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Alignment and Pulse-Duration Effects in Two-Photon Double Ionization of H2 by

Femtosecond XUV Laser Pulses

Xiaoxu Guan1,∗ Klaus Bartschat1, Barry I. Schneider2, and Lars Koesterke31Department of Physics and Astronomy, Drake University, Des Moines, Iowa 50311, USA

2Applied and Computational Mathematics Division, Information Technology Laboratory,

National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA and3Texas Advanced Computer Center, University of Texas at Austin, Austin, Texas 78758, USA

(Dated: September 5, 2014)

We investigate the dependence of theoretical results for two-photon double ionization (DI) of H2

on the relative orientation of the linear laser polarization and the nuclear axis, as well as the lengthof the pulse, for a femtosecond laser in the direct DI domain, i.e., for central photon energies betweenabout 26 eV and 34 eV in the fixed-nuclei approximation at the equilibrium distance. In contrast tothe parallel geometry studied earlier [Guan et al., Phys. Rev. A 88, 043402 (2013)], in the vicinityof the photon energy of 30 eV the effect of the pulse duration is almost negligible for the case whenthe two axes are perpendicular to each other. This is due to a different symmetry (1Πu) accessed inthe latter case, compared to 1Σ+

uin the parallel geometry. When the photon energy approaches the

threshold of sequential DI, a sharp increase of the generalized cross section with increasing pulseduration is also observed in the perpendicular geometry, very similar to the case of the molecularaxis being oriented along the laser polarization direction. Our results differ from those of Colganet al. [J. Phys. B 41, 121002 (2008)] and Morales et al. [J. Phys. B 42, 134013 (2009)], but are inexcellent agreement with the generalized total cross sections of Simonsen et al. [Phys. Rev. A 85,063404 (2012)] over the entire domain of direct DI.

PACS numbers: 33.80.-b, 33.80.Wz, 31.15.A-

I. INTRODUCTION

Double ionization (DI) induced by photon andcharged-particle impact provides a unique way to un-veil the intriguing electron correlation in atoms [1–3],molecules [4–6], and surfaces [7]. With the advent ofnew light sources and the development of effective nu-merical approaches, we are now able to investigate funda-mental two-electron processes on an unprecedented sub-femtosecond time scale. Generally, the preferred emissiondirection of the second electron depends in a highly sen-sitive way on where the first electron is going and also onthe sharing of the available energy between the two.In a recent communication [8], we reported poten-

tial resonance effects on theoretical predictions for two-photon DI processes in molecular hydrogen (H2). Thosecalculations were performed in the fixed-nuclei approxi-mation (FNA) at the equilibrium distance of 1.4 bohr,with the linear laser polarization parallel to the inter-nuclear axis. Although the nuclei are not stationaryin nature, it is important to provide well-characterizedbenchmarks for this theoretically very challenging prob-lem, for which vastly different predictions exist in the lit-erature [9–11]. For the parallel geometry, we followed upon an earlier suggestion [6] that resonance effects mightbe responsible for these differences, in light of the factthat a number of doubly excited states with 1Σ+

u symme-try are located in the vicinity of 30 eV above the ground-

∗ Current Address: Department of Physics and Astronomy,

Louisiana State University, Baton Rouge, Louisiana 70803, USA

state potential energy curve.

An even more computationally challenging problem ispresented when the polarization of the electric field vec-tor is perpendicular to the internuclear axis. In this in-stance, the azimuthal symmetry is broken and additionalangular-momentum states must be accounted for in thecomputation. For this case, too, there are major differ-ences among theoretical predictions in the literature.The principal goals of the present work are twofold.

Firstly, while alignment effects in the perpendicular ge-ometry were investigated previously [6, 12], these studieswere limited to relatively short pulse durations and asingle photon energy. Hence we considered it essentialto extend that work by generating a set of benchmarkresults that can be used as a reference in future stud-ies, where the nuclear motion should be accounted for insome way. As mentioned above, this will be necessarybefore a realistic comparison with any experimental databecomes possible.

Secondly, we address the question of whether or notdoubly excited states might affect the two-photon DI ofH2 driven by a laser pulse with the polarization vectornormal to the molecular axis. This study should fur-ther narrow down possible sources for the large discrep-ancies found in the literature, due to either the physi-cal effects included or to numerical issues in the varioustreatments of the problem. For the geometry of interestin the present work, two earlier studies [9, 10] concen-trated on a single photon energy (30 eV).

As a function of photon energy, angle- and energy-integrated generalized total cross sections in the photon-energy regime from 26.4 eV to 32.7 eV were com-

2

puted in the recent work by Simonsen et al. [11], whoused B-spline bases and the FNA. In the present work,also within the framework of the FNA, we employ afull-dimensional ab initio approach to investigate theabove problem in the photon-energy regime from 26.5 eVthrough 34.0 eV. In addition to total cross sections, wealso present a variety of results for energy- and angle-resolved kinematics. While our emphasis is placed onthe perpendicular geometry, some results for the parallelgeometry will be presented in order to allow for a com-parison between the two cases.

II. NUMERICAL METHOD

As outlined in a number of previous papers [5, 6, 12],we formulate the problem in prolate spheroidal coordi-nates. The wave function is expanded in a finite-elementdiscrete-variable representation (FE-DVR). The solutionof the time-dependent Schrodinger equation (TDSE) ispropagated in time under the influence of the laser pulseusing the short iterative Lanczos method [13]. We ex-tract “cross sections” from the probability for double ion-ization by projecting onto two-center Coulomb functionswith the appropriate energies.Although, strictly speaking, the concept of a cross sec-

tion requires infinitely long “pulses” and weak intensities,it makes sufficient physical sense for the cases discussedin this work. The peak intensity of the laser pulses usedhere is 1013 W/cm2, which is indeed weak for the ex-treme ultraviolate (XUV) regime. Also, the pulses con-tain enough cycles for the details of the envelope functionto be of relatively small importance, provided the appro-priate effective interaction time is used. Projecting ontoa product of Coulomb waves is valid as long as one propa-gates for sufficiently long times and large distances. Com-putationally, this can be tested by examining the predic-tions of the projection at different times after the laserhas been switched off and monitoring the convergenceof the results (see, for example, Refs. [5, 14, 15]). Theresults presented below were found to be stable againstreasonable variations in the numerical details, such asthe number of and distribution of the DVR functions,the size of the box, the number of partial waves retainedin the expansion of the wave function, and the time afterthe pulse at which the information is extracted.

All the calculations presented below employed a laserpulse with a well-defined integer number of optical cy-cles (o.c.). A sin2 envelope function was used from theinitial time Ti = 0 to the final time Tf , when the laseris turned off. Hence, in contrast to a convention oftenused in experiments, the pulse length is not defined bythe full width at half maximum of the temporal intensity.Furthermore, the carrier-envelope phase was set to zero.

(a)(b)

FIG. 1. (Color online) Angle-integrated two-photon general-ized cross section for DI of the H2 molecule. In panel (a),the molecular axis is oriented perpendicular to the linear po-larization vector (represented by a double-headed arrow) ofthe laser pulse. Our present results were obtained for pulselengths of 10, 20, and 30 optical cycles. Also shown are the abinitio and the model results of Simonsen et al. [11], as well asthe cross sections of Colgan et al. [9] and Morales et al. [10]at 30 eV. Panel (b) shows a comparison between our ab ini-

tio results obtained in the perpendicular (⊥) and parallel (‖)geometries, with the corresponding predictions from a simplemodel suggested by Simonsen et al. [11].

III. RESULTS AND DISCUSSION

The angle-integrated generalized cross sections for DIof the H2 molecule for various cases are displayed inFig. 1. At hν = 30 eV, all previously published time-dependent results [6, 9, 11] employed an interaction timebetween 1.4-2.1 femtoseconds (fs), i.e., 10-15 o.c. For theperpendicular geometry shown in Fig. 1(a), the depen-dence of our results on the pulse length is small for centralphoton energies up to about 33 eV, as shown by examin-ing predictions obtained with 10, 20, and 30 o.c. Whenapproaching the threshold for sequential DI, on the otherhand, the rapid increase of the generalized cross sectionis not at all captured by the 10 o.c. pulse. Our results for20 o.c. are in excellent agreement with the ab initio cal-culations by Simonsen et al. [11], which were performedusing an entirely independent method, but with similarlaser parameters (15 o.c. as the pulse duration). The sim-plified model of Ref. [11] also does reasonably well, butover most of the energy range it predicts slightly lowergeneralized cross sections than the direct solution of theTDSE. The time-dependent close-coupling (TDCC) re-sults of Colgan et al. [9] and the exterior complex scaling(ECS) predictions of Morales et al. [10] for a photon en-ergy of 30 eV are about a factor of two larger than ourTDSE results at this energy.

As discussed earlier in some detail [8], major discrepan-cies between the TDCC and ECS predictions, and sub-sequently with our own calculations [6], also occurredfor the parallel geometry. There it was conceivable thatresonance effects, which were investigated by changingthe pulse length, play a role. As discussed above, how-ever, this is an unlikely explanation for the discrepan-cies in the perpendicular case. There are no doublyexcited states with 1Πu symmetry located in the vicin-ity of 30 eV for the equilibrium internuclear separation,and hence in this case the cross section at 30-eV pho-ton energy is well defined. As a consequence, the lackof dependence on the pulse length seen in Fig. 1(a) isunderstandable. In fact, our results at this photon en-ergy are well converged with respect to the pulse length,to the value of 2.40× 10−52 cm4 s. Increasing the pulseduration from 20 cycles to 30 cycles changes the resultby less than 0.5%. The total cross section obtained in

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Ref. [11] is 2.349× 10−52 cm4 s at 29.9 eV, in excellentagreement with our number. The TDCC and ECS resultsat 30.0 eV, on the other hand, are 4.7× 10−52 cm4 s [9]and 5.76× 10−52cm4 s [10], respectively.

At the internuclear distance of 1.4 bohr, the two lowestdoubly excited states of the 1Πu symmetry are located,respectively, about 33.1 eV and 33.8 eV [16] above theground state. By properly choosing the photon energy,these states are, in principle, accessible through single-photon absorption if the electric field has a componentnormal to the molecular axis. These autoionizing stateslie, however, very close to the threshold for sequential DIand, most importantly, their energy widths are extremelynarrow. For instance, the width of the lowest doublyexcited 1Πu state is only about 8 × 10−3 eV [16], cor-responding to a long lifetime of 82 fs. The pulse dura-tions set in the present work, therefore, would not allowus to resolve possible resonance effects related to theseautoionizing states on the generalized cross sections fortwo-photon DI. Furthermore, the sharp rise of the crosssection in the vicinity of 33-eV photon energy would makeit very difficult to isolate the effects of the doubly excitedstates.

Figure 1(b) shows a comparison of the results obtainedin the perpendicular and parallel geometries. In Ref. [11],the total cross section obtained with the simple modelwas shown up to a photon energy of 32.6 eV. Here we ex-tend those results to 34.8 eV, i.e., just below the thresh-old for sequential DI. For the parallel case, the resonancestructure around 30 eV was discussed in detail by Guan et

al. [8]. Here we only note that the generalized cross sec-tion is overall about an order of magnitude larger in theperpendicular compared to the parallel geometry, thusindicating a very strong alignment effect. The simplemodel of Simonsen et al. [11] yields very reasonable re-sults for both cases, except that it cannot reproduce theresonance effect. A further difference compared to thetime-dependent calculations is the rapid rise of the crosssection with photon energy as one approaches the thresh-old for sequential DI. Effects due to the energy resolutionof the finite length pulses in ab initio solutions of theTDSE have been seen before and are most likely alsothe cause of the sharp increase observed here. In otherwords, when the photon energy approaches the thresholdof sequential ionization, the sharp increase in the crosssection is a general feature, which is independent of therelative orientation of the molecular axis and the laserpolarization vector. The absolute values of the cross sec-tion, however, may differ significantly if this orientationis changed.

In Fig. 2, we display the angular distribution for a pho-ton energy of 30.0 eV. For equal-energy sharing, when thefixed electron is observed along the molecular axis, thetwo electrons are predominantly ejected back-to-back.Comparing the predictions for the parallel geometry inrows (a,b) and the perpendicular geometry in rows (c,d),we see strong similarities in the two cases, especially whenone of the electrons is observed along (θ1 = 0◦, first col-

umn) or near (θ1 = 30◦, second column) the laser po-larization axis. Consequently, alignment affects manifestthemselves in the magnitude (each panel is individuallyscaled, see below) but not so much in the shape of theangular distribution. Both alignment and pulse-lengtheffects are clearly visible for θ1 = 60◦ (third column) andθ1 = 90◦ (fourth column). We will see, however, thatthe magnitude of the angular distributions drops rapidlywhen increasing θ1 from 0◦ towards 90◦. This is anotherexample of the common experience that predictions forsmall cross sections are more sensitive to details in thephysics and the numerical model than those for relativelylarge cross sections.Figures 3 and 4 show the corresponding results for

the perpendicular case only, but for incident energies of26.5 eV and 34.0 eV, respectively. We note that pulse-length effects are only significant for large values of θ1,where the magnitudes of the angular distributions areagain relatively small. The energy dependence in theshape of the angular distribution is also weak, once againmostly visible at θ1 = 60◦ and θ1 = 90◦.

The generalized triple-differential cross section(TDCS) in the coplanar geometry, where the internu-clear axis, the laser polarization vector, and the linearmomenta of the two ejected electrons all lie in thesame plane, is shown in Figs. 5 (parallel case) and 6(perpendicular case) for the central photon energy of30.0 eV and symmetric energy sharing. For a numberof detection angles θ1 of one electron, measured relativeto the direction of the linear laser polarization, thedetection angle θ2 of the other electron is varied. Ourresults are again compared with earlier predictions byColgan et al. [9] and Morales et al. [10].

For the parallel case (c.f. Fig. 5), we note qualitativeagreement in the angular dependence of the TDCS pre-dicted by the various methods, except perhaps for theoverall smallest values occurring at θ1 = 90◦. Anotherexception is the increase in the TDCC results in the re-gion where the two active electrons are traveling in aboutthe same direction. This seems improbable for equalsharing of the excess energy and is most likely a numericalartifact in the early calculation [17]. We also see a sig-nificant effect of the pulse length, with the 40-cycle pulseproducing larger values near the maxima of the angu-lar distribution. This may, at least qualitatively, explainthe differences between our results and those from theECS calculation, which was effectively performed withan infinitely long pulse, i.e., no width in the frequencydistribution.For the perpendicular case (c.f. Fig. 6), on the other

hand, the agreement with the TDCC predictions [9] isquite satisfactory, whereas serious discrepancies, againmostly in the magnitude, are seen with the ECS re-sults [10]. Given the very small dependence on the pulselength for this case, it seems highly unlikely that thesedifferences can be explained by the fixed frequency usedin the ECS model.

Comparing the results for various values of θ1, we note

4

FIG. 2. (Color online) Angular distributions of the electrons ejected from the H2 molecule at equal energy sharing(E1 = E2 = 4.3 eV) for the central photon energy of 30.0 eV in the FNA at the equilibrium distance of 1.4 bohr. The laserpolarization axis is represented by a double-headed arrow. Row (a): 10-cycle pulse in the parallel geometry. Row (b): 40-cyclepulse in the parallel geometry. Row (c): 10-cycle pulse in the perpendicular geometry. Row (d): 30-cycle pulse in the perpen-dicular geometry. One electron (k1) is observed, respectively, at fixed angles (θ1) of 0◦, 30◦, 60◦, and 90◦ with respect to thelaser polarization axis. Each panel has been individually scaled.

FIG. 3. (Color online) Angular distributions of the electrons ejected from the H2 molecule at equal energy sharing(E1 = E2 = 2.5 eV) for the central photon energy of 26.5 eV in the FNA at the equilibrium distance of 1.4 bohr. Row (a):10-cycle pulse in the perpendicular geometry. Row (b): 30-cycle pulse in the perpendicular geometry.

a strong dependence of the predicted TDCS on this an-gle. For the parallel case (c.f. Fig. 5), this dependenceis seen in both the angular dependence and the magni-tude, with the latter dropping by about a factor of four inthe maxima of the TDCS, with these maxima also beingshifted to different θ2 and even split in a variety of ways.In contrast, the maximum near θ2 = 180◦ in the per-pendicular case (c.f. Fig. 6) remains more or less at thisangle, but its magnitude drops by nearly a factor of 50when going from θ1 = 0◦ to θ1 = 90◦. For the largervalues of θ1, a second maximum develops. The patternsdiscussed above, except for the magnitude change, canalso be seen as a cut through the appropriate plane inFig. 2.

The significant drop in the magnitude of the TDCS forthe perpendicular geometry, as well as the overall energydependence, is further illustrated in Fig. 7 for 26.5 eV and34.0 eV, respectively. Once again the smallest cross sec-tions exhibit the most detailed features, while the domi-nant cross section (θ1 = 0◦) simply shows a peak aroundθ2 = 180◦. Again, these patterns in the angular distribu-tion can also be seen in Figs. 3 and 4.

Even though the angle- and energy-integrated totalgeneralized cross section is much larger for 34.0 eV thanfor 26.5 eV (c.f. Fig. 1), we note that it is the other wayaround for the maximum in the coplanar TDCS. Thisis a consequence of (i) the increased excess energy avail-able for the higher photon energies, and (ii) the 34.0-eVcase being dominated by asymmetric sharing of the ex-cess energy, while symmetric energy sharing prevails atthe lower excess energies. Figures 6 and 7 show that themagnitude of the TDCS in the dominant emission mode(θ1 = 0◦) at equal energy sharing decreases with increas-ing photon energy.

The latter dependence on the sharing of the excess en-ergy is demonstrated in Fig. 8, where we chose a polar-plot representation of the results for the coplanar geome-try. For 30 eV (see the panels in the left column of Fig. 8),we only notice a minor dependence of the TDCS predic-tions on the relative fraction of sharing the excess energy,with the symmetric (50%:50%) case overall representingthe least likely scenario. Once again, the largest effects

due to varying the relative portions of the excess energyare seen for θ1 = 90◦. The TDCS for this case, however,is very small compared to that for the dominant emissionmode along the direction of the laser polarization. Wenote that Ivanov and Kheifets [18] also considered a caseof unequal energy sharing, E1 : E2 = 63% : 37%, for thephoton energy of 30 eV in the perpendicular geometry.They did not notice significant changes in the TDCSseither, in agreement with our observation here.

As suggested above, asymmetric energy sharing be-comes increasingly important when the threshold for se-quential double ionization is approached. This is exhib-ited in the panels in the right column of Fig. 8. Also,the scale factors show that the drop in the magnitude ofthe TDCS when going from θ1 = 0◦ to 90◦ is more pro-nounced for 34.0 eV than for 30.0 eV. While the patternslook most interesting for the larger values of θ1 and couldbe used to test theoretical predictions most thoroughly,it seems unlikely that they would be experimentally ob-servable — independent of the validity (or lack thereof)of the fixed-nuclei approximation.

IV. CONCLUSIONS AND OUTLOOK

We have extended our previous investigation of align-ment effects [5], this time in combination with a potentialinfluence of the pulse length, on theoretical predictionsfor the two-photon DI of H2 at the level of the fixed-nuclei approximation. Due to the different symmetryof the one-photon intermediate state in the parallel andperpendicular geometries, resonance and pulse-length ef-fects, which were clearly identified in the former case [8],are virtually non-existent for a photon energy near 30 eVwhen the linear laser polarization vector and the inter-nuclear axis are perpendicular to each other. The dis-crepancies between our results, those of Colgan et al. [9],and those of Morales et al. [10] could thus not be re-solved in this case. Given the very challenging natureof the present problem, however, it is satisfying to seethat our calculations support the results presented in themost recent work by Simonsen et al. [11], which was per-

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FIG. 4. (Color online) Angular distributions of the electrons ejected from the H2 molecule at equal energy sharing(E1 = E2 = 6.3 eV) for the central photon energy of 34.0 eV in the FNA at the equilibrium distance of 1.4 bohr. Row (a):10-cycle pulse in the perpendicular geometry. Row (b): 30-cycle pulse in the perpendicular geometry.

FIG. 5. (Color online) TDCSin the coplanar geometry forelectrons ejected from the H2

molecule at equal energy sharing(E1 = E2 = 4.3 eV) in the parallelcase for the central photon energyof 30.0 eV. The detection angle θ1of one electron is held fixed at0◦ (a), 30◦ (b), 60◦ (c), or 90◦ (d),while the detection angle θ2 is var-ied. The results from the presentcalculation for 10 o.c. and 40 o.c.are compared with predictions fromearlier TDCC [9] and ECS [10]calculations. Note the scale factorsin the legend of panel (a), whichapply to all panels of this figure.

formed using an entirely independent method, but withvery similar laser parameters.

In order to provide a more extensive dataset than iscurrently available in the literature, we presented detailedenergy- and angle-resolved results for three different en-ergies, chosen to be (i) close to the non-sequential double-ionization threshold as well as (ii) about half-way towardsand finally (iii) close to the sequential double-ionizationthreshold. Furthermore, we showed results obtained forasymmetric energy sharing, which is the dominant modefor photon energies approaching the sequential DI thresh-old. This pattern in the energy distribution is also thereason for the sharp rise of the total cross sections in thisenergy regime. The present work in the perpendicular ge-ometry hence complements and further supports our ear-lier conclusions, then drawn for the parallel case [8], re-garding the general energy dependence of the total crosssection near the threshold for sequential DI.

Our results are available in electronic form upon re-quest. In the future, we hope to relax the restrictionsin the FNA and account for the nuclear motion at least

in some approximate way. With presently available com-putational resources, however, this is not a short-termproject. In the meantime, it would be highly desirableto investigate the sources of the remaining discrepanciesamong the predictions of various theoretical groups byjointly studying a number of well-defined cases with ex-actly the same laser parameters and exactly the samekinematics, for both symmetric and asymmetric energysharing as well as in-plane and out-of-plane geometries.

ACKNOWLEDGMENTS

We thank Drs. J. Colgan, M. Førre, and F. Moralesfor sending results in electronic form. This work wassupported by the United States National Science Foun-dation under grants No. PHY-1068140 and No. PHY-1430245 (XG and KB) and supercomputer resources onKraken at Oak Ridge National Laboratory, Stampedeat the Texas Advanced Computer Center, and Gordonat the San Diego Supercomputing Center through theXSEDE Allocation No. PHY-090031.

[1] B. Bergues, M. Kubel, N. G. Johnson, B. Fischer,N. Camus, K. J. Betsch, O. Herrwerth, A. Senftleben,A. M. Sayler, T. Rathje, T. Pfeifer, I. Ben-Itzhak,R. R. Jones, G. G. Paulus, F. Krausz, R. Moshammer,J. Ullrich, M. F. Kling, Nat. Commun. 3, 813 (2012).

[2] X. Guan, O. Zatsarinny, C. J. Noble, K. Bartschat andB. I. Schneider, J. Phys. B 42, 134015 (2009).

[3] X. Guan, K. Bartschat, and B. I. Schneider, Phys. Rev. A77, 043421, (2008).

[4] H. Sann, T. Jahnke, T. Havermeier, K. Kreidi, C. Stuck,M. Meckel, M. S. Schoffler, N. Neumann, R. Wal-

lauer, S. Voss, A. Czasch, O. Jagutzki, Th. Weber,H. Schmidt-Bocking, S. Miyabe, D. J. Haxton, A. E. Orel,T. N. Rescigno, and R. Dorner, Phys. Rev. Lett. 106,133001 (2011).

[5] X. Guan, K. Bartschat, and B. I. Schneider, Phys. Rev.A 83, 043403 (2011).

[6] X. Guan, K. Bartschat, and B. I. Schneider, Phys. Rev. A82, 041404(R) (2010).

[7] L. Avaldi and G. Stefani, in Fragmentation Processes:

Topics in Atomic and Molecular Physics, edited byC. T. Whelan, (Cambridge University Press, Cambridge,

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FIG. 6. (Color online) TDCSin the coplanar geometry forelectrons ejected from the H2

molecule at equal energy sharing(E1 = E2 = 4.3 eV) in the perpen-dicular case for the central photonenergy of 30.0 eV. The results fromthe present calculation for 10 o.c.and 30 o.c. are compared withpredictions from earlier TDCC [9]and ECS [10] calculations. No scalefactors were applied here.

FIG. 7. (Color online) Same as Fig. 6 for central photonenergies of 26.5 eV (a) and 34.0 eV (b), respectively. Theresults are from the present calculation for 30 o.c. No TDCCor ECS results are available for these cases. Note the scalefactors for θ1 = 60◦ and 90◦.

FIG. 8. (Color online) Coplanar TDCS for central photonenergies of 30.0 eV (left column, panels (a)−(d)) and 34.0 eV(right column, panels (e)−(h)). The results are from thepresent calculation for 30 o.c. in the perpendicular case, forfixed angles θ1 = 0◦, 30◦, 60◦, and 90◦ for different por-tions of the excess energy shared between the two outgo-ing electrons. The radius of the outer circle corresponds to3× 10−54 cm4 s/(sr2 eV). Note the scale factors in the panelsfor θ1 6= 0◦.

2012), Chap. 1.[8] X. Guan, K. Bartschat, B. I. Schneider, and L. Koesterke,

Phys. Rev. A 88, 043402 (2013).

[9] J. Colgan, M. S. Pindzola, and F. Robicheaux, J. Phys. B41, 121002 (2008).

[10] F. Morales, F. Martın, D. A. Horner, T. N. Rescigno, andC. W. McCurdy, J. Phys. B 42, 134013 (2009). The totalcross section was not published in this paper, but wasobtained via private communication from Dr. Morales.

[11] A. S. Simonsen, S. A. Sørngard, R. Nepstad, andM. Førre, Phys. Rev. A 85, 063404 (2012).

[12] X. Guan, K. Bartschat, and B. I. Schneider, Phys. Rev. A84, 033403 (2011).

[13] T. J. Park and J. C. Light, J. Chem. Phys. 85, 5870(1986).

[14] L. B. Madsen, L. A. A. Nikolopoulos, T. K. Kjeldsen,and J. Fernandez, Phys. Rev. A 76, 063407 (2007).

[15] J. Feist, S. Nagele, R. Pazourek, E. Persson, B. I. Schnei-der, L. A. Collins, and J. Burgdorfer, Phys. Rev. A 77,043420 (2008).

[16] I. Sanchez and F. Martın, J. Chem. Phys. 106, 7720(1997).

[17] J. Colgan, private communication (2014).[18] I. A. Ivanov and A. S. Kheifets, Phys. Rev. A 87, 023414

(2013).

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55cm

4s/sr2eV

)

360300240180120600

70

60

50

40

30

20

10

0

(b)

θ1 = 30◦

θ2 (deg)

TDCS(10−

55cm

4s/sr2eV

)

360300240180120600

40

30

20

10

0

(c)

θ1 = 60◦

θ2 (deg)

TDCS(10−

55cm

4s/sr2eV

)

360300240180120600

10

8

6

4

2

0

(d)

θ1 = 90◦

θ2 (deg)

TDCS(10−

55cm

4s/sr2eV

)

360300240180120600

1.0

0.8

0.6

0.4

0.2

0.0

θ1 = 90◦θ1 = 60◦θ1 = 30◦θ1 = 0◦

×8

×4

(a)

θ2 (deg)

TDCS(10−

55cm

4s/sr2eV

)

360300240180120600

60

50

40

30

20

10

0

×8

×4

(b)

θ2 (deg)

TDCS(10−

55cm

4s/sr2eV

)

360300240180120600

16

12

8

4

0

1

(a) θ1 = 0◦

k1

1

(e) θ1 = 0◦

k1

1

E1 : E2 = 50% : 50%

E1 : E2 = 40% : 60%

E1 : E2 = 30% : 70%

E1 : E2 = 20% : 80%

(b) θ1 = 30◦ (×1.5)

k1

1

(f) θ1 = 30◦ (×1.5)

k1

1

(c) θ1 = 60◦ (×5.2)

k1

1

(g) θ1 = 60◦ (×6.2)

k1

1

(d) θ1 = 90◦ (×18.5)

k1

1

(h) θ1 = 90◦ (×38.0)

k1

1