Dissociative ionization of H 2 by 400 ev circularly polarized photons

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1 Dissociative ionization of H 2 by 400 ev circularly polarized photons Vladislav V. Serov 1 and A. S. Kheifets 2 1 Department of Theoretical Physics, Saratov State University, 83 Astrakhanskaya, Saratov , Russia 2 Research School of Physical Sciences, The Australian National University, Canberra ACT 0200, Australia Abstract. Calculations of single photon one-electron ionization-dissociation of H 2 are performed ab initio with accurate quantum mechanical description of the nuclear motion. The clear interference pattern is observed in the photoelectron angular distribution detected in the molecular frame. An effect of this interference on the dissociation process is found. PACS numbers: b, Ca Submitted to: J. Phys. B: At. Mol. Phys.

2 Dissociative ionization of H 2 by 400 ev circularly polarized photons 2 1. Introduction Dissociative photoionization of the H 2 molecule has been exploited recently as a convenient tool to study the fundamental concepts of quantum mechanics such as entanglement, decoherence and delocalization. Coincident momentum imaging techniques make it possible to study photoelectron angular distribution (PAD) in the molecular frame and to identify the two-centre interference effects. Single-photon double ionization of H 2 is naturally dissociative as the bare nuclei explode by the Coulomb repulsion and the molecule becomes fully fragmented. The two photoelectrons in full fragmentation of H 2 form an entangled pair. The correlated momenta of the entangled electron pair exhibit quantum interference while interference fringes observed in the angular distribution of a single electron are lost through its Coulomb interaction with a second electron [1, 2]. Similar effects can be detected in dissociative photoionization of heavier diatomic molecules via Auger decay of the inner vacancies. Signature of entanglement and delocalization was observed in the interference between emitted core electrons from CO and N 2 molecules [3]. Single photoionization of H 2 can become dissociative as well via a small overlap of the Frank-Condon region of the ground state with the dissociative continuum of the H + 2 ion. This opens an avenue of detecting the photoelectron angular distribution (PAD) in the molecular frame and to observe localization of the remaining electron on either of the protons. Such a localization is a result of forming non-gerade electron states of the H + 2 ion. It can proceed via mixture of direct and autoionizing ionization channels [4, 5]. A related process of breaking up of the H + 2 ion is also a useful tool to study electron delocalization. In this process, the electron should be localized, intuitively, on just one of the protons when they are separated by the distance R much larger than the atomic Bohr orbit. Quantum delocalization of the electron over two protons can be revealed by a proposed pump-probe experiment utilizing ultra-short laser pulses [6]. The persistence of two-center interferences for large delay between the pump and the probe times would show that even at large internuclear distance the electron can be well delocalized over two separated nuclei. In the present work, we consider ionization-dissociation process ω+h 2 e +H+p by high-energy circularly polarized photons ω = 400 ev. Our prime interest is to obtain the PAD in the molecular frame. Particular emphasis in our work is placed on the full account of the nuclear motion. The motivation of the present study is the following. Quantum interference interpretation of the double photoionization experiments [1, 2] was questioned in Ref. [7]. They argued that the energy of the circularly polarized photons between 130 to 240 ev, employed in these experiments, was not sufficient. Much larger energy, close to 400 ev, was needed to observe the interference effects unequivocally. At such photon energy, the cross-section of double photoionization of H 2 is small which makes it difficult to measure experimentally. In the meantime, single ionization-dissociation of H 2 by high-energy circularly polarized photons can reveal

3 Dissociative ionization of H 2 by 400 ev circularly polarized photons 3 the same interference and delocalization effects. Such experiments are being planned [8] at the PETRA III synchrotron radiation source [9]. Particular emphasis in these experiments will be placed on the role of the electron recoil on the nuclear motion and a possible breaking of the p H symmetry in the PAD. To elucidate these effects, a full account for the nuclear motion has to be included in the theoretical model. Such a model is developed in the present work. A related study of the p H symmetry breaking in ionization-dissociation of H 2 close to the threshold by low energy photons is reported in a separate publication [10]. We use the adiabatic approximation, i.e. consider the initial and final state wave functions as the respective products of the nuclear and electron parts. This approximation is particularly justified for large photon (and final electron) energies because the ionization process itself is very fast in comparison with a characteristic time of the nuclear motion. No further approximations are made in the theoretical model and hence we refer to the present calculations as ab initio. The electron part of the wave function is obtained by the Prolate Spheroidal coordinates implementation of the Exterior Complex Scaling method (PSECS) as described in our earlier work [11]. 2. Theory The amplitude of ionization-dissociation of H 2 can be written as a matrix element between the initial and final molecular states (see, e.g. Eq. (1) of [12]): F n (k,k) = χ ( ) nk (R) f n(k;r) χ 0 (R). (1) Here the index n refers to a dissociative state of the H + 2 molecular ion, k is the electron momentum and K is the relative momentum of the protons. The electron part of the amplitude is expressed as a dipole matrix element f n (k;r) = Φ ( ) nk (r 1,r 2 ;R) e d Φ 0 (r 1,r 2 ;R) (2) between the initial electron state Φ 0 (r 1,r 2 ;R) and the final state Φ ( ) nk (r 1,r 2 ;R) corresponding to a fixed inter-nuclear distance R and the molecular axis direction n R = R/R. The dipole operator in the length gauge is a scalar product of the dipole momentum d = r 1 +r 2 and the polarization vector of light e. The information about the nuclear recoil momentum is contained in the dependence of the phase of the amplitude f n on R. We use the adiabatic approximation, i.e. consider the initial and final electronproton wave functions as the respective products i = Φ 0 (r 1,r 2 ;R)χ 0 (R); (3) f = Φ ( ) nk (r 1,r 2 ;R)χ ( ) nk (R). We consider H 2 initially in the para-hydrogen ground rotational and vibrational state χ 0 (R) = Y 00 (n R ) χ 00(R) R. (4)

4 Dissociative ionization of H 2 by 400 ev circularly polarized photons 4 The radial function χ 00 (R) is the lowest bound state solution of the equation [ 1 ] 2 2µ R + U 2 H 2 (R) E 00 χ 00 (R) = 0, (5) where µ = m p /2 is the reduced proton mass, U H2 (R) is the effective internuclear potential for H 2 in its ground electronic state. The final dissociative molecular state, corresponding to the H + 2 ion in the n-th electronic state, can be presented as χ ( ) nk (R) = 4π (2π) 3/2 LM YLM(n K )Y LM (n R )i L e iδ χ L nkl(r). (6) R The radial function χ nkl (R) is the solution of the equation [ 1 ] 2 2µ R + U n(r) E 2 K χ nkl (R) = 0, (7) where U n (R) is the effective internuclear potential for H + 2 in the n-th electronic state, E K = E R + U n ( ), E R = K 2 /2µ is the kinetic energy release. Asymptotics of the continuum wave function is χ nkl (R ) = 1 K sin (KR Lπ/2 + δ L), (8) where δ L (K) is the scattering phase. Both the final and the initial vibrational wave functions are calculated numerically. These functions are plotted in Figure 1. (nk (nk Figure 1. The wave function χ 00 (R) of the ground vibrational state of H 2 (black line) and the wave function χ nkl (R) of the continuous vibrational state of H + 2 (1σ g) for L = 0 and E R = 1 ev (red line). For circular polarization in the xoz plane, the polarization vector is e = e z + ie x. In this case, the triple differential cross section (TDCS), resolved with respect to the photoelectron and proton momenta and the KER, is d 3 σ = µk d3 σ = 2π2 ω µk F n (k,k) 2. (9) de R dω k dω K dkdω k c To see clearly the effect of the nuclear motion, we also performed simplified calculations in the Born-Oppenheimer approximation. In this approximation, σ (3) = 2π2 ω µk c du n dr [R(E R)] 1 f n (k;r(e R )) 2, (10)

5 Dissociative ionization of H 2 by 400 ev circularly polarized photons 5 where R(E R ) is the solution of the turning point equation U n (R) = E R + U n ( ). (11) For testing purposes, we also peformed calculations with a fixed molecular axis direction. This is implemented by replacing the sum over the angular momenta L in Equation (6) by a single term L F. This reduces the function (6) to χ ( ) nk (R) = 4π f e iδ χ L nklf (R) f δ(n (2π) 3/2iL R n K ). (12) R We used L f = 2 in examples below, because for given electron ejection energy this component in Equation (6) is dominant Electronic amplitudes The R-dependent single photoionization amplitudes f n (k;r) were calculated using an ab initio PSECS method [11]. The energy of ejected electron was fixed at 382 ev, corresponding to 400 ev photon energy, as in the experimental set up proposed by [8], minus the threshold energy of ionization-dissociation (18.1 ev). The parameters of angular basis (see [11]) were chosen as N l = 7, N m = 9. For the quasi-radial coordinates, the 4th-order B-splines were used on a uniform grid with the step r = 0.2 and the size r max = 20 a.u. ( r = R(ξ 1)/2, ξ is the quasi-radial spheroidal coordinate), exterior complex scaling starts from r ECS = 18 a.u. with an angle θ ECS = 45. The H 2 ground state wave function Φ 0 (r 1,r 2 ;R) was calculated numerically on the same basis with the same grid parameters. The calculations were performed for R from 0.4 a.u. to 2.8 a.u. with the step 0.4 a.u. For integration in (1), the amplitude was interpolated using its Fourier expansion over R. For testing purposes, the amplitudes were also calculated using an approximate method [13]. This method is based on a separable form of the final state electronic wave function Φ ( ) nk (r 1,r 2 ;R) = 1 [ ] ϕ ( ) k (r 1)ϕ n (r 2 ) + ϕ n (r 1 )ϕ ( ) k (r 2), (13) 2 where the wave functions ϕ n (r) and ϕ ( ) k (r) describe the bound and continuum states of the H + 2 ion, respectively. We performed calculations for two different effective nuclear charges: Z + = 1 (equal to the net charge of the ion H + 2 ) and Z + = 2 (equal to the charge which the ejecting electron feels near the nuclei). 3. Results In the Born-Oppenheimer (BO) approximation, the nuclear recoil momentum is neglected. So, comparison of the BO results with analogous results with correct quantum description of the nuclear motion (NM) should demonstrate the influence of the recoil momentum on the dissociation process. Comparison of NM results with results of nuclear motion with no rotation (NMnR) should demonstrate the effect of the rotation of the molecular axis induced by recoil angular momentum.

6 Dissociative ionization of H 2 by 400 ev circularly polarized photons 6 (a) RTs ) sd R R R R r/2r (b) RTs θ sd R R R R r/θr RTD RTD (c) RTs ) sd R R R R r/gr (d) RC2 d 2S R R R R θr RTD RCS Figure 2. The TDCS σ (3) as a function of the angle θ er between electron and proton ejection, in NM (9) (solid lines), BO (10) (dashed lines), NMnR (12) (dot-dashed lines) and two-center interference formula (14) (dotted lines): (a) E R = 0.1 ev; (b) E R = 0.2 ev; (c) E R = 0.5 ev. In case of circular polarization, due to the rotational symmetry in the xoz plane, the TDCS depends on the proton ejection angle θ R and the electron ejection angle θ e only via their difference θ er = θ e θ R. In Figure 2, the TDCS is plotted as a function of θ er for several different values of KER. We also plot the curves given by a simple two-center interference formula (IF) [1] σ (3) cos 2 [kr(e R ) cos(θ er )/2], (14) where k 5.3 a.u. at the photon energy of 400 ev. In Figure 2, the IF curves were rescaled to the best fit of the BO peaks near θ e = 90. The IF formula describes qualitatively all the main features of the TDCS. It confirms the interference origin of TDCS angular patterns in the present case. The close shapes of angular distributions for different KER is the sequence of a small difference between the values of dimensionless interference parameter kr(e R )/2. For the KER E R = 0.1 ev, the internuclear distance R(E R ) = 1.1 a.u., it is close to R(E R ) = a.u. for the KER E R = 1 ev. As is seen from Figure 2, the NM TDCS is larger then BO TDCS, especially in the peaks near θ e = 0 and 180 and for small KER, while in the peaks near θ e = 90

7 Dissociative ionization of H 2 by 400 ev circularly polarized photons 7 C2θ 0 θe C R C R C R C R C2e Figure 3. The TDCS σ (3) as the function of KER for the ejection angles close to the PAD maxima: NM (solid line) and BO (dashed line) for θ e = 0, NM (dotted line) and BO (dot-dashed line) for θ e = 90. and 270 the difference is much less (see also the Figure 3). All these details are understandable if we re-derive Equation (14) with all the approximations used in its derivation [1] except the BO approximation. The IF gives the amplitude of molecular ionization f(k;r) 2f H (k) cos[kr cos(θ er )/2], (15) where f H (k) is the amplitude of ionization of a single atom. By plugging Equation (15) into Equation (1), we obtain σ (3) χ ( ) nk (R) cos[kr cos(θ er)/2] χ 0 (R) 2. (16) For θ er = 90, this equation is reduced to the simple Frank-Condon factor which does not depend on k. That can be interpreted as the result of the equil sharing of the recoil momentum between the nuclei. In such case, the recoil momentum can not affect the relative nuclear motion and, consequently, the dissociation probability. In the case of θ er = 0, the influence of the recoil momentum on the relative nuclear motion is maximal. This causes a strong difference between the NM and BO results near θ er = 0. Interestingly, this influence appears due to the modulation of the integrand of the Frank-Condon factor by the electron interference factor. Such an effect can be possibly used for a check of occurence of two-center electron interference without a measure of electron angular distribution, just from the KER spectrum. Another interesting feature of angular distributions displayed in Figure 2 is the shift of peaks near θ er = 0 and 180 relative to the simple interference pattern. The origins of this shift are a secondary Coulomb scattering of electron on the nuclei and a spin of the circularly polarized photon. If we consider ionization of a single atom by a circularly polarized photon, the classical trajectory of an electron ejected in the polarization have a non-zero impact parameter due to a finite angular momentum transfer m = 1. In result, when there is scattering on the second nucleus in the molecule, the angular distribution of the scattered electron is asymmetric relative to rotation around the molecular axis.

8 Dissociative ionization of H 2 by 400 ev circularly polarized photons 8 A side effect of asymmetric scattering should be a transfer of a part of the angular momentum of the absorbed photon to a rotation of the internuclear axis. For tracing this effect, we present in Figure 2 the NMnR results also. As it is seen in Figures 2a and 2b, the differences in positions (and forms) of peaks between the BO and NM are maximal for a small KER. But position of peaks in the NM curves actually coinside with the ones for NMnR. It means that a difference between peak positions between the NM and BO for the small KER are consequence of the same effect as a difference in peak magnitudes: the influence of a recoil is maximal for θ er = 0 and this amplifies the dissociation. As the result, the peak position is shifted to θ er = 0 relative to the BO. On the other hands, for larger KER (see Figures 2c and especially 2d) the peaks in the NMnR curves are closer to θ er = 0 and 180 then in the NM curves. Therefore, the nuclei get the rotatation kick that deflects the internuclear axis beyond dissociation from a molecular axis orientation during the ionization process. S S S)M θ S)θ Figure 4. The DDCS σ (2) as the function of angle θ er in the BO (dashed line) and the NM (solid line) approaches. As the PAD for different KER does not differ strongly, the double differential cross section (DDCS) integrated over KER σ (2) = 0 σ (3) de R (17) is more informative because the DDCS is simpler to measure experimentally. The DDCS is shown in Fig.4. An additional interest presents the average KER E R = 1 σ (2) 0 E R σ (3) de R. (18) The average KER dependence on the angle θ er is shown inn Fig.5. Near the main maxima of the angular distribution (θ er = 0 and 180 ) the average KER is equal 0.4 ev. In Fig.6, the DDCS curves are shown calculated using different final electron wave functions (13). It is clear from this figure that results of calculations with the effective charge Z + = 2 are much closer to ab initio results, both in shape and magnitude, that

9 Dissociative ionization of H 2 by 400 ev circularly polarized photons 9 Eθd E E Eθd Figure 5. The average KER as the function of angle θ er in BO (dashed line) and NM (solid line). (a) (b) Drg n 2 DD Di Dn Drg n 2 DD Di Dn Dr2 Dr2 Figure 6. The σ (2) as the function of angle θ er in BO (a) and NM (b). The ab initio results (black lines) and results with approximated function (13) for effective charge Z + = 1 (red lines) and Z + = 2 (green lines). ones with Z + = 1. This can be interpreted as the main contribution to ionization matrix element arises from the region near the nuclei where the screening from the second electron, residing in the ion, is absent. It is consequence of the fact that for large ejected electron momentum (k 5.3 a.u.) the electron should be near the nucleus during ionization process to facilitate the transfer of a large recoil momentum. We note that in our analysis we assume KER E R to be the total energy of a proton and a neutral hydrogen atom. The ejection angles are given in the center of mass frame of the rest ion. Experimentally [14], only the proton is detected and its kinetic energy is measured in the laboratory frame. Due to the ejection of the fast electron, the rest ion absorbs the recoil momentum Q = k. The momentum of the ejected proton in the laboratory frame is p p = K k/2, and the energy is E p = E R /2 + E e /8µ E R E e /4µcos θ ep. As the result, in the laboratory frame, the number of detected protons should have an asymmetry respective to the electron ejection direction. For instance, the maximum of the proton ejection probability (that corresponds to E R = 0 ev) will appear near E p = E e /8µ = 0.05 ev in the direction

10 Dissociative ionization of H 2 by 400 ev circularly polarized photons 10 opposite to electron ejection direction. Generally, for θ ep = 180 the number of registered protons should be larger than for θ ep = 0. But this visible asymmetry does not mean the p H asymmetry in dissociation of the H + 2 ion, because the number of the ejected atoms will coincide with the number of ejected protons for each fixed direction in the laboratory frame. 4. Conclusion In the present work, we considered theoretically the process of dissociative photoionization of the H 2 molecule by 400 ev circularly polarized photons. At this high photon energy, the photoelectron angular distribution displays a clear two-center interference pattern, which is affected by the recoil momentum transferred to the H + 2 ion. To account for this effect, we implemented in our theoretical model a full quantum mechanical description of the nuclear motion. By making a comparison with the Born- Oppenheimer approach, we are able to elucidate the recoil effects most clearly. It is shown, that two-center electron interference strongly affectes on dissociation probability. Acknowledgments The authors are thankful to Reinhard Dörner for many useful and stimulating discussions. One of the authors (VVS) wishes to thank the Australian National University (ANU) for hospitality. His stay at ANU was supported by the Australian Research Council Discovery grant DP VVS also acknowledges support from the Russian Foundation for Basic Research (Grant No a). References [1] Akoury D, Kreidi K, Jahnke T, Weber T, Staudte A, Schöffler M, Neumann N, Titze J, Schmidt L P H, Czasch A, Jagutzki O, Fraga R A C, Grisenti R E, Muiño R D, Cherepkov N A, Semenov S K, Ranitovic P, Cocke C L, Osipov T, Adaniya H, Thompson J C, Prior M H, Belkacem A, Landers A L, Schmidt-Böcking H and Dörner R 2007 The simplest double slit: Interference and entanglement in double photoionization of H 2 Science [2] Kreidi K, Akoury D, Jahnke T, Weber T, Staudte A, Schöffler M, Neumann N, Titze J, Schmidt L P H, Czasch A, Jagutzki O, Costa Fraga R A, Grisenti R E, Smolarski M, Ranitovic P, Cocke C L, Osipov T, Adaniya H, Thompson J C, Prior M H, Belkacem A, Landers A L, Schmidt-Böcking H and Dörner R 2008 Interference in the collective electron momentum in double photoionization of H 2 Phys. Rev. Lett. 100(13) [3] Zimmermann B, Rolles D, Langer B, Hentges R, Braune M, Cvejanovic S, Geszner O, Heiser F, Korica S, Lischke T, Reinkoster A, Viefhaus J, Dorner R, McKoy V and Becker U 2008 Localization and loss of coherence in molecular double-slit experiments Nat Phys 4(8) [4] Martín F, Fernández J, Havermeier T, Foucar L, Weber T, Kreidi K, Schöffler M, Schmidt L, Jahnke T, Jagutzki O, Czasch A, Benis E P, Osipov T, Landers A L, Belkacem A, Prior M H, Schmidt-Böcking H, Cocke C L and Dörner R 2007 Single photon-induced symmetry breaking of H 2 dissociation Science [5] Billaud P, Géléoc M, Picard Y J, Veyrinas K, Hergott J F, Poullain S M, Breger P, Ruchon T, Roulliay M, Delmotte F, Lepetit F, Huetz A and Dowek B C D 2012 Molecular frame

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