INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS

J. Phys. B: At. Mol. Opt. Phys. 37 (2004) 4423–4433 PII: S0953-4075(04)83008-6

Doubly differential cross sections for ionization ofxenon by spin-polarized electrons

A Prideaux and D H Madison

Physics Department, University of Missouri-Rolla, Rolla, MO 65409, USA

Received 24 June 2004, in final form 5 October 2004Published 8 November 2004Online at stacks.iop.org/JPhysB/37/4423doi:10.1088/0953-4075/37/22/003

AbstractEven though spin-dependent effects are naturally associated with relativisticeffects, it has been known for some time that significant spin asymmetries forelectron-impact ionization are possible in a completely non-relativistic modelif the J-state of the residual ion can be experimentally resolved. In the lowestorder implementation of the same model, the spin asymmetry would vanishif the J-states of the ion are not experimentally resolved (i.e., summed over).Consequently, it is perhaps possible to search for relativistic effects by lookingat asymmetries for which the final ion J-state is not resolved. There is alsosome experimental evidence that relativistic effects might be important forlarge scattering angles which are inaccessible to current experimental set-ups.If this is the case, relativistic effects might be seen in a double differential crosssection measurement which integrates over all angles. Very recently, somesignificant experimental spin asymmetries for electron–xenon scattering havebeen reported for a doubly differential spin-asymmetry measurement in whichthe final J-state was not resolved. The purpose of this paper is to investigatewhether or not these experimental results indicate relativistic effects.

Introduction

Investigations of atomic ionization by electron impact, normally referred to as (e, 2e), haveprovided very sensitive tests for theoretical models particularly when differential cross sectionsare examined. The standard cross section differential in energy and angular location of the twofinal-state electrons is typically called the triple differential cross section (TDCS) in spite ofthe fact that this cross section is actually five-fold differential. The TDCS are not completelydifferential since the spins of the continuum electrons and target atoms are not resolved.Measurements of atomic ionization using spin-polarized electrons were first reported in theearly 1990s and the most differential measurement made to date was for a spin-polarizedincident electron beam ionizing polarized atoms (Baum et al 1992, Lower et al 2001).

0953-4075/04/224423+11$30.00 © 2004 IOP Publishing Ltd Printed in the UK 4423

4424 A Prideaux and D H Madison

Experiments performed so far have not attempted to analyse the spin of the final-stateelectrons. A parameter associated with these experiment in which only the spin of the incidentelectrons is known is the spin-asymmetry parameter which is defined as the difference betweenthe spin up and spin-down cross sections divided by the sum. The two physical effects thatcan cause the spin-asymmetry parameter to be non-zero are electron-exchange and relativisticeffects. Relativistic effects are clearly important for high incident-energy electrons and theyhave been studied for 300 keV polarized electrons ionizing the K-shell of silver and uraniumat relativistic energies (Ast et al 1994, Prinz et al 1995).

It has been known for over 20 years that spin up–down asymmetries can result from non-relativistic electron-exchange effects for atomic excitation (Dummler et al 1993, Hanne 1983,Bartschat and Madison 1987, Padial et al 1990) if the incident electrons are spin polarized andthe final J-state of the atom is resolved. This effect has been called the fine-structure effect.Jones et al (1994) predicted that significant spin asymmetries should also be seen for atomicionization as a result of the non-relativistic fine-structure effect. The measurements of Granitzaet al (1996), Guo et al (1996) and Hanne (1996) demonstrated significant fine-structure effectspin asymmetries for low to intermediate incident-energy electrons ionizing xenon.

The fine-structure effect results from electron exchange. For atomic ionization, thereare two different possible exchanges that can take place: (1) the two final-state continuumelectrons exchanging with each other and (2) any of the initial or final-state continuumelectrons exchanging with an atomic electron. The second type of exchange is sometimesreferred to as ‘exchange distortion’ since this exchange modifies the differential equationfor the continuum electron and ‘distorts’ the continuum electron wavefunction. The initialcalculations only included the first exchange possibility and the agreement between experimentand theory was sometimes good and sometimes terrible (Madison et al 1996a, 1996b). It wassubsequently shown that the cases of bad agreement resulted from neglecting the secondexchange possibility—exchange distortion (Madison et al 1998, Mette et al 1998). One ofthe interesting issues concerns the importance of relativistic effects for low to intermediateincident-electron energies. Although most of the evidence indicates that relativistic effects arenot important for the kinematical conditions that have been measured so far (Granitza et al1996, Guo et al 1996, Hanne 1996, Mette et al 1998), Dorn et al (1997) found some indicationsthat relativistic effects might be important for larger scattering angles. Mazevet et al (1998)performed a semi-relativistic distorted wave Born approximation (DWBA) calculation forthe ionization of closed shell atoms at intermediate energies, and they found that, despitethe fact that relativistic effects are important at low and intermediate energies for elastic andinelastic electron–xenon scattering, the relativistic interactions were most important for thespin-asymmetry parameter only in the kinematic region where the TDCS is small. In thesemi-relativistic approach, relativistic corrections are included in the radial the Schrodingerequation.

One possibility for searching for relativistic effects at moderate incident-electronenergies lies in the spin-asymmetry parameter summed over the possible J-states. Presentmeasurements indicate that the unresolved asymmetry is quite small (Granitza et al 1996,Guo et al 1996, Dorn et al 1997). Consequently, if relativity is a small effect, then it may beobservable in these measurements. Further, a lowest order non-relativistic treatment of thefine-structure effect predicts that the unresolved spin asymmetry should be zero so that anynon-zero spin asymmetry might come from relativistic effects.

We define the scattering plane as the plane that contains the incident and faster final-state electron momentum vectors. This definition restricts the incident and fast outgoingelectrons be in this plane, but makes no restriction on the slow outgoing electron. In thiswork, we have considered both in-plane and out-of-plane contributions from the slow electron.

Doubly differential cross sections for ionization of xenon by spin-polarized electrons 4425

Ernsting (2003) recently reported a measurement of the unresolved spin-asymmetry parameterfor double differential cross sections (DDCS). In the DDCS measurement, the energy andangular location of the faster final-state electron is measured. By energy conservation,one knows the energy of the slower electron but the angular position is not determined.Consequently, the DDCS represents the integration of the TDCS over all ejection angles(in and out of the scattering plane) for the slow-ejected electron. If relativistic effects aremore important for the larger scattering angles not measured in the TDCS (Dorn et al 1997),they might be observable in the DDCS. Consequently, the possibility exists that the DDCSunresolved spin-asymmetry parameter might represent a better test of relativistic effects forlow to intermediate energies than the resolved TDCS measurements. The purpose of thispaper is to investigate this possibility.

Theory

The details of the theory have been given previously (Jones et al 1994, Madison et al 1996a,1996b, 1998) so here we present only the essential points. Let us define σJ (↑) as the crosssection for an electron beam that is spin polarized up perpendicular to the plane containing theincident electron and the fast outgoing electron (the scattering plane). We use the so-callednatural coordinate system where the x-axis lies along the incident beam direction and the z-axisis pointing up perpendicular to the scattering plane. The fully differential cross section (FDCS)σJ (↑) is a function of the incident energy, the final ion state with total angular momentum J,the energies of the two final-state electrons and the directions of the fast and slow outgoingelectrons. Likewise σJ (↓) is the corresponding cross section for incident spin-down electrons.The spin asymmetry is defined as

AJ = σJ (↑) − σJ (↓)

σJ (↑) + σJ (↓). (1)

Here, we are concerned with ionization of p-shell electrons of heavy inert gases. This meansthat there are only two possible states of J, either 1

2 or 32 . If the fast outgoing final-state

electron is the same as the incident electron, the process is called ‘direct’ ionization and thecorresponding scattering amplitude will be referred to as f. If, however, the fast outgoingelectron is the ejected atomic electron, the process is called ‘exchange’ ionization and thecorresponding scattering amplitude is denoted by g. Assuming the final-state ions can bedescribed by the LS coupling scheme, the four possible cross sections can be expressed as(Madison et al 1996a, 1996b, 1998)

σ1/2(↑) = 23 (|f−1|2 + |g−1|2 + |f+1 − g+1|2) (2)

σ1/2(↓) = 23 (|f+1|2 + |g+1|2 + |f−1 − g−1|2) (3)

σ3/2(↑) = 13 (|f ′

−1|2 + |g′−1|2 + |f ′

+1 − g′+1|2) + (|f ′

+1|2 + |g′+1|2 + |f ′

−1 − g′−1|2) (4)

σ3/2(↓) = 13 (|f ′

+1|2 + |g′+1|2 + |f ′

−1 − g−1|2) + (|f ′−1|2 + |g′

−1|2 + |f ′+1 − g′

+1|2). (5)

Here, the subscripts +1 and −1 on f and g denote the M value of the orbital angular momentumof the final state of the ion in the natural coordinate system (i.e., the projections are for thez-axis perpendicular to the scattering plane). The scattering amplitudes corresponding to theion being left in a J = 3

2 state are differentiated from the J = 12 equivalents by marking them

with a prime (i.e.,f ′M and g′

M ).

4426 A Prideaux and D H Madison

In this work, we will compare with an experiment that does not resolve the different J-statesthat could potentially be distinguished. As these final states are, in principle, distinguishable,the cross sections are added coherently. Consequently, what is measured experimentally is

A = σ(↑) − σ(↓)

σ (↑) + σ(↓)(6)

where

σ(↑) = σ1/2(↑) + σ3/2(↑) = 23 (|f−1|2 + |g−1|2 + |f+1 − g+1|2)

+ 13 (|f ′

−1|2 + |g′−1|2 + |f ′

+1 − g′+1|2) + (|f ′

+1|2 + |g′+1|2 + |f ′

−1 − g′−1|2) (7)

and

σ(↓) = σ1/2(↓) + σ3/2(↓) = 23 (|f+1|2 + |g+1|2 + |f−1 − g−1|2)

+ 13 (|f ′

+1|2 + |g′+1|2 + |f ′

−1 − g′−1|2) + (|f ′

−1|2 + |g′−1|2 + |f ′

+1 − g′+1|2). (8)

It should be noted that, if f ′M = fM and g′

M = gM , σ(↑) = σ(↓). Consequently, whensumming over the J-state, the spin asymmetry will vanish if L-dependent direct and exchangeamplitudes are the same for both J-states.

The important question then concerns what can make these amplitudes different. First, itis clear that relativistic effects would make the amplitudes different. As a result, a non-zerospin asymmetry summed over the J-states could represent a direct measurement of relativisticeffects. However, there is a second effect that could also cause (f ′, g′) to differ from (f, g)

and that is the energy effect. Since the energy loss is slightly different for the two J-states,the final-state wavefunction for the ejected electron will be slightly different in the scatteringamplitudes. In this calculation, we will neglect relativistic effects and see if the energy effectsalone are sufficient to explain the experimental data.

All cross sections discussed so far are functions of both �a and �b, the directions of thefast and slow electrons, respectively. To compare with the doubly differential cross sections(DDCS) of Ernsting (2003), we have to integrate over �b since only the energy and angle �a

are experimentally resolved. Consequently, we define

σ1/2(↑,�a) =∫

σ1/2(↑,�a,�b) d�b (9)

σ3/2(↑,�a) =∫

σ3/2(↑,�a,�b) d�b (10)

σ1/2(↓,�a) =∫

σ1/2(↓,�a,�b) d�b (11)

σ3/2(↓,�a) =∫

σ3/2(↓,�a,�b) d�b. (12)

The doubly differential cross section summed over J-states is given by

σ(↑,�a) = σ1/2(↑,�a) + σ3/2(↑,�a) (13)

σ(↓,�a) = σ1/2(↓,�a) + σ3/2(↓,�a) (14)

and the experimentally measured doubly differential asymmetry parameter is

A(�a) = σ(↑,�a) − σ(↓,�a)

σ (↑,�a) + σ(↓,�a). (15)

Doubly differential cross sections for ionization of xenon by spin-polarized electrons 4427

In the present work, a distorted wave Born approximation (DWBA) approach is used togenerate the cross sections (Madison et al 1996a). In this formalism, the direct scatteringamplitude is given by

fM = 〈χ−a χ−

b ψI(M)|V − Ui

∣∣ψAχ+i

⟩(16)

where χi is the wavefunction of the incoming electron, χa is the wavefunction for the fastscattered electron, χb is the wavefunction for the slow-ejected electron, ψA is the initial atomicwavefunction, ψI is the wavefunction for the final-state ion with orbital angular momentumLM, V is full interaction between the incident electron and the atom and Ui is the sphericallysymmetric Hartree–Fock potential for the ground-state atom (Prideaux and Madison 2003,Biava et al 2002a, 2002b). The exchange amplitude

gM = 〈χ−b χ−

a ψI(M)|V − Ui

∣∣ψAχ+i

⟩(17)

is similar to equation (16) except that the scattered electron is now the slow electron χb and theejected electron is the fast electron χa . The wavefunction for the incident electron is obtainedas a solution of the Schrodinger equation

(T0 + Ui − εi)χ+i = 0. (18)

Here, T0 is the kinetic energy operator and εi is the energy of the incident electron. In likemanner, the final-state electron wavefunctions χa and χb are solutions of

(T0 + Uf − εa(b))χ−a(b) = 0. (19)

The final-state distorting potential Uf is chosen to approximate the final-state electron–electroninteractions (Jones et al 1994):

Uf = zUion + (1 − z)Ui (20)

where Uion is the Hartree–Fock potential of the residual ion. The effective charge z is given by

z = 1 − 1

2 sin(θab/2). (21)

Here, θab is the angle between the two final-state electrons. The distorted waves were alsoorthogonalized to the atomic wavefunctions so that amplitudes involving two atomic electrons,such as capture, will vanish.

It has been found (Madison et al 1998, Mette et al 1998) that it can be important to include‘exchange distortion’ in the calculation of the distorted waves. Following the above works,the Furness–McCarthy (Furness and McCarthy 1973) approximation is used for the ‘exchangedistortion’

(T0 + Ui − εi)χ+i (r) = Uexχ

+i (r) (22)

where

Uex = − 12 {

√(εi − Ui)2 + 2ρ − (εi − Ui)}. (23)

Here, ρ is half the charge density for the atomic electrons.

Results

Ernsting (2003) reported doubly differential cross sections (DDCS) and doubly differentialasymmetry parameters (DDAP) for electron-impact ionization of the 5p-shell of xenon forincident energies of 40 eV and 50 eV. The DDCS measurements are compared with presenttheoretical calculations in figure 1. In each part of figure 1, the incident energy and fasterfinal-state energy is noted. The energy of the slower final-state electron is determined by the

4428 A Prideaux and D H Madison

Figure 1. Doubly differential cross section for electron-impact ionization of the 5p-shell of xenon.The incident energy E0 and fast outgoing energy Ea are noted in each part of the figure. Thesolid line is DWB(01), the long-dash short-dashed line is DWB(ZE) and the dashed dot line isDWB(11). The experimental data of Ernsting (2003) were normalized to a best visual fit tothe theory. The same normalization factor was used for all experiments with the same incidentenergy. The horizontal axis is the angle at which the fast outgoing electron is observed, measuredcounterclockwise relative to the beam direction as viewed from above. In all models no exchangedistortion is included.

final J-state of the ion (which is not experimentally resolved). The ionization energies of thetwo possible final states are 12.13 eV for J = 3

2 and 13.44 eV for J = 12 . Consequently, for

the (E0, Ea) = (40, 25) results of the top part of figure 1, the slower electron has an energy of2.87 eV for J = 3

2 and 1.56 eV for J = 12 (we will denote these energies as(

32 , 1

2

) = (2.87, 1.56)). For the (E0, Ea) = (50, 30) case,(

32 , 1

2

) = (7.87, 6.56); for the(E0, Ea) = (50, 36.56) case,

(32 , 1

2

) = (1.31, 0.0) and finally for the (E0, Ea) = (50, 37.87)

case,(

32 , 1

2

) = (0,−). Consequently, for the bottom part of figure 1, there is not enoughenergy to ionize the J = 1

2 and it is just possible to ionize J = 32 with the ejected electron

having zero energy far from the atom. For the second panel up from the bottom, both statescan be ionized but the J = 1

2 electron will have zero energy asymptotically. The third panelup from the bottom has the most energetic final-state electrons of all the measurements. Forthe experimental points with no visible error bars, the error bars are smaller than the size ofthe points.

Doubly differential cross sections for ionization of xenon by spin-polarized electrons 4429

The first issue concerns how to compare with experiments set up for the slow-ejectedelectron to have zero energy. The experiment was performed for a spectrometer with anenergy resolution of 1.4 eV FWHM. To make a proper comparison, one should convolute overthe energy resolution of the spectrometer taking into account the beam profile of the source.However, such a convolution is impractical even if we know the necessary experimental profilessince we have significant numerical instabilities very close to threshold. If one ignores thebeam profile and assumes a Gaussian distribution for the spectrometer, a distribution with1.4 eV FWHM would correspond to a weighted average energy of 0.5 eV for positive energiesso we used this energy for the ejected electron when the experimental conditions correspondto zero energy.

We have performed three different calculations using different final-state distortingpotentials. In the very first distorted wave calculation for atomic ionization (Madison et al1977), the neutral atom distorting potential was used for the faster final-state electron and thedistorting potential for an ion was used for the slower ejected electron. The idea here is thatthe faster electron is effectively gone before the slower electron leaves and this is probablysensible if there is a significant energy difference between the two final-state electrons such aswe have here. Chen et al (2004) recently found that this model works best in a second-orderdistorted wave approximation. In terms of the effective charges of equation (20), z = 0 forthe incident and faster final-state electron and z = 1 for the slower ejected electron. We labelthis model as DWB(01). In the second model, both ejected electrons are calculated usingthe distorting potential of the final-state ion. This is a commonly used model for distortedwave calculations for ionization and the argument is that the two indistinguishable final-stateelectrons should be treated symmetrically. For equation (20), z = 0 for the incident electronand z = 1 for both final-state electrons and so we label this DWB(11). The final model is touse the angle-dependent effective charges given by equation (21) which we label as DWB(ZE)for Z-effective. The angle-dependent effective charges were chosen to approximate the final-state electron–electron interaction which is contained only to first order in a distorted wavecalculation (Jones et al 1994).

The experimental differential cross section data of figure 1 are relative. Consequently,two normalization factors are used—one for the 40 eV incident energy and a second one forthe 50 eV incident-energy results. It is seen from figure 1 that DWB(ZE) and DWB(01) arein significantly better agreement with experiment than DWB(11) and overall, DWB(01) isperhaps the best. It is clear that the agreement with experiment is better for the higher energyfinal-state electrons (top two panels) than the lower energies of the bottom two panels whenone of the slow-ejected electrons has an experimental energy of zero. For these cases, theexperimental results for the smallest measured angles tend to be larger than the theoreticalpredictions (for this normalization). Recall that for the bottom panel, only the J = 3

2 stateis energetically allowed. It is interesting to note that the shape of the experimental J = 3

2DDCS is almost the same as the other cases for which both J = 1

2 and J = 32 are energetically

possible. Comparing the theoretical cross sections on the bottom two panels indicates a moredramatic change in the shape than is seen in the experiment.

Figure 2 compares the theoretical and experimental results for the DDAP. Since thisparameter is a ratio of cross sections, the experimental results are now absolute. Similar tothe DDCS, the DWB(11) DDAP results are in the worst agreement with the experimentaldata of the three models we tried. Again, DWB(01) is in the best agreement with experimentexhibiting reasonable qualitative agreement for the two upper panels for which both final-state electrons have positive energies (

(32 , 1

2

) = (2.87, 1.56) for (E0, Ea) = (40, 25) and(32 , 1

2

) = (7.87, 6.56) for (E0, Ea) = (50, 30)). None of the theories are very good for thebottom panel

((32 , 1

2

) = (0,−))

and DWB(ZE) appears to have an incorrect sign. The only

4430 A Prideaux and D H Madison

Figure 2. Doubly differential asymmetry parameter for electron-impact ionization of the 5p-shellof xenon. The incident energy and fast outgoing energy are noted in each graph. The solid line isDWB(01), the long-dash short-dashed line is DWB(ZE) and the dashed dot line is DWB(11). Theexperimental data are those of Ernsting (2003). The horizontal axis is the angle at which the fastoutgoing electron is observed, measured counterclockwise relative to the beam direction as viewedfrom above. In all models no exchange distortion is included.

case where DWB(ZE) is in reasonable agreement with experiment is the second panel fromtop (highest ejected electron energies of

(32 , 1

2

) = (7.87, 6.56)). On the other hand, a straightline at zero would also be in reasonable agreement with that data! On the more positive side,only DWB(ZE) predicts the fairly large negative dip seen for the

(32 , 1

2

) = (1.31, 0.0) case(second panel from bottom).

As mentioned in the theory section, Madison et al (1998; see also Mette et al (1998))showed that ‘exchange distortion’ can be very important in getting the sign on the asymmetryparameter correct for triple differential cross sections. Consequently, we decided to investigatethe importance of exchange distortion by using the Furness–McCarthy approximation ofequations (22) and (23) for exchange (this is the same approximation that was used in theabove references). Figures 3 and 4 show the effect of exchange distortion on the DWB(01)results. It is seen that exchange distortion tends to improve the shape agreement for theDDCS and makes the agreement for the DDAP worse if anything. We performed the samecalculations for the other two models with similar results except that larger changes were found

Doubly differential cross sections for ionization of xenon by spin-polarized electrons 4431

Figure 3. Doubly differential cross section for electron-impact ionization of the 5p-shell ofxenon. The incident energy and fast outgoing energy are noted in each graph. The solid lineis DWB(01) with no exchange distortion and the dashed line is DWB(01) including exchangedistortion multiplied by 0.5. The experimental data of Ernsting (2003) were normalized to a bestvisual fit to the theory. The same normalization factor was used for all experiments with the sameincident energy. The horizontal axis is the angle at which the fast outgoing electron is observed,measured counterclockwise relative to the beam direction as viewed from above.

for the DDAP. On the other hand, we do not believe that this comparison accurately representsthe importance of exchange distortion for these low-ejected electron energies. Baiva et al(2002a, 2002b) examined the accuracy of using Furness–McCarthy local exchange for inertgases by comparing distorted waves calculated using the continuum Hartree–Fock methodwith those calculated using the Furness–McCarthy local exchange approximation. It wasfound that the Furness–McCarthy approximation was accurate for s-shells if the core wastreated as triplet and the active electron shell was treated as singlet. Several different possiblemethods for using the Furness–McCarthy potential were tried for p-shells and none of themwere as accurate as was found for s-shells. Consequently, the comparison shown here onlyserves as an indication of the possible effects of exchange distortion. Unfortunately, using thecontinuum Hartree–Fock distorted waves for the calculation of DDCS and DDAP is beyondour capabilities at the present.

4432 A Prideaux and D H Madison

Figure 4. Doubly differential asymmetry parameter for electron-impact ionization of the 5p-shellof xenon. The incident energy and fast outgoing energy are noted in each graph. The solid lineis DWB(01) with no exchange distortion and the dashed line is DWB(01) including exchangedistortion. The experimental data are those of Ernsting (2003). The horizontal axis is the angleat which the fast outgoing electron is observed, measured counterclockwise relative to the beamdirection as viewed from above.

Conclusions

We have examined DDCS and DDAP for electron-impact ionization of xenon by spin-polarizedelectrons. Whenever spin effects are examined, the importance of relativity immediately comesinto question. One of the intriguing aspects of an experiment which does not distinguishbetween the residual J-states of the ion lies in the fact that the spin asymmetry should be zeroin the lowest order non-relativistic approximation if the energy difference between the twoJ-states can be neglected in the evaluation of the scattering amplitudes. For the ionizationof the outer p-states of xenon, the energy difference between the two J-states is 1.31 eV.Consequently, if this difference is small enough that the direct and exchange amplitudes forthe J = 1

2 state are the same as the direct and exchange amplitudes for the J = 32 state, then

the non-relativistic spin-asymmetry parameter would be zero. Consequently, an experimentalmeasurement of a non-zero spin-asymmetry parameter would be a direct consequence ofrelativistic effects.

Ernsting (2003) measured non-zero DDAP for electron–xenon ionization. The primarypurpose of our investigation was to determine if the experimental results could be explained in

Doubly differential cross sections for ionization of xenon by spin-polarized electrons 4433

terms of small energy differences in the direct and exchange amplitudes or if the measurementsindicated important relativistic effects. Our non-relativistic DWB(01) calculations yieldedreasonably good agreement with the experimental DDAP measurements for the case of finitenon-zero energies for the slow-ejected electron. The agreement was not as good for the twocases in which one of the ejected slow electrons had the experimental equivalent of zeroenergy. However, we do not think that this is a strong indication of relativistic effects since (1)exchange distortion will certainly be important at these energies and our treatment is very likelyto not be accurate and (2) our approach is a first-order approach and it is also known that higherorder effects will be important for very low energies such as this. In conclusion, we believethat the agreement we found with the experimental DDAP measurements is surprisingly goodfor the near threshold non-zero energy cases and this strongly indicates that the experimentalresults can be explained by purely non-relativistic physics. An improved theoretical modelwill have to be used for ejected electrons with zero energy.

Acknowledgment

The support of the NSF under grant PHY-0070872 is gratefully acknowledged.

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