oven describedetail below. The interaction region lies in the scatteriplane defined by the momentum vectors of the sodium beand the incident and detected scattered electrons, the lthree quantities constrained to share a common plane uthe coplanar reaction kinematics employed for the preswork. The intersection of the sodium and electron beadefines the interaction region from which (e,2e) events aremeasured. The interaction region also lies on the axisrotation of three rotatable turntables. On one of the turntabthe sodium oven is mounted. On each of the remainingturntables an electron spectrometer is mounted.

Each of the two electron spectrometers comprises a 1hemispherical electrostatic electron analyzer preceded bfive-element cylindrical lens system. The combination ofenergy selecting analyzer and the angle resolving lenstem defines the momentum and energy of measured scatelectrons. The lens system is similar in design to thatscribed in Ref.@21# and comprises two three-element zoolenses in series sharing a common central lens, allowingdependent control over energy and angular resolution toachieved. Position-sensitive detectors are mounted at the

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plane of each analyzer, behind a slit in the radial directienabling simultaneous measurement of scattered electover a 6 eVenergy band with an energy resolution of arou200 meV and at an angular resolution of around 2° fwidth at half maximum.

The sodium oven, constructed of stainless steel, is shschematically in Fig. 2 and is similar in design to that dscribed in Ref.@22#. It is comprised of three distinct components, namely, the reservoir, the nozzle, and the recirculaThe reservoir consists of a cylindrical vessel containingdium around which heating wire is wrapped to proviohmic heating. Reservoir temperatures of between 400450 °C are necessary to produce the required target densfor our measurements. The temperature of the reservoirother key components is monitored by use of thermocoupVapor leaving the reservoir is formed into a beam of sodiatoms within the nozzle stage, which is terminated by1-mm-bore capillary of around 10 mm length. The separatheated nozzle is maintained at a temperature at least 1higher than that of the reservoir to prevent blockage duethe buildup sodium within its narrow exit capillary. The sdium beam emerging from the nozzle is skimmed throuthe action of an aperture positioned at the exit of the ovrecirculation stage. Atoms not contained within the centcone of the nozzle beam are recondensed to the liquid pupon impact with the recirculator walls, which are maitained at a temperature of between 150 and 200 °C, andsequently returned to the sodium reservoir. The inclusionthe recirculation stage extends the operating lifetime betwsodium refills and, by maintaining the collimating apertusurrounds at a temperature above that required for themation of solid sodium, circumvents the problems of tclosing over of cold collimation apertures with time.

The whole oven assembly is enclosed within a wacooled rectangular copper box whose purpose is twofoFirst, it provides a heat sink for radiation emitted by the ov~around 150 W! and, secondly, it offers additional protectio

FIG. 2. Outline of recirculating sodium oven and interactiregion. The copper walls of the oven surround are water cooThe exit aperture and beam dump are cooled by liquid nitrogen

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for delicate vacuum components from residual backgrosodium vapor or accidental spills of liquid sodium undvacuum. The box is kept compact in design by incorporatV-shaped channels through each of its 8-mm-thick sidewthrough which cooling water is passed. The compact naof the design maximizes the angular range through whichspectrometers can rotate within the scattering plane.

Mounted off the box is a final stage liquid nitrogen coolcollimating aperture and sodium beam dump to provide fther collimation of the beam and additional protectiagainst contamination of the vacuum environment by geous sodium. Both are thermally isolated from the boxceramic insulators and cooled by means of a flexible copbraid connected to a cold finger protruding through the wof the scattering chamber.

B. Laser preparation of the target

Intersecting the scattering plane at right angles and cpletely encompassing the interaction region is a frequemodulated 589 nm laser beam used to excite and orienthe sodium target atoms by preparing them in a specificperfine magnetic substate. The initially linearly polarizedser light is converted to circularly polarized radiationtransmission through a quarter wave plate, the rotationwhich by 90° reverses the helicity of the radiation field athus the orientation of the excited-state atoms. The labeam preparation is shown in Fig. 3, which shows schemcally the experimental arrangement.

In order to reach a high fraction of laser excited atomsthe interaction region, the laser excitation is performedmeans of two frequencies produced by frequency modulaof a single-mode dye laser beam. Frequency modulatioachieved by passing the laser beam through an elecoptical modulator which is similar in design to that reportin @23#. It consists of a rectangular LiTaO3 crystal embeddedwithin a copper foil resonator. Power is coupled into t

FIG. 3. Schematic of the experimental arrangement showingcoplanar arrangement of the incident and outgoing electronsodium beams intersecting at the collision region with the perpdicular laser beam.

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modulator by mutual induction with a single-loop drive coThe application of a resonator ensures that for modespower inputs, a large voltage is built up across adjacent faof the LiTaO3 crystal. To first order, the refractive index oLiTaO3 varies linearly with strength of the applied field.follows then from Fourier analysis that for a sinusoidal dring frequencyvm and for an incident laser beam of frequencyv0 passing through the crystal, the output field cosists of the primary frequencyv0 and additional sidebandseparated by multiples ofvm . In the present case for sodiumthe laser beam is modulated at a frequency of 856 MHz,first two sidebands being separated by 1712 MHz. One sband is used to pump the transition 3s1 2S1/2(F51)→3p1 2P3/2(F52) and the other the 3s1 2S1/2(F52)→3p1 2P3/2(F53) transition. In this way a relative fractiona of around 40% 3p1 2P3/2 excited-state atoms can be rotinely achieved@24#. After a few excitation/decay cycles thatoms gather exclusively in the two-level syste3s1 2S1/2(F52,mF512),3p1 2P3/2(F53,mF513) forpumping by right-hand circularly polarized light, or in thsystem 3s1 2S1/2(F52,mF522),3p1 2P3/2(F53,mF523)for pumping by left-hand circularly polarized light. This iconfirmed numerically by solving the corresponding raequations. The degree of orientation of the excited statethe interaction region was estimated to be between 96%100% by comparing superelastic scattering data with thof previous authors@22#.

C. Experimental procedure

In order to discriminate between ionization events resing from the removal of 3s and 3p electrons, respectivelythe binding energye I of the ejected electron is determinefrom the energies of both outgoing continuum electronsEfandEs and the energy of the incoming electronE0 from therelation

e I5E02~Ef1Es!. ~1!

For each measured coincidence event the summed enspectrum of both detected outgoing electrons is stored omultichannel analyzer yielding a binding-energy spectrwith a binding-energy resolution given by the convolutionthe energy spreads of the incoming beam and the appafunctions of both of the electron analyzers. In the prescase, an (e,2e) binding-energy resolution of around 0.9 ewas achieved, more than sufficient to resolve events cosponding to ionization of the ground- and excited-stateoms, which are separated by 2.14 eV in binding energy.

It should be noted, however, that an energy averagebeen performed at each binding energy over all combinatiof values forEf andEs within the 6 eV acceptance band oeach analyzer that satisfy Eq.~1!. This was undertaken toenable the data to be displayed in a compact form andimprove statistics. Thus, although a high value for bindinenergy resolutionDe I of 0.9 eV is achieved, the energy reslution for both slow and fast scattered electrons,DEs andDEf , respectively, is 6 eV for the experimental data psented in this paper.

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For each chosen angular setting of the analyzers, specorresponding to positive and negative target orientationrecorded, as well as random coincidence background spewhich are subtracted from the binding-energy spectra inusual manner@25#. A typical binding-energy spectrum fothe two helicities of the incident radiation is shown in Fig.

The experiments consisted of measuring (e,2e) binding-energy spectra for a fixed scattering angleu f of the fastemitted electron as a function of the scattering angleus forthe slow electron and for both positive (mF513) and nega-tive (mF523) orientations of the excited state. These staconsist of maximal projections of the nuclear spin (I 53/2),the orbital angular momentum (l 51), and the electron spinalong the laser beam direction. Neglecting the spin-orbitteraction and for unpolarized incident electrons and nolarization analysis of the final electron, the cross sectionpends only on the orbital orientationml561, and not on thenuclear and electron spin orientations.

Many scans were performed in each experiment to aage over the effects of instrumental drifts. Cross normalition of the experimentally derived ground- and excited-stcross sections for each particular kinematic arrangementachieved in the following manner. For any arbitrary reactkinematics, the relative fractiona of excited-state atoms idetermined by measuring the (e,2e) count ratesN3s

on andN3s

off , corresponding to the ionization of ground-state atowith the pump laser on and off, respectively, throughrelation

a512N3son/N3s

off . ~2!

Having determined the value ofa, the ratio of ground- toexcited-state cross sectionss3s /s3p , for a given target ori-

FIG. 4. Sodium (e,2e) binding-energy spectra forE0590 eV,Es520 eV,u f520°, us565°, with left- and right-hand circularlypolarized optical pumping light, showing the ground-state aexcited-state transitions.

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entation (mf513 or mf523), is derived from the corre-sponding (e,2e) count ratesN3s

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ionization of ground- and excited-state atoms, respectivmeasured with the pump laser beam on. This is achiethrough the relation

s3s /s3p5N3s

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In the present series of measurements,a50.4560.04 for the151 eV measurements and 0.3560.05 for the remainderleading to an error in the relative normalizations of grounand excited-state data of around 20%.

III. THEORY

A. Formulation

In the most general case of state-selected (e,2e) experi-ments, the atomic beam may be oriented and/or aligwhile the incident electron beam may be partially polarizperpendicular to the collision plane defined by the momeof the incoming and scattered~fast! electrons, respectivelydenoted ask0 andk f . Following the methodology used fothe description of state-selected elastic and inelastic scaing experiments@26–28#, we use the density matrix formalism to describe such (e,2e) experiments and to take intaccount the information relevant to the initial-state prepation @29#. Within this formulation, the atomic and electrobeams are characterized by density operators whose mrepresentations reflect the statistical mixture of pure staeach corresponding to the state of one atom or elecpresent in the beam. The density operatorr in describing theinitial state of the electron-atom system is given by the dirproduct of the density operatorsre and ra that represent,respectively, the electron and atomic beams. This is becathe electron and atomic beams are initially prepared far frthe interaction region, and as such are assumed to be unrelated before the interaction begins@26#. Thus we write

r in5re3ra. ~4!

In the present study, as the initial electron beam is unpoized, the reduced density matrix of the electron beam is sply the unit matrix, the normalization coefficient reflectinthe dimension of the incoming electron spin space,

r in51

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un0 ,J,M &^n0 ,J,M 8urM ,M8J . ~5!

The superscripta has been dropped from the density matdescribing the atomic beam. Furthermore, for the pure iniatomic stateuF,MF&, we assume that the nonzero nuclespin of the atom plays no dynamical role in the collisioprocess@30#. The atom is thus described as being prepareda quantum stateuJ,M &, whereJ is the total angular momentum of the atom andM its projection along the quantizatioaxis. The effect of the nonzero nuclear spin will appear o

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through recoupling coefficients which are dropped in thelation ~5!. The quantum numbern0 in Eq. ~5! labels theelectron spin projections.

The atomic beam is most easily described in the phoframe where the quantization axiszph is parallel to the direc-tion of propagation in the case of circularly polarized ligand to the direction of the electric field in the case of lineapolarized light@31#. In the photon frame the density matrof the excited atomic state and the ground state becodiagonal.

The differential cross section for the ejection of two eletrons from an atomic target, initially prepared in a particuquantum stateuJLM&, upon the impact of an unpolarizeelectron beam can be expressed as@2,16#

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whereEs is the energy of one emitted electron@the energy ofthe other electron is deduced from Eq.~1!# anddV f anddVsare the solid angles associated withk f andks. Equation~6!assumes no resolution of the spin projectionsn f andns of thetwo outgoing electrons and no detection of the orientatand/or alignment for the residual ion. The wave functionthe atom ~residual ion! in a particular quantum statuJLM& (uJiLiM i&) is Fatom (F ion) whereasT is the transi-tion operator. The kinematical factork is given by

k5~2p!4kfks

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As shown in recent articles@11,15#, the effect of theinitial-state preparation on the collision dynamic is most coveniently seen when state multipolesrKQ are used rathethan density matrices. The state multipolesrKQ are related tothe density matrix elementsrMM

J by @27#

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K~21!K2J2M^J2MJMuK0&rKQ50 , ~8!

where K and Q in relation ~8! stand, respectively, for therank of the tensor and its projection along the quantizataxis. The relation~8! indicates that only theQ50 compo-nents of the state multipoles are nonvanishing as the denmatrix describing the atomic beam is diagonal in the phoframe. Using relations~6! and ~8! the cross sections can bexpressed in terms of irreducible tensor components as@15#

d5s

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Expression~9! shows that the use of the state multipolrenders possible a separation of geometrical properties~de-

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scribed by the state multipoles! and reaction dynamics~con-tained in the tensorial parametersL0

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as sodium, one can neglect the relativistic interactionsmight alter the spin projections of the continuum electrowithout conservation of the total spin of the system. In thsituation, the exchange process is the only spin-depenprocess that needs to be taken into account. The spin pathe T-matrix elements appearing in relation~6! can then befactored out. In the present case of a sodium target, Eq.~9!can be expanded as@12,15#

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The tensorial components along the quantization axisthe target are expressed in terms of the state-resolved csectionssL,mL

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L0(1)5

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L0(2)5

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The cross sectionssL,mLare themselves expressed as fun

tions of the symmetrizedT-matrix elements as

sL,ml5k(

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These relations allow the calculation of the state-resolcross section for arbitrary angles between the momentransfer vectorq5k02k f and the quantization axiszph. Inthe experimental arrangement under consideration the atare prepared using circularly polarized light propagating ppendicular to the scattering plane@12#. In this situation thezph quantization axis coincides withzn, the quantization axisof the natural frame. As for the case of inelastic scatterstudies@32#, we define, for the present study, the natuframe as a right-handed coordinate system. In the natframe, the quantization axiszph is defined aszph5 k03 k f .

The cross sections are calculated usually in the collisframe, where the quantization axiszc is defined along theincident electron momentumk0 and where, in coplanar kinematics, the fast electron is scattered in the (xy) plane. Thecollision and natural frames are related as

xn5zc, yn5xc, and zn5yc. ~15!

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The natural frame cross sectionssL,mLappearing in the ex-

pressions of the tensor components, Eqs.~11!–~13!, can beexpressed as functions of the collision frame scatteringplitudes by using the appropriate frame transformatthrough Euler angles@33,34#. The natural and collision framescattering amplitudes, respectivelyf mL

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c !#, ~16!

f 0n52 iA 1

2 ~ f 1c1 f 21

c !. ~17!

For the case of ionization, as for the case of inelastic stering, reflection in the scattering plane also imposes thelowing relation for the collision frame scattering amplitude

f 1c52 f 21

c . ~18!

This reduces relations~16! and ~17! to

f 61n 57A 1

2 @ f 0c6 i f 1

c#, ~19!

f 0n50. ~20!

Using relations~18! and ~19! in the expressions of thetensorial components~11!–~13!, we see that the alignmenparameterL (2) is related to the tensorial componentL (0) as

L0(2)5A 1

2 L (0). ~21!

This indicates that in the present experimental arranment the measurement of the initial-state averaged and sselected cross sections allows determination of all threerameters describing the process, the scalar componentL (0),which is the quantity usually measured in a conventio(e,2e) experiment, and the vector componentL (1), which isa measure of the change in the cross sections resultingan inversion of the initial target orientation.

B. Scattering methods

To investigate the influence of the short- and long-raninteractions, as well as the electron-electron interaction infinal state, we use two different scattering methods to evate the state-resolved cross sections. The first is the distowave Born approximation~DWBA! @16,17#, which accountsfor the short- and long-range interactions in both the iniand the final states but treats the two outgoing electronindependent particles. The second is the dynamicscreened three-Coulomb-wave method~DS3C! @18#, whichaccounts explicitly for the electron-electron correlation in tfinal state but in contrast neglects the short-range interacin both the initial and final states.

In both approximations, the scattering from the Na atis reduced to a three-body problem by considering onlyactive ~valence! electron of the Na atom. The total HamitonianH of the projectile electron and target is

H5h11h21v12, ~22!

01270

-n

t-l-:

e-te-a-

l

m

ee

u-d-

lasly

n

e

whereh1 andh2 are the Hamiltonians of the active electronconsisting of the kinetic energy operatorKi and potentialVi .v12 is the electron-electron interaction potential not includin Vi .

The DWBA approximation is formulated by partitioninthe collision Hamiltonian into two parts@16,17#,

H5~K11U11K21V2!1~V11v122U1!, ~23!

5K1V, ~24!

whereU1 is the distorting potential, which needs to be dfined @35#. WhenLS coupling is assumed to be valid for th(N11)-electron state, the exact unsymmetrizedT-matrix el-ements are approximated, within the DWBA, as

^k fksFJiLi Mi

ion uTuFJLMatomk0&

[^x (2)~k f !x(2)~ks!uVufLMx (1)~k0!&. ~25!

In relation~25!, fLM is the one-electron orbital of the activtarget electron, the electron ejected from the atom duringionization process. The distorted wavesx (6)(k) are one-electron state solutions of the channel HamiltonianK, sepa-rable in the electron coordinates. The DWBAT-matrix ele-ments ~25! can be derived by considering a post or priform derivation @16,17#. In the former case, the entrancchannel distorted wavex (1)(k0), which is the solution of theelectron scattered by a central local potentialU1, approxi-mates a formal distorted waveX(1)(k0) obtained by project-ing the one-electron orbitalfLM from an exact collision stateCa

(1)(k0). In this situation, the distorted wavesx (6)(k) areobtained by solving the elastic scattering problem in the fiof the atom for the incoming and scattered electrox (1)(k0) and x (2)(k f), and in the field of the ion for thecalculation of the distorted waves describing the scatteringthe slow outgoing electron,x (2)(ks). In the prior formula-tion, the product of the two outgoing distorted wavx (2)(k f) and x (2)(ks), calculated in the field of the ionapproximates the exact collision state with final-state bouary conditionC (2)(k f ,ks). In the next section we will useboth of these formulations. We will refer to the post-forformulation as DWBAS and the prior-form formulation aDWBAI.

In both cases, the radial part of the distorted wavesderived as a solution of a radial equation of the type

S d2

dr22

l ~ l 11!

r 222v~r !1k2D ul~r !50. ~26!

In Eq. ~26!, the potentialv(r ) corresponds to the distortinpotential U1(r ) chosen as the equivalent local statiexchange potential of Furness and McCarthy@36# when scat-tering in the field of the atom is considered~e.g., for theincident and scattered electron distorted waves!. The corre-sponding local static-exchange potential for the ion is csen, in addition to the Coulomb potential, when the distorwaves are considered as electron-ion states.

6-7

eiorocaon

ho

momitete

re

m

emi

dyfers

alo

dsanFi

e

heanebiisinrttin

tedperandlcu-the

ntelinein-the

asolute,nea-

tionsonrior-a-ki-ultsedof

s ofionf the

thewthe. Inhilel-

orever,-thet of

er-ri-isitsrossr-of

n inntalnsinri-.32hatne-


We remark that in both alternatives of the DWBA thelectron-electron interaction is not included in the calculatof the outgoing distorted waves. Thus, the bound-electorbital and the distorted waves representing the slow esing electron are orthogonal. Therefore, only the electrelectron interaction potentialv12 contributes to Eq.~25!.

In the second method considered here, the DS3C metthe exactT-matrix elements are approximated as

^k fksFJiLi Mi

ion uTuFJLMatomk0&[^C f

(2)~k f ,ks!uVufL,MLk0&.

~27!

In relation ~27!, the initial state of the electron-atom systeconsists of the product of a plane wave describing the incing projectile and a bound state describing the laser-excatom state. The final state is reduced to a three-body sysby assuming that the residual ion (Na1) acts as a pointcharge on the two escaping electrons. The state of this thbody system is approximately represented by

C f(2)~r f ,r s!5Neik f•r feiks•rs

1F1@ ib f ,1,2 i ~kfr f1k f•r f !#

31F1@ ibs,1,2 i ~ksr s1ks•r s!#

31F1@ ib f s,1,2 i ~kf sr f s1k f s•r f s!#, ~28!

where r f s5r f2r s and k f s5(k f2ks)/2. 1F1@a,b,c# is theconfluent hypergeometric function whileN is a normaliza-tion factor. The form of the dynamical Sommerfeld paraetersb i can be found in@18,37#. The interpretation of thewave function~28! is straightforward. It assumes that ththree-body system consists of three two-body subsysteThe interaction strength within each individual subsystemdictated by the coupling to the other remaining two-bosubsystems. This coupling is contained in the Sommerparameters. The final-state wave function within the fiBorn approximation is readily obtained from Eq.~28! uponthe replacementb f[0[b f s , bs52ZNa1 /ks , whereZNa1 isthe charge of the residual ion. If we consider the two finstate electrons to be moving independently in the fieldNa1, the Sommerfeld parameters in the wave function~28!reduce tobs52ZNa1 /ks , b f52ZNa1 /kf , b f s[0.

IV. RESULTS AND DISCUSSION

To establish the validity of the two scattering methopresented in the previous section for the particular kinemics considered here, we first concentrate on the descriptiothe ionization process of ground-state sodium atoms. In5, we present DWBA calculations for the ionization of2S1/2ground-state sodium for four different kinematical arrangments.

Within the DWBA method, a Slater representation of tsodium Hartree-Fock orbitals, as tabulated by ClementiRoetti @38#, is used for the target radial orbitals. This reprsentation is used to generate the bound-electron radial oras well as the static potential in the calculation of the dtorted waves. As mentioned earlier, the equivalent spaverage local exchange potential of Furness and McCa@36# is used to generate the exchange part of the distor

01270

nnp--

d,

-dm

e-

-

s.s

ldt

-f

t-ofg.

-

d-tal--

hyg

potential in both the entrance and exit channels. It is nohere that the DWBA calculations presented in this pahave not been energy averaged over the 6 eV energy bover which scattered electrons are detected, but rather calated at the mean value of this energy band for each oftwo analyzers separately.

Figure 5 shows DWBA calculations for the two differederivations of the DWBAT-matrix elements presented in thprevious section compared to measurement. The dashedcorresponds to the post-form derivation, where both thecident and scattered distorted waves are calculated indistorting potentialU1, and the solid line to the prior-formderivation where both outgoing electrons are calculatedelectron-ion states. Since the measurements are not absthe only valid comparison of theory with experiment is oof shape. Having chosen the form of the DWBA, normaliztion of the data to theory is via the 3p cross section asdiscussed later. This gives the ground-state normalizafactors indicated in the caption of Fig. 5. Shape compariwith the experimental measurements indicates that the pform derivation provides a better description of the ioniztion process of the ground-state sodium atom for all thenematics considered here. The post-form formulation resin a cross section where the maximum is slightly shifttoward lower values of the slow electron angle. This shiftthe maximum of the cross sections toward smaller valuethe slow electron angle increases in the DWBAS calculatas the incident energy decreases. For the lowest value oincident electron energy considered here,E0560 eV, thebinary peak of the cross sections when calculated usingDWBAS formulation is shifted to lower values of the sloelectron angle by almost 20° when compared to bothDWBAI calculation and the experimental measurementsterms of the magnitudes of the ionization cross section, wthe DWBAS and DWBAI calculations predict values of amost identical magnitude for the incident energyE05151eV, the DWBAI formulation predicts smaller magnitudes fthe cross sections as the incident energy decreases. Howsince the measurements~a! to ~d! have not been crossnormalized experimentally, the energy dependence oftwo calculations cannot be directly compared against thaexperiment.

Normalization of experiment to theory has been pformed in the following manner. The excited-state expemental 3p cross sections, which form the main focus of thpaper, are normalized to the DWBAI theory based onsuperior description of the shapes of the ground-state csections. Given that the 3s/3p cross-section ratios are detemined experimentally for each kinematics, within an error20% @see Eqs.~2! and ~3!#, multiplicative coefficients areapplied to the ground-state theoretical calculations showFig. 5 for best comparison with the shape of the experimecross sections. The coefficients for the DWBAI calculatioreflect how the 3S/3P cross-section ratio, as predicted withthe DWBAI model, compares to that derived from expement. The values of these coefficients, which vary from 1at 151 eV to 2.15 at the lowest energy, clearly indicate tthe branching ratio is underestimated for three of the kimatic regimes considered here@graphs~a!, ~b!, and ~d!#. A

6-8

f

in


FIG. 5. Variation of the crosssections as a function of the slowelectron angle for the ionization oground-state sodium. DWBAI~solid line!, DWBAS ~dashedline!. ~a! E05151 eV, Es520.5eV, andu f520°; ~b! E0590 eV,Es520 eV, andu f520°; ~c! E0

590 eV, Es510 eV, and u f

515°; and ~d! E0560 eV, Es

520 eV, andu f520°. The DW-BAI calculations are multiplied by1.32 ~a!, 1.46 ~b!, 1 ~c!, and 2.15~d!. The DWBAS calculations aremultiplied by 1.2~a!, 1.1 ~b!, 0.78~c!, and 1.2~d!. The data are nor-malized to the DWBAS excited-state cross sections as discussedthe text.

sf

st

folaicort

bs

de

rtree

t

the

n-ther

ateasl isag-

ki-t theionmosts, inyc-he

wm-se.

coefficient of 1.0 is applied to the DWBAI calculationshown on graph~c!, indicating that in this case the ratio othe 3S/3P cross section now appears to be slightly overemated.

Nevertheless, the figure does show that the prior-formmulation is superior in describing quantitatively the angubehavior of the experimental data over the whole kinematrange considered here. The comparison between prior-fresults and experimental measurements suggests thaproximating the exact final-state wave functionC (2)(k f ,ks)as the product of two Coulomb waves gives a reasonadescription of the position of the binary peak of the crosections over the range of kinematics considered here.

We now turn to the main focus of the present study,scription of the ionization of laser-excited 3P sodium atoms.In the DWBA calculations, the excited sodium atoms aalso described using a Slater representation of the HarFock orbitals. The Hartree-Fock orbitals are in this case gerated using the Hartree-Fock program of Fischer@39#. Forthe DS3C calculations, a Klapisch-type potential is usedgenerate the excited bound electron orbital@40,41#.

01270

i-

r-ralmap-

les

-

ee-

n-

o

The thick solid and dashed curves of Fig. 6 showsstate-resolved cross sectionss1,1 ands1,21, when calculatedusing the prior form of the DWBA method. The experimetal measurements have not been normalized to one anoand hence are individually normalized to the excited-stDWBAI theory for each of the kinematics considered,discussed earlier. The figure shows that the DWBAI modereasonably successful in predicting the correct relative mnitude of the state-resolved cross sections at each of thenematics presented. The most obvious discrepancy is aincident energy of 151 eV, where the theoretical predictindicates that the state-resolved cross sections are of althe same magnitude while the experimental cross sectioncontrast, show thes1,1 cross section to be significantlsmaller than thes1,21 cross section. The double-peak struture in thes1,1 cross section is also not reproduced in tcalculation. Recently, it has been argued@42# that the originof this structure lies in the final-state interaction of the sloelectron with the residual ion. However, the scattering geoetry investigated was slightly different from the present ca

For the lower value of the incident electron energy,E0

6-9

f,

-e

.


FIG. 6. Variation of the crosssections as a function of slowelectron angle for the ionization o3P excited sodium. Filled circlesmeasureds1,21 (Na 32P3/2, mF

523); open circles,s1,1 (mf513). DWBAI calculationss1,21

~thick solid line! and s1,1 ~thickdashed line! and DWBACF calcu-lations s1,21 ~thin solid line! ands1,1 ~thin dashed line!. See textfor details. The experimental measurements are normalized to thmaximum of the DWBA s1,21

calculations for~a!–~c! and to themaximum of DWBAs1,1 for ~d!.The kinematics are as in Fig. 5The arrow indicates the directionof the momentum transferq.

ose,

,

att

erta

yt in

ghtioingl

thec

te-thetheame. In

tothe

hinthesort-teredini-sslved

hes.

tions

590 eV, the relative magnitudes of the experimental crsectionss1,21 ands1,1 are also clearly different. In this casfor a slow electron ejected at an energyEs of either 10 or 20eV and at a scattering angleu f of 15° and 20°, respectivelythe magnitude ofs1,1 is consistently smaller thans1,21. Ascan be seen from Fig. 6, the relative magnitudes of the stresolved cross sections are predicted reasonably well byDWBA method; however the ratio of peak heights is undestimated by the theoretical model while the experimenmeasurements ofs1,1 suggest a slightly narrower binarpeak for both kinematics. Further, a mild shoulder evidenthe experimentals1,1 is not reproduced by the theory whethe slow electron is ejected with an energy ofEs510 eV. AtE0560 eV, the relative magnitude ofs1,21 ands1,1 is stillreasonably well described by the DWBA model althouagain underestimated. However, the theoretical predicfor the position and relative height of the binary peaks1,21 is now at variance with the experimental data, its anbeing shifted by around 5° toward lower angles.

To further investigate the mechanism responsible forbreaking of symmetry around the momentum transfer dir

01270

s

e-he-l

n

n

e

is-

tion we used two additional models to calculate the staresolved cross sections. In the first model, DWBAPF,scattered electron is described by a plane wave whileincident and ejected electron are described using the sdistorting potential as described in the previous sectionthe second model, DWBACF, the distorting potential usedcalculate the fast-electron distorted wave is replaced byCoulomb potential only.

The DWBACF calculations are represented by the tsolid and dashed curves in Fig. 6. Comparison betweenDWBA and DWBACF calculations in this figure indicatethat, for the kinematics considered in this study, the shrange interactions, static and exchange, between the scatelectron and the remaining electrons of the ion have a mmal influence on the position of the maximum of the crosections or on the relative magnitudes of the state-resocross sections. We note in the figure that thes1,1 cross sec-tions calculated within the DWBACF are normalized to tcorresponding DWBA calculations for each kinematicNonetheless, we can see that the short-range interactend to increase the difference in magnitude betweens1,1

6-10

d

e-

k

.


FIG. 7. Influence of the Cou-lomb interaction for the scattereelectron on the variation of thecross sections as a function of thslow electron angle for the ionization of 3P sodium. Same legenddetails as in Fig. 6; the DWBAIcalculations are shown as thiclines, the DWBAPF calculationsas thin lines. The DWBAPF cal-culations are multiplied by 0.87~a!, 0.80 ~b!, 0.83 ~c!, and 0.75~d!. Same kinematics as in Fig. 5

ns

atect

theioe

thoeraras

ecei

ks

ardgly-

the

is

ryec-isac-

n-e-ted

ns

er-

ands1,21 by the same amount for all cases.In contrast, the comparison between the DWBAPF a

DWBA calculations, shown in Fig. 7, indicates an increaingly strong influence of the Coulomb interaction of the sctered electron with the residual ion when the incident eltron energy decreases. This is comprehensible, asdichroism as such vanishes when all interactions ofejected electrons with the ion are neglected. At lower engies the interactions of both electrons with the residualbecome of the same order and hence we can expect thchroism to be influenced by both of these interactions.

For an incident electron energy of 151.6 eV and atkinematical arrangement of the present experiment, thesuggests that the Coulomb interaction between the scattelectron and the residual ion has a minor effect on the shand magnitude of the dichroism. This situation changes dtically when the incident energy is lowered to 90 eV. Adeduced from Fig. 7 the Coulomb interaction of the fast eltron with the ion is largely responsible for the break in rflection symmetry of the state-resolved cross sections wrespect toq. Furthermore, the positions of the binary pea

01270

d---heer-ndi-

eryedpes-

--th

in both of the state-resolved cross sections are shifted towlarger ejection angles. The dichroism is then correspondinmodified. At the moment it is not clear why this shift proceeds in that direction when the final-state interaction ofprojectile electron is taken into account.

At yet lower energies (E0560 eV!, the situation becomesmore delicate. Obviously the magnitude of the dichroismunderestimated by the DWBA calculations~cf. Fig. 7!. Nev-ertheless, the DWBA model predicts a shift of the binapeak positions with respect to the momentum transfer dirtion in the direction shown by the experiments. Again thshift can be associated with the Coulomb final-state intertion of the scattered electron with the residual ion~cf. Fig.8!.

In earlier work @12# we suggested that the electroelectron final-state interaction, missing in the DWBA, bcomes important at this low energy. This conjecture is tesby the DS3C calculations shown in Fig. 8~d! in which wesystematically switch on the various final-state interactioand explore their influence.

For all DS3C calculations presented in this work an av

6-11

ed

Ans at all theBA results

itionald


FIG. 8. The same kinematical arrangement as in Fig. 5. In~a! the incident energy isE05151.62 eV, the median energy of the ejectelectron isEs521.5 eV ~additional 1.0 eV energy due to weaker binding energy of 3p electron!, and the scattered electron angle isus

520°. The calculations fors1,21 ~solid curve! and s1,1 ~dashed curve! are performed within the DS3C model including exchange.average has been performed over the 6 eV range of energies over which the outgoing electrons are detected in all the calculationkinematics. For best shape comparison, the experimental results have been normalized to the DS3C theory. The inset shows the F~light solid curve fors1,21 and light dashed curve fors1,1) and the results~solid curve representss1,21 and dashed curve iss1,1) when thetwo escaping electrons move independently in the field of the residual ion, i.e., we set in Eq.~28! b f s[0,bs52ZNa1 /ks , b f52ZNa1 /kf . For inset in~a! only the experimental results, after normalization to the DS3C theory, have been multiplied by an addfactor of 1.5 to facilitate comparison with the FBA calculations. In~b! the same labeling of curves as in~a! but the incident energy is lowereto E0590 eV. In ~c! the same as in~b! but the scattering angle is fixed tou f515° and the ejected electron energy is chosen asEs510 eV.The FBA calculations are not shown in the inset.~d! the 60 eV data with the same descriptions of curves.

012706-12


FIG. 8 ~Continued!.

oot

ten

p-

ard

age has been performed over the 6 eV range of energieswhich outgoing electrons are detected. For best shape cparison, the experimental results have been normalized toDS3C theory. If all final-state interactions are neglec@b f[bs[b f s[0 in Eq. ~28!# the dichroism vanishes. Upo

01270

verm-hed

replacing in Eq.~28! b f[0[b f s , bs52ZNa1 /ks ~the FBAcase! we obtain a finite dichroism with the symmetry proerties mentioned above. If we set in Eq.~28! b f s[0,bs52ZNa1 /ks , b f52ZNa1 /kf ~independent Coulomb particles!the binary peaks in the cross sections are shifted tow

6-13

inr

eothigtounonais

nt-al-thineeroreeetg

thndtei

hndutthe-roak

ssm

It isthean-et-

bergyere,ossthe

theastle.stillntsV,

, be-rononin

byrgyven-esegs.

n atronalr in

eus

naldant


larger ejection angles, as in the case of Fig. 7. Now takthe electron-electron final-state interactions into accountsults in a further shift of the binary peaks and an increasthe dichroism. The direction of the shift, however, is nsimply one to larger angles as would be expected onbasis of mutual repulsion of the two escaping electrons. Fure 8~d! shows the direction and magnitude of the shiftdepend upon the initial magnetic quantum state of the boelectron state. This implies a dynamical role of the electrelectron interaction beyond a simple repulsion effect. In pticular, the ionization via direct scattering from the coregreatly enhanced when the interelectronic correlationtaken into account. This is because, in the FBA and withifrozen core approximation, ionization following direct scatering from the nucleus is not allowed due to the orthogonity of initial and final target states. This situation is not atered when the interaction of the scattered electron withresidual ion is taken into account, since the two escapelectrons are still decoupled. It is the coupling between thtwo electrons that makes possible an ionization event aftdirect projectile-core scattering. At yet lower energies, mdrastic effects of the electron-electron interaction have banticipated@15#. All DS3C calculations shown in Fig. 8 arenergy averaged over the 6 eV range over which the ouing electrons are detected.

For the higher-incident-energy cases@Figs. 8~a!–8~c!# theeffect of the electron-electron final-state interaction onshapeof the cross sections becomes less pronounced uthe kinematics of concern here, which are primarily dictaby the final-state interactions of the escaping electrons wthe residual ion. This in accord with the conclusions of tDWBA calculations. From the results depicted in Fig. 7 aFigs. 8~a!–8~d! we can conclude, however, that the absolvaluesof the cross sections are considerably affected byinterelectronic final-state interaction. This is in line with rcent conclusions for the electron-impact ionization of isotpic targets@43#. It is interesting to note that the double pein thes1,1 cross section for the 151 eV kinematics@Fig. 8~a!#is reproduced in the DS3C calculations.

V. CONCLUSIONS

In this work we have presented fully differential crosections for the electron-impact ionization of the sodiu

W

ld

01270

ge-inte-

d-

r-

isa

l-

eg

sea

en

o-

eer

dthe

ee

-

atom laser pumped into a selected magnetic sublevel.demonstrated, experimentally and theoretically, thatspectra of the emitted electrons depend in a nontrivial mner on the helicity of the absorbed photon. For the asymmric kinematics considered in this work, our findings cansummarized as follows. At an intermediate incident eneof 151.6 eV and for the scattered angles presented htheory suggests that the dichroism and the individual crsections are essentially determined by the interactions ofslow outgoing electron with the residual ion, whereasinterelectronic coupling as well as the interaction of the foutgoing electron with the residual ion play a minor roHowever, discrepancies between theory and experimentexist under these conditions, precluding definitive statemeon the nature of the dichroism. At yet lower energies, 90 ethe scattering dynamics, and hence the cross sectionscome sensitive to the interaction of the fast outgoing electwith the residual ion while the final-state electron-electrcorrelation is of less importance. This situation is reflectedincreasing deviations from reflection symmetry~with respectto the momentum transfer direction! of the state-resolvedcross section. The strong symmetry breaking is confirmedexperimental measurements. At the lowest incident eneconsidered here~60 eV!, the cross sections become sensitito all interactions, and, in particular, the final-state electroelectron interaction needs to be taken into account. Thconclusions are substantiated by the experimental findinAs a result of the present study, it is anticipated that evehigher incident energy, the inclusion of the electron-electinteraction will prove indispensible for certain kinematicsituations where the two electrons are close to each othevelocity space or when a strong scattering from the nuclis unavoidable, e.g., in backscattering geometry.

ACKNOWLEDGMENTS

One of the authors~S.M.! wishes to thank the Atomic andMolecular Physics Laboratories at the Australian NatioUniversity for hospitality while this work was performed analso to acknowledge financial support from the NSF GrNo. PHY-9722055.

.

.

allli-.

P.

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dichroism in the electron-impact ionization of excited and

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