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Complex Fano asymmetry parameters for helium L = 0, 1, 2 autoionizing levels excited by

electron impact

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2009 J. Phys. B: At. Mol. Opt. Phys. 42 225201

(http://iopscience.iop.org/0953-4075/42/22/225201)

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

J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 225201 (7pp) doi:10.1088/0953-4075/42/22/225201

Complex Fano asymmetry parameters forhelium L = 0, 1, 2 autoionizing levelsexcited by electron impactN L S Martin1, B A deHarak2 and K Bartschat3

1 Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506-0055, USA2 Physics Department, Illinois Wesleyan University, PO Box 2900, Bloomington, IL 61702-2900, USA3 Department of Physics and Astronomy, Drake University, Des Moines, IA 50311, USA

Received 19 August 2009, in final form 29 September 2009Published 30 October 2009Online at stacks.iop.org/JPhysB/42/225201

AbstractTheoretical Fano q-parameters for He autoionizing states excited by electron impact arepresented, as extracted from first- and second-order hybrid distorted-wave Born + R-matrixcalculations. It is found that the q-parameters for (2s2)1S, (2p2)1D, and (2s2p)1P from thefirst-order calculations are essentially real quantities, while those from the second-ordercalculations are complex but with a real part similar to the first-order values. These findingsare interpreted in terms of the relative phases of the first- and second-order amplitudes.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

We recently reported (e,2e) experiments [1] on Heautoionization for the process

He(1s2)1S + e0 → He+(1s)2S + esc + eej, (1)

in which the incident electron e0 is sufficiently fast that thetwo outgoing electrons may be described as a fast scatteredelectron esc and a much slower ejected electron eej. Animportant quantity of the kinematics is K = k0 − ksc, whichis the momentum transferred to the atom by an electronof incident (scattered) momentum k0(ksc). The data werepresented as angular distributions of ejected electrons fromthe three autoionizing levels (2s2)1S, (2p2)1D and (2s2p)1Pand exhibited two well-known features, the binary and recoilpeaks. Whereas for direct (non-resonant) ionization the recoilpeak essentially vanished for our kinematical situation, thepresence of autoionization resulted in a clearly visible recoilpeak with a shape that was strongly dependent on the orbitalangular momentum L of the resonance. It was found that therecoil peak (relative to the binary peak) could be accuratelyreproduced by a second-order hybrid distorted-wave Born + R-matrix with pseudo-states calculation (DWB2-RMPS), but notby an equivalent first-order calculation (DWB1-RMPS), whichsignificantly underestimated the size of the recoil peak. It wasalso found that a first-order plane-wave Born approximation

(PWB1) calculation could reproduce the results, but onlyif anomalously large values of the Fano q-parameters wereassumed; this is discussed in appendix A.

In this paper, we present effective q-parameter valuesextracted from our first- and second-order distorted-wavecalculations. In what follows, we have isolated the amplitudescalculated for the resonant channels and used them to formpartial cross sections whose line profiles yield q-values for theresonant channel. These q-parameter values can in principlebe experimentally determined in a charged-particle scatteringexperiment, but this is non-trivial since the line profile inthe observed cross section corresponds to a coherent sumover scattering amplitudes for resonant and all non-resonantchannels [2, 3].

2. Theory

In Fano’s theory of autoionization [4], the asymmetryparameter qr of a resonant state r is defined in terms of theexcitation amplitude f r of the state, the ionization amplitudeF r of the open channel r with which the resonance interacts,and the width �r of the resonance, as

qr = fr

Fr√

π�r/2. (2)

Fano’s original expression was in terms of matrix elements;here, we have followed the more general definition in terms

0953-4075/09/225201+07$30.00 1 © 2009 IOP Publishing Ltd Printed in the UK

J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 225201 N L S Martin et al

of amplitudes used by Kheifets [5]. The physical significanceof equation (2) is that the quantity 1

2πq2r is the ratio of the

resonance cross section to the direct ionization cross section(for the open channel r) integrated over the width of theresonance. The total ionization amplitude for energy E is

Frqr + εr

εr + i= Fr

qr + εr(1 + ε2

r

) 12

ei�r , (3)

where εr = (E − Er)/(�/2) is the energy away from theresonance position Er in multiples of the halfwidth and �r =tan−1(−1/εr) is the additional phase due to autoionization,which increases from 0 to π across the resonance. (Thisextra phase appears in the asymptotic form of the ejectedelectron wavefunction. It also appears in the perturbationof a Rydberg series by a single state as a change in thequantum defect of �r/π on either side of the perturber.) Forphotoionization the q-parameters are real, and the partial crosssection associated with the resonant channel (after integratingover ejected electron directions) is then given by

σr = σ (0)r

(εr + qr)2

1 + ε2r

, (4)

where σ (0)r is the direct ionization cross section for channel

r. The above relationships assume that each autoionizinglevel only interacts with one open channel, i.e., what Fanoreferred to as ‘one discrete state and one continuum’. This isappropriate for helium ionization provided the residual ion isin the ground state. We shall also assume that all quantitiesremain essentially constant in the region of the resonance.

For photoionization of helium, within the validity of thedipole approximation, there is only one open channel andthe well-known asymmetric Fano profile corresponding toequation (4) is observed where the cross section vanishes atεr = −qr. For a PWB1 calculation of ionization by chargedparticle impact, qr is also real, and the partial cross sectionis again given by equation (4). In general, however, moresophisticated scattering theories involve complex amplitudesfor f r and Fr that have different phases. As a result, qr will bea complex quantity [2, 5], which we shall write as

qr = qRr + iqI

r (5)

in equation (3) with the corresponding partial cross section

σr = σ (0)r

(εr + qR

r

)2

1 + ε2r

+ σ (0)r

(qI

r

)2

1 + ε2r

. (6)

This is the sum of a normal asymmetric Fano profile,

σRr = σ (0)

r

(εr + qR

r

)2

1 + ε2r

, (7)

which vanishes at εr = −qRr , and a symmetric Lorentzian

profile,

σ Ir = σ (0)

r

(qI

r

)2

1 + ε2r

, (8)

which does not vanish for finite εr. Thus the most noticeabledifference between equations (4) and (6) is that the latter isnever zero in the region of a resonance. Note that σ I

r doesnot depend on the sign of qI

r . Complex q-parameters, and

hence non-zero minima, also occur in multiphoton ionization[6], and coherent transport through quantum dots [7].

The first- and second-order calculations presented hereuse a hybrid distorted-wave plus R-matrix with pseudo-statesapproach (RMPS) [8–11]. The general idea is to describea (fast) projectile by a distorted wave, with the Coulombinteraction between this electron and the target electron actingas a perturbation on the initial bound state as well as theejected-electron−residual-ion collision system. The latterstates are treated via a convergent RMPS close-coupling-typeexpansion, in which a sufficiently large number of pseudo-states is included to get accurate results for both the He(1s2)1Sground state and the e–He+(1s) collision in the final state.

Note that this method does not explicitly calculate separateamplitudes for the open channels (direct ionization) andclosed channels (autoionization). Instead, the calculationsyield amplitudes that depend on the energy of the ejectedelectron and automatically include the effects of the resonantautoionizing states. In this paper, we analyse the results ofthese calculations in terms of effective q-parameters that aredetermined by fitting partial and total cross sections for thethree autoionizing levels to equation (6).

The calculations were carried out for an incidentelectron energy of 488 eV and a scattering angle of 20.5◦,corresponding to the experimental data of [1]. Exchangeeffects between the projectile and the target were ignored,which is a good approximation at such a high projectile energyand ejected-electron energies around 35 eV. For ionization ofground-state He(1s2)1S leading to a ground-state He+(1s) ion,there is a direct correspondence between a multipole expansionof the Coulomb interaction between the projectile and thetarget and a partial-wave expansion of the ejected electron.The resonant channels may then be labelled r ≡ LM , whereL = 0, 1, 2 are the angular momenta of the three autoionizinglevels and M denotes a magnetic sublevel. We define a summedcross section, for an isolated partial wave L,

σL(E0, θsc) =L∑

M=−L

σLM(E0, θsc), (9)

which represents the ‘scattering cross section’ (for an electronof incident energy E0 scattered through θsc) associated withan autoionizing level after integrating over all ejected-electrondirections and summing over the sublevels.

3. Results

Figure 1 exhibits the summed cross sections σL ofequation (9) for the (2s2)1S, (2s2p)1P and (2p2)1Dautoionizing states, as calculated in the DWB1-RMPS andDWB2-RMPS models, respectively. (Throughout this paper,results from the first-order calculations are shown in the left-hand panels, while the corresponding second-order predictionsare shown in the right-hand panels of each figure.) Thefigure also shows fits of these results to equation (6), andthe corresponding values of qR

L and qIL. For the first-order

calculations, the q-values are essentially real for all threeresonances, and each cross section passes through zero. Forthe second-order calculations, on the other hand, all three

2

J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 225201 N L S Martin et al

qR = −1.10 |qI | = 0.04

σL

(arb

.un

its)

He(2s2)1S DWB1-RMPS

0

0.1

32.5 33 33.5

qR = −1.32 |qI | = 0.77He(2s2)1S DWB2-RMPS

0

0.1

32.5 33 33.5

qR = −1.81 |qI | = 0.11

σL

(arb

.un

its)

He(2s2p)1P DWB1-RMPS

0

1

35 35.5

qR = −2.12 |qI | = 1.36He(2s2p)1P DWB2-RMPS

0

1

35 35.5

qR = −0.59 |qI | = 0.01

σL

(arb

.un

its)

He(2p2)1D DWB1-RMPS

0

0.1

0.2

35 35.5Ejected Electron Energy (eV)

qR = −0.62 |qI | = 1.38He(2p2)1D DWB2-RMPS

0

0.1

0.2

35 35.5Ejected Electron Energy (eV)

Figure 1. Solid circles: the summed cross sections σL for the (2s2)1S, (2s2p)1P and (2p2)1D autoionizing states calculated in theDWB1-RMPS (left panels) and DWB2-RMPS (right panels) models. Solid lines (green): fits of equation (6) to the calculations. Dashed(red) and dotted (blue) lines: contribution to the fits due to the real (equation (7)) and imaginary (equation (8)) parts of qL. The fitted valuesof qR

L and |qIL| are shown in each panel. All cross sections are normalized relative to the maximum value of the second-order 1P calculation.

q-values are complex with significant imaginary parts, thusresulting in non-zero cross sections at all energies. The effectis particularly dramatic for 1D. For these complex q-values,the fits are also displayed as the separate contributions σR

L ofequation (7) and σ I

L of equation (8).We now turn to examining the individual sub-level partial

cross sections σLM . The calculations were carried out in thecollision frame, i.e., with the quantization axis chosen alongthe incident electron beam direction: z = k0. At the relativelyhigh incident energy of the present kinematics, however, it isexpected that the rotational symmetry around the momentum-transfer direction K, predicted by the first-order plane-waveBorn approximation, should be valid. Thus, if the results areexpressed in terms of cross sections with a quantization axisz = K (the momentum-transfer frame), it is expected thatthe cross section for M = 0 will dominate, especially for theDWB1 calculations.

This is indeed the case, as can be seen in figures 2 and 3.Figure 2 shows the first- and second-order P-state partial crosssections σ1M in both the collision frame (upper panels) and themomentum-transfer frame (lower panels). The scale in eachpanel is such that σL = ∑M

−M σLM = 1 at the maximum of thecross section. For both first- and second-order calculations inthe collision frame all M are important, but for the momentum-transfer frame M = 0 dominates to the extent that M = ±1has been magnified by a factor of 5 in order to be visible

in the figure. Similar results hold for the D-state partialcross sections shown in figure 3, except that the dominanceof M = 0 is even more pronounced; the M = ±1 (±2) crosssections were magnified by a factor of 100 (10) in order to bevisible.

We therefore choose the momentum-transfer frame as the‘natural’ frame in which to examine the sublevel cross sections.Figures 4 and 5 show fits of qLM to the individual sublevel crosssections σLM of the P-state and the D-state in this frame. Thereare noticeable similarities in the two figures: the first-ordercalculations yield essentially real values of q, which are verysimilar for each value of M for a given L. Regarding the second-order calculations, the M = 0 cross sections have the largestimaginary parts to the q-values—but with a real componentqR

L0 that is close to the first-order value. Note that this is alsotrue for the S-state shown in the top panel of figure 1. (For a1S state, σ00 = σ0, and the results are the same in the collisionand the momentum-transfer frames.) On the other hand, thesecond-order M �= 0 cross sections are very different fromtheir first-order counterparts, both in absolute magnitude andshape: for M = ±1 the magnitude

∣∣qRL±1

∣∣ is large, but it is ofopposite sign for L = 1 and L = 2, while we see a windowresonance for L = 2,M = ±2. Table 1 summarizes the fittedvalues of qLM in the momentum-transfer frame.

Table 2 lists the positions and widths of the three levels,as determined from the fits of the calculated cross sections

3

J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 225201 N L S Martin et al

M = 0

ΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔ

ΔΔΔΔ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

ΔΔΔΔΔΔΔ

ΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔ

Δ M = ±1

σLM

/σL

( MA

X.)

He(2s2p)1P DWB1-RMPS (z = k0)

0

0.1

0.2

0.3

0.4

0.5

35 35.5

M = 0

ΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔ

ΔΔΔΔ

Δ

Δ

Δ

Δ

ΔΔ

Δ

Δ

Δ

ΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔ

Δ M = ±1He(2s2p)1P DWB2-RMPS (z = k0)

0

0.1

0.2

0.3

0.4

35 35.5

M = 0

ΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔ

Δ M = ±1 (×5)

σLM

/σL

( MA

X.)

He(2s2p)1P DWB1-RMPS (z = K)

0

1

35 35.5Ejected Electron Energy (eV)

M = 0

ΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔ

Δ M = ±1 (×5)He(2s2p)1P DWB2-RMPS (z = K)

0

1

35 35.5Ejected Electron Energy (eV)

Figure 2. Relative contribution of the partial cross sections σLM to the total cross section σL for the He (2s2p)1P state. The upper panelsshow the first- and second-order results in the collision frame, the lower panels are the corresponding results transformed into themomentum-transfer frame. The data have been normalized to the maximum value of σL. Note that the M = ±1 data in themomentum-transfer frame have been multiplied by a factor of 5 in order to make them visible.

M = 0

ΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔ

Δ

Δ

Δ

ΔΔΔΔΔΔΔΔΔ

ΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔ

Δ M = ±1

x x x x x x x x x x x x x x x x x x x x x x x x x x x xx

x

x

x

x

x x

x

x

x

x

x

xx

xx

xx

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

x M = ±2

σLM

/σL

( MA

X.)

He(2p2)1D DWB1-RMPS (z = k0)

0

0.1

0.2

0.3

35 35.5

M = 0

ΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔ

ΔΔΔΔΔ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

Δ

ΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔ

Δ M = ±1

x x x x x x x x x x x x x x x x x x x x x x xxx

xx

x

x

xx

x

x

x

x

xx x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

x M = ±2

He(2p2)1D DWB2-RMPS (z = k0)

0

0.1

0.2

35 35.5

M = 0

ΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔ

Δ M = ±1 (×100)

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

x M = ±2 (×10)

σLM

/σL

( MA

X.)

He(2p2)1D DWB1-RMPS (z = K)

0

1

35 35.5Ejected Electron Energy (eV)

M = 0

ΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔΔ

Δ M = ±1 (×100)

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

x M = ±2 (×10)

He(2p2)1D DWB2-RMPS (z = K)

0

1

35 35.5Ejected Electron Energy (eV)

Figure 3. As figure 2 but for the He (2p2)1D state. Note that the M = ±1 (±2) data in the momentum-transfer frame have been multipliedby a factor of 100 (10) in order to make them visible.

Table 1. Complex q-parameters, qLM = qRLM + iqI

LM , for heliumautoionizing levels, obtained from fits to the momentum-transferframe DWB1-RMPS and DWB2-RMPS calculations. Values havebeen rounded to the nearest 0.01.

DWB1-RMPS DWB2-RMPS

Level L, M qRLM |qI

LM | qRLM |qI

LM |

2s2 1S 0, 0 −1.10 0.04 −1.32 0.772s2p 1P 1, 0 −1.78 0.04 −1.97 1.29

1, ±1 −2.20 0.00 −4.03 0.452p2 1D 2, 0 −0.56 0.00 −0.67 1.38

2, ±1 −0.42 0.25 +5.40 0.522, ±2 −0.55 0.06 −0.07 0.00

to equation (6). To convert ejected-electron energies to levelpositions we used a first ionization potential of 24.587 eV.

Table 2. Positions and widths of helium autoionizing levels foundfrom fitting the distorted-wave RMPS calculations. The ejectedelectron energy is Eej, the corresponding energy above the groundstate is EL, where L is the orbital angular momentum of the leveland �L is the level width. The values have been rounded to thenearest 0.001 eV.

Level Eej (eV) EL (eV) �L (eV)

2s2 1S 33.255 57.842 0.1252s2p 1P 35.564 60.151 0.0382p2 1D 35.327 59.914 0.064

The fitted values for each level varied by less than 1 meVfor the total and partial cross sections. They are in excellentagreement with values found in the literature [12].

4

J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 225201 N L S Martin et al

qR10

= −1.78 |qI10

| = 0.04

σLM

(arb

.un

its)

He(2s2p)1P M = 0 DWB1-RMPS

0

1

35 35.5

qR10

= −1.97 |qI10

| = 1.29He(2s2p)1P M = 0 DWB2-RMPS

0

1

35 35.5

σLM

(arb

.un

its)

qR1±1

= −2.20 |qI1±1

| = 0He(2s2p)1P M = ±1 DWB1-RMPS

×5

0

0.02

0.04

0.06

0.08

35 35.5Ejected Electron Energy (eV)

qR1±1

= −4.03 |qI1±1

| = 0.45He(2s2p)1P M = ±1 DWB2-RMPS

0

0.02

0.04

0.06

0.08

35 35.5Ejected Electron Energy (eV)

Figure 4. As figure 1 but for the He (2s2p)1P state partial cross sections σ1M in the momentum-transfer frame. Note that the first-orderM = ±1 cross section has been multiplied by a factor of 5 in order to make it visible on the same ordinate scale as the second-ordercalculations. The cross sections were normalized relative to the maximum value of the second-order 1P calculation.

σLM

(arb

.un

its)

qR20

= −0.56 |qI20

| = 0He(2p2)1D M = 0 DWB1-RMPS

0

1

35 35.5

qR20

= −0.67 |qI20

| = 1.38He(2p2)1D M = 0 DWB2-RMPS

0

1

35 35.5

σLM

(arb

.un

its)

qR2±1

= −0.42 |qI2±1

| = 0.25He(2p2)1D M = ±1 DWB1-RMPS

×100

0

0.01

0.02

0.03

35 35.5

qR2±1

= +5.40 |qI2±1

| = 0.52He(2p2)1D M = ±1 DWB2-RMPS

0

0.01

0.02

0.03

35 35.5

σLM

(arb

.un

its)

qR2±2

= −0.55 |qI2±2

| = 0.06He(2p2)1D M = ±2 DWB1-RMPS

×5

0

0.01

0.02

35 35.5Ejected Electron Energy (eV)

qR2±2

= −0.07 |qI2±2

| = 0He(2p2)1D M = ±2 DWB2-RMPS

0

0.01

0.02

35 35.5Ejected Electron Energy (eV)

Figure 5. As figure 1 but for the He (2p2)1D state partial cross sections σ2M in the momentum-transfer frame. Note that the first-orderM = ±1 (±2) cross sections have been multiplied by a factor of 100 (5) in order to make them visible on the same ordinate scale as thesecond-order calculations. The cross sections were normalized relative to the maximum value of the second-order 1D calculation.

4. Discussion

Since the autoionizing levels of helium are doubly excitedstates, excitation from the ground state by charged particleimpact requires the promotion of both 1s electrons. In the

hybrid DWB1-RMPS model, this can be achieved as follows.The projectile interacts with the target once and ejects oneelectron. The outgoing electron, however, interacts with theresidual ion and can temporarily be captured by promoting theremaining electron, thus forming a doubly excited state. This

5

J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 225201 N L S Martin et al

is the classic mechanism for the corresponding photoionizationprocess. For charged-particle impact, it is sometimes labelled‘TS1’ [13]. In addition, the ‘shake-up’ mechanism (seebelow) can also occur. This is due to the fact that, in amulti-configuration expansion [14], the initial He(1s2)1S isknown to have considerable contributions from doubly excitedconfigurations such as (2s2) and (2p2). In the second-ordermodel, on the other hand, the projectile can interact with thetarget twice, thereby forming a doubly excited state. Thismechanism is often referred to as ‘TS2’ [13].

In an attempt to understand the results from thesesophisticated calculations, and in particular to see thedevelopment of a complex q-parameter in the transition fromfirst-order to second-order theory, we now use the independentparticle model with plane waves for the incident and scatteredelectron. As suggested by the results from the abovetransformations to the momentum-transfer frame, and also byrepeating the numerical RMPS calculations with plane wavesrather than distorted waves, we expect similar results at highincident projectile energies for this simpler model.

In the first-order Born approximation (PWB1) the q-parameter for a discrete autoionizing state φr, a final continuumstate ψr, an interaction V between the discrete and thecontinuum state, and an initial state φi may be written as

qr = 〈φr| eiK·r|φi〉π〈φr|V |ψr〉〈ψr| eiK·r|φi〉 . (10)

As pointed out by Kheifets [5], qr is real because the sametransition operator appears in both the numerator and thedenominator.

To find the overall phase of an amplitude to a state oforbital angular momentum L, with z component M, we use thestandard expansion of the exponential as a sum over Besselfunctions and spherical harmonics,

eik·r = 4π∑λ,μ

iλjλ(kr)Yλμ(r)Yλμ(k)∗, (11)

and take the quantization axis along K, so that only the termwith λ = L and μ = M = 0 is non-zero. The bound-statewavefunctions are therefore real and the overall phase is thengiven by iL. Note that any extra phase in a continuum state doesnot enter the value of qr because of |ψr〉〈ψr| in the denominatorof (10).

It is shown in appendix B that the overall phase of thesecond-order contribution to the plane wave Born amplitudefor excitation of an autoionizing resonance is iL+1; thus thefirst- and second-order contributions to the Born amplitudesare π/2 out of phase. There are then two possible cases toconsider: M = 0 and M �= 0. If M = 0 the direct ionizationamplitude is dominated by the first Born contribution; thesecond Born term only provides a small correction. It thenfollows that since the first Born contribution to qr is real, thesecond Born contribution is purely imaginary, precisely as isseen in our M = 0 DWB2-RMPS calculations at high incidentenergy. If M �= 0, all first Born amplitudes exactly vanishand the second-order direct ionization amplitude has the samephase as the autoionizing state amplitude. The second-ordercontribution to qr is therefore real. Again, this is precisely

what we see in our M �= 0 DWB2-RMPS calculations at highincident energy.

It remains to explain the relative magnitudes of thecontributions of the first- and second-order calculations tothe real and imaginary parts of qr. In the first-order theory,excitation of both 1s electrons is achieved by collisionalexcitation of one electron accompanied by shake-up of theother electron. This occurs because helium is a highlycorrelated system and in a configuration interaction description[14] the 1S ground state may be represented as 0.996|1s2〉 −0.062|2s2〉 + 0.062|2p2〉 + · · ·, which allows for the excitationof all three autoionizing levels. Thus, for example, theexcitation amplitude for 2s2p has a component which is theproduct of the dipole amplitude 2s → 2p and the factor−0.062〈2s|2s〉, where the overlap integral is slightly lessthan unity because the ground- and excited-state 2s electronsmove in different potentials. There are also other possiblemechanisms due to excited-state configuration interaction, andthe TS1 mechanism referred to above. The first-order Bornamplitude, for excitation to an autoionizing state 2�a2�b ofangular momentum L, is thus given by a sum of such smallterms.

As mentioned above, in the second-order model it isnot necessary to invoke the mixing coefficients. In fact, thedominant term is due to the double collision 1s2 → 1s2�a →2�a2�b, and the amplitude contains the product of two singleparticle matrix elements 1s → 2�a and 1s → 2�b.

5. Conclusions

We have analysed first- and second-order hybrid distorted-wave Born + R-matrix with pseudo-states calculations ofhelium autoionization in terms of Fano theory. We findthat the non-zero minima in the partial cross sections of thethree autoionizing levels are caused by effective qr parametersthat are complex quantities, with a real part due to the first-order contribution and an imaginary part due to the second-order contribution. Furthermore, the first- and second-ordercontributions to qr for M = 0 are of equal importance becausethe first-order amplitudes become of second-order importancevia the shake up mechanism, i.e., due to the small mixingcoefficients of 2s2 and 2p2 in the ground state.

It has long been known that second-order effects areimportant even at high incident energies for direct ionizationof helium [15]. Our calculations demonstrate that becausethey lead to complex q-parameters, second-order effects arealso extremely important for the doubly excited autoionizingstates.

Acknowledgments

This work was supported by the United States National ScienceFoundation under grants No. PHY-0855040 (NLSM) andPHY-0757755 (KB).

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J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 225201 N L S Martin et al

Appendix A

The PWB1 calculations in [1] yielded fitted values q0 = −15,q1 = −6.3 and q2 = −4.8, respectively. The calculationused plane waves for the incident, scattered, and ejectedelectron wavefunctions, and the standard Gram–Schmidtorthogonalization procedure was used to force the ejected-electron wavefunction to be orthogonal to the ground-stateHe 1s wavefunctions. The contribution of each partial wavefor L = 0, 1, 2 was then extracted from the resultant directionization scattering amplitude and used in Balashov et al ’s[16] model for autoionization.

The orthogonalization procedure is expected to affect theL = 0 state the most. In order to verify this, we carriedout new PWB1 calculations for direct ionization, in which theejected electron wavefunction was a distorted wave, with adistorting potential due to the helium ion. Using the resultantpartial-wave amplitudes in the Balashov et al ’s model yieldednew fitted values q0 = −3.0, q1 = −6.0 and q2 = −4.4. Thusq0 is smaller by a factor of 5, while q1 and q2 are virtuallyunaffected.

In fact the same value q0 = −3.0 results if the Gram–Schmidt orthogonalization is not carried out when a planewave is used for the ejected electron. Tweed and Langlois[17] examined the issue of orthogonalization in some detail.They found that orthogonality problems affect the theoreticalresults for the triple-differential cross section more seriouslyin the recoil region than in the binary region. Our findingsare in agreement with this. The value of qL determines themagnitude of the recoil peak, but it has little effect on thebinary peak.

Appendix B

The second-order plane-wave amplitude is [10, 18, 19]

f B2r ≈ 1

(2π)3/2

∑n

ikn

π

×∫ 〈φr| ei(K ′

n)·r|χn〉〈χn| ei(Kn)·r|φi〉K ′2

n K2n

dkn, (B.1)

where K ′n = kn − ksc and Kn = k0 − kn are the two

momentum transfers when going through an intermediate stateχn, and kn is the electron momentum after the first collision.This form of the second-order amplitude neglects a principalintegral under the assumption that all quantities vary slowlyin the neighbourhood of an autoionizing resonance—which isthe usual approximation in the theory of autoionization.

Using (11) introduces sums over λ′, μ′, λ, μ of productsof matrix elements that are integrated over all directions of theintermediate momentum kn. The integrand of each term is ofthe form

Yλ′μ′(K ′n)

∗Yλμ(Kn)∗

K ′2n K2

n

= (real part) × e−i(μ′φ′n+μφn)

K ′2n K2

n

, (B.2)

where φn(φ′n) is the azimuthal coordinate of the unit vector

Kn(K′n).

The scattering plane contains the vectors k0,ksc and K.In what follows, we define the scattering plane as the xz-planewith the z-axis along K. In general, kn lies out of this plane,and the six momentum vectors form an irregular tetrahedron,whose base is formed from k0,ksc and K and whose vertex atyn is given by the common point of the remaining three vectorskn,Kn and K ′

n. Reflection in the xz-plane gives another suchtetrahedron with the same base and the same magnitudes ofall vectors. Such a reflection corresponds to y → −y, andhence to φ → 2π − φ. Thus for every value of the integrand(B.2) there is a corresponding value with φn → −φn andφ′

n → −φ′n. Since the sum of these two values is real, the

integral over kn is real as well. The overall phase of eachterm with given values of λ′, μ′, λ and μ is therefore iλ

′+λ+1.Since the ground state of helium is 1S, conservation of angularmomentum requires that L = λ′ + λ and hence the overallphase of the second Born amplitude in this approximationis iL+1.

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