Photodouble ionization of He with circularly polarized synchrotron radiation: complete experiment and dynamic nodes

P Bolognesi¹, V Feyer¹, A Kheifets², S Turchini³, T Prosperi³, N Zema³ and L Avaldi^1,4

¹ CNR-IMIP, Area della Ricerca di Roma 1, 00016 Monterotondo Scalo, Italy
² Research School of Physical Sciences and Engineering, ANU, Canberra, Australia
³ CNR-ISM, Area della Ricerca di Roma 2, Via Fosso del Cavaliere, Roma, Italy
⁴ CNR-INFM-TASC, Gas Phase Photoemission Beamline@Elettra, Trieste, Italy

Received 22 January 2008, in final form 1 February 2008
Published 26 February 2008

The understanding of the correlated motion of electron pairs is of fundamental interest for the knowledge of the structure and properties of matter. In the last ten years it has been proved that the study of photodouble ionization, PDI, of He, the simplest process dominated by electron correlations, can provide valuable and unique information on this topic [1]. The deepest understanding of PDI is achieved in coincidence experiments where either the two photoelectrons or a photoelectron and the recoiling ion are detected. In these experiments the triple differential cross section, d³σ/dΩ₁ dΩ₂ dE₁ (TDCS) is measured. Ω₁  =  (θ₁, ₁) and Ω₂  =  (θ₂, ₂) are the angles of emission of the two electrons and E₁ is the energy of one of them. The energy of the second electron is determined by energy conservation hν-IP^2 +   =  E₁  +  E₂, where hν is the photon energy and IP^2 +  the double ionization potential. By considering the ¹P° symmetry of the electron pair continuum wavefunction and the invariance by rotation about a preferential symmetry axis, the full separation of the geometrical factors and the dynamic parameters in the TDCS is achieved [2]. The formers come from the description of the photon–atom interaction in the dipole approximation; the latters include all the information on the dynamics of the process in two complex amplitudes a_g(E₁, E₂, θ₁₂) and a_u(E₁, E₂, θ₁₂) [2], symmetric (gerade) and antisymmetric (ungerade), respectively, in the exchange of the two electrons. These basic quantities, calculated by the theories, are used to reconstruct any particular TDCS. The extraction of the moduli and relative phase, δ, of the amplitudes from experiments is of considerable interest, because it allows a direct comparison among different sets of data and between theory and experiment. After some attempts relying upon parametrizations with various degree of complexity and approximation [3, 4], Bolognesi et al [5] proposed a procedure that (i) does not rely on any approximation, (ii) needs only three determinations of the TDCS at the same relative angle θ₁₂ between the photoelectrons and (iii) can be applied to any set of experimental data. The method, applied to measurements with linearly polarized incident radiation, provides |a_g|², |a_u|² and cos δ [5], while the sign of δ remains undetermined. When two sets of measurements at the same hν, E₁ and E₂ with linearly and circularly polarized radiation are combined then also the sign of δ is determined and the complete description of the process achieved. The He TDCS measured with circularly polarized radiation display a helicity dependence, i.e. a non-vanishing circular dichroism, CD [6, 7]. The observation of chirality in an electron pair ejected by a target with a spherically symmetric ground state (¹S^e) might surprise, because in the common wisdom chirality is associated with objects not identical to their mirror image. However Berakdar and Klar [6] showed that, due to parity conservation, the helicity of the incident photon is transferred to the three body-system, the He atom, and the continuum spectrum of this excited system depends on the helicity of the absorbed photon. CD is proportional to sin δ [1]. Thus the combination of the CD measurement and three TDCS measured with linearly polarized radiation provides the full information for a complete experiment [5]. Another procedure [8], based on COLTRIMS experiments, can provide analogous results [9]. However the four experimental geometries needed in that procedure are not achieved by most of the set-ups used in PDI experiments. By contrast our procedure can be applied with no restrictions to any set of experimental data.

The first calculations of CD in He [7] predicted that for certain values of E₁  +  E₂, R  =  E₂/E₁ and θ₁₂, CD can vanish in a non-trivial way, i.e. due to the vanishing neither of the amplitudes nor of the geometrical factors. Later calculations in the convergent close coupling, CCC, model [10] and recently in the lowest-order perturbative approach [11] supported this result. Some experiments (see table 1 in [1]) provided an indirect evidence of these `dynamic nodes', while a recent measurement at E₁  +  E₂  =  60 eV and R  =  5 and 11, supported by the predictions of the hyperspherical R matrix with semiclassical outgoing wave model, did not display such nodes [12].

The aim of this work is twofold: (i) to achieve a complete description of the PDI process via measurements with linearly and circularly polarized radiation and (ii) to prove definitely the existence of the dynamic nodes.

The experiments have been done with the multicoincidence end-station [13] at the GAPH [14] and CIPO [15] beamlines of the Elettra storage ring. The two rotatable turntables of the end-station host ten electrostatic hemispherical analyzers in the plane perpendicular to the direction, z, of the incident radiation. The three analyzers of the smaller turntable have been set to detect low energy electrons (E₁  =  3.5 eV), while the other seven detect electrons with 20 ≤ E₂ ≤ 50 eV depending on the measurement. The set-up and experimental procedure have been described elsewhere [13].

The light source of the GAPH beamline is an undulator which provides completely linearly polarized radiation. The light source of the CIPO beamline is an electromagnetic elliptical undulator/wiggler, which provides radiation of variable polarization state. In the energy range of interest the radiation is characterized by Stoke parameters S₁  =  0.24, S₃  =  ±0.97 (S₃ > 0 corresponds to a radiation whose polarization vector rotates clockwise for an observer looking towards the source) and the major axis of the polarization ellipse lays at λ  =  ±34° for S₃  =  ±0.97, respectively, with respect to the orbital plane of the electrons in the ring [15].

In the case of elliptically polarized radiation, the TDCS in the plane perpendicular to the direction of the radiation can be expressed as [12]

where ₁ is referred to the major axis of the polarization ellipse and θ₁₂ is the relative angle between the two photolectrons. The reduction of TDCS in equation (1) to the case of fully linearly polarized radiation (S₁  =  1, S₃  =  0) is straightforward [1]. The circular dichroism, CD, which is obtained by the TDCS measured with radiation of both helicities, is defined by

where TDCS_L (TDCS_R) is the TDCS for PDI by left (right), i.e. S₃  =  –1( + 1), circularly polarized radiation. From the experimental point of view it is convenient to define the normalized circular dichroism, CD_n,

equations (1)–(3) and the procedure described in [5] show that measurements with linearly and circularly polarized radiation provide the full information on the complex amplitudes, i.e. their moduli and relative phase.

Three TDCS measured at hv  =  127 eV for E₁  =  3.5 eV, E₂  =  44.5 eV, S₁  =  1 and θ₁  =  0°, 30° and 60° are shown in figure 1(a)–(c). The experimental data have been compared with the predictions of the CCC model [10]. The three TDCS of figure 1(a)–(c), being measured simultaneously, are on the same relative scale, therefore in the comparison with the theory they have been rescaled by a common factor. The CCC predicts correctly both the shape and the relative intensity versus θ₁ of the TDCS. The TDCS measured with the two helicities of the light are reported in the insets of figure 1(d). In these insets the TDCS measured at the three θ₁ have been added up. This procedure is strictly valid only for fully circularly polarized radiation. However in our cases neither a simulation with the CCC considering the actual characteristics of light nor the measurements of the TDCS at the three θ₁ display variations larger than the experimental uncertainties. Thus we concluded that the second term in equation (1) does not contribute significantly to the measured TDCS and therefore derived the CD_n (figure 1(d)) according to equation (3). A reasonable agreement between the CCC theory and experiment is observed also for the measurements with the two helicities and the CD_n (figure 1(d)).

Following the procedure by Bolognesi et al [5] the three independent determinations of the TDCS at the same E₁, E₂ and θ₁₂ are inserted in a nonlinear system based on equation (1), whose solution gives |a_g|², |a_u|² and cos δ. The uncertainties in these quantities have been calculated using equations (4)–(6) of [5] under the assumption that the uncertainties in the three experimental TDCS are statistically independent. These quantities when used to reproduce the measured CD_n provide us with the sign of the relative phase δ, too. Due to the geometrical constraints of our set-up the amplitudes and the phase are measured only in the region θ₁₂ > 90°. The results are shown in figure 2, where they are compared with the CCC predictions and the previous data of [5] measured at hv  =  119 eV (E₁  =  5 eV and E₂  =  35 eV). Good agreement with the theoretical predictions is observed also for the shape of the amplitudes and phase measured in the two experiments. However the different R of the two sets of data results in a different |a_g|²/|a_u|² ratio, indeed |a_u|² of [5] in figure 2 has been scaled up by a factor 1.6. The knowledge of a_g, a_u and δ allows to calculate other observables of the process. The linear dichroism, Δ_lin  =  ΔTDCS(E₁ E₂), i.e. the difference between two TDCS measured with linearly polarized radiation in which the energies of the two electrons are exchanged, has been calculated at θ₁  =  0° and compared with CCC in figure 3(a). Then by integrating equation (1) over all the directions of one of the two photoelectrons [16] the asymmetry parameter, β, of the angular distribution of the fast (slow) photoelectron, d²σ/dE dΩ_i 1  +  β_i P₂(cos θ_i) i  =  1, 2, can be obtained. The calculated β (0.68 ± 0.14 and –0.12 ± 0.10 for the fast and slow photoelectron, respectively) from the present amplitudes and phase well compare with the CCC values (0.80 and –0.18, respectively). These values can be compared with previous experimental [17] and theoretical [16] values at hν  =  120 eV. At the same R  =  E₁/E₂ ratio of the present experiment, good agreement is found for the β of the fast photoelectron, while the experimental value [17] of the slow photoelectron appears to be higher (0.5 ± 0.2). Finally also the He^2 +  recoil momentum distribution, dσ/dK, where K  =  k₁  +  k₂ is the centre of mass of the two-electron subsystem and therefore the opposite of the ion recoil momentum, has been calculated [18] and compared in figure 3(b) with the predictions of CCC, because no experiments exist at this hν. This latter finding shows for the first time that it is possible to reconstruct from an electron–electron experiment with fixed analyzers a typical observation of a COLTRIMS experiment. None of the observables of PDI studied up to now, apart from the CD, relies on the sign of δ. However by similarities with photoionization [19], we expect that this quantity plays a role in the determination of the spin asymmetry parameters of the two electrons of PDI.

In figure 1(d) the CD_n crosses zero at about θ₁₂  =  85°. It can be easily checked in equation (3), that this is a `dynamic node' due to a zero in sin δ. Indeed in figure 2 the theory predicts δ  =  π at this θ<sub>12</sub>. Recently Istomin et al [11] predicted that `at all excess energies up to 50 eV' two dynamic nodes should be observed in the ranges 14.6° ≤ θ₁₂ ≤ 40.1° and 81.3° ≤ θ₁₂ ≤ 88.4°. We confirm the existence of a node in the range of the larger θ₁₂. No experimental data cover the other range, however our calculations do not support the prediction of a node in that region. To investigate better the existence of dynamic nodes other measurements have been done at 102.5 ≤ hν ≤ 142.5 eV. E₁ was kept at 3.5 eV, while E₂ was incremented according to the variation of hν. In figure 4(a) the CD_n measured at six E₂ is reported versus θ₁₂. While a change in sign of the CD_n at certain E₂ values is observed for θ₁₂ < 127°, in the measurements at ₁₂ ≥ 127° all the measured CD_n are negative. To emphasize this result the CD_n at θ₁₂  =  67°, 97° and 127° is reported versus E₂ in figure 4(b). Consistently with the CCC predictions the CD_n changes sign at θ₁₂  =  97° when E₁  =  3.5 eV and E₂ ≈ 35 eV. This definitely proves the existence of the dynamic nodes, whose occurrence depends on E₁/E₂ and θ₁₂, as firstly predicted by Berakdar et al [7]. From equation (2) it is evident that the appearance of a dynamic node corresponds to the mathematical condition of linear dependence of the two amplitudes. This results in a reduction of the dimensionality of the process. Dichroism is intrinsically a 3D phenomenon, thus the (E₁, E₂, θ₁₂) sets with δ  =  0, π correspond to dynamic conditions where the PDI process looses in a non-trivial way its 3D characteristics.

Figure 4. CDn versus 12 at E1  =  3.5 eV and six different 20.5 ≤ E2 ≤ 59.5 eV (a) and versus E2 at θ12  =  67° (dots), 97° (triangle) and 127°(diamond) (b). The solid lines in (b) are the CCC predictions. The data at E2  =  44.5 eV are the weighted average of the CDn obtained in this measurement and the one from the measurement of the full TDCS shown in figure 1(d).

The achievement of complete experiments is one of the main goal in atomic physics. Photoelectron-Auger electron [20] and photoelectron-ion [21] coincidence experiments have been proved to be an ideal tool for complete photoionization experiments. Here electron–electron coincidence experiments with linearly and circularly polarized radiation enabled us to obtain the determination of the amplitudes needed for a complete quantomechanical description of the PDI in He. It has been shown that the data measured with fixed analyzers can be used to determine the He^2 +  momentum distribution, a typical result of COLTRIMS. Another result is the observation of a node in the CD when the relative phase between the two amplitudes is π. Analytical methods [22] showed that the vanishing of CD at certain (E₁, E₂, θ₁₂) is completely dependent on electron correlations and the position of the dynamic nodes varies depending on the used wavefuntions [7]. The definite prove of their existence and their clear identification in the multidimensional (E₁, E₂, θ₁₂) space may be used to discriminate among different correlated wavefunctions and may be of prominent importance in the understanding of one-photon two-electron emission in highly correlated materials [23]. The existence of dynamic nodes has been also correlated to a non-trivial reduction of dimensionality of the process.

Acknowledgments

Work partially supported by INTAS Project No 0–51-4706, MIUR FIRB project `Probing the microscopic dynamics of chemical reactivity'. V F thanks the ICTP for a TRIL scholarship and A K for the CNR-Short Term Mobility program.

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