Photoelectron diffraction from laser-aligned molecules with X-ray free-electron laser pulses

We report on the measurement of deep inner-shell 2p X-ray photoelectron diffraction (XPD) patterns from laser-aligned I2 molecules using X-ray free-electron laser (XFEL) pulses. The XPD patterns of the I2 molecules, aligned parallel to the polarization vector of the XFEL, were well matched with our theoretical calculations. Further, we propose a criterion for applying our molecular-structure-determination methodology to the experimental XPD data. In turn, we have demonstrated that this approach is a significant step toward the time-resolved imaging of molecular structures.

. Illustration of X-ray photoelectron diffraction of a single aligned molecule. In a single aligned I 2 molecule, a 2p photoelectron wave emitted from the left I atom (a) and a scattered wave by the right I atom (b) cause a fringe pattern due to interference between the two waves (c). The fringe pattern depends on an internuclear distance and photoelectron energy, although this simulation was done under the condition of an equilibrium internuclear distance and photoelectron energy of 140 eV.
Scientific RepoRts | 5:14065 | DOi: 10.1038/srep14065 selected a mean photon energy of 4697 eV for the probing XFEL pulses, which is above the I 3/2 ionization threshold of 4557 eV. This creates I 2p photoelectrons with a mean energy of 140 eV, although the photon energies of the XFEL fluctuate within a band-pass range of Δ / = × − E E 5 10 3 shot-by-shot 28 . We neglect the interaction between the spin and the orbital motion of each electron hereafter. For further reference, the experimental details are described in the Methods section.
Photoemission from laser-aligned molecules. 2p photoelectron momentum images and fragmention momentum images of the I 2 molecules under different experimental conditions are shown in Fig. 3, along with illustrations of relevant polarization geometries. Each image was obtained from alternating measurements of the images with and without the I 2 molecular beams, which was operated at half of the XFEL repetition rate.
The 2D momentum images of the fragment ions created by the probing XFEL (bottom row) provide the basic experimental information. The observation of circularly symmetric rings, in Fig. 3(d), shows that the initial orientation of the molecules is random. The angular distribution of I m+ fragment ions, with ≤ ≤ m 4 6, is localized around the polarization direction of the Nd:YAG laser, see Fig. 3(e). This is an effect of the molecules being aligned by the laser pulse along its field vector. From the angular distribution of a prominent ring bounded by the radii r = 7.5 mm and 10.5 mm, which originates from the Coulomb explosion of fragment-ion pairs of I m+ and I n+ ions, with ≤ , ≤ m n 4 6, we determined the degree of alignment 16  , where θ m is the angle between the polarization vector and the molecular axis being parallel to fragment-ion momentum direction (see Methods). In Fig. 3(f), the prominent ring in Fig. 3(e) nearly disappears, indicating that there are few molecules left in the xz plane, which is parallel to the detector surface. Thus, the observed results for the fragment ions ensure that the photoelectron-momentum image in Fig. 3(a) comes from the randomly oriented molecules, and that the images in Fig. 3(b,c) are from the aligned molecules.
On this basis, we turn to a qualitative examination of the momentum images of the 2p photoelectrons. The electron momentum images in Fig. 3(a-c) in the upper row are composed of the extremely strong central ring and the outer ring of the 2p photoelectrons. In deep inner-shell photoionization, as with I 2p, great quantities of low-energy electrons are produced frequently via shake-off processes induced by Auger transitions, so that intense signals appear in the central region of the VMI detector. This condition does apply for our experiment; however, thanks to the high photoelectron energy (140 eV), the outer-ring photoelectron image (between r = 27.5 and 35 mm) is distinguished from the central-ring image of low-energy electrons, although the outer-ring image is blurred by the photon-energy shot-by-shot fluctuation, ∆ ≈ , E 24 eV and by the azimuthal-angle distribution.  Figure 2. Schematic drawing of the experimental set-up. Two laser beams propagating in a collinear arrangement intersect a supersonic pulsed molecular beam at the center of a vacuum chamber. A Nd:YAG laser is used to align the sample molecules probed by the XFEL. XPD images of photoelectrons are recorded by the upper VMI. The degree of alignment is quantified using 2D momentum distributions of ionic fragments, which are registered by the lower VMI.
between the outer-ring images in Fig. 3(a,b), without and with the alignment laser, respectively. This difference is minor but clearly visible: the lengths of the bright arc segments in Fig. 3(b) are slightly shorter than those in Fig. 3(a). It should be noted that the expected interference structure in Fig. 3(b) may be smeared out by molecular-axis distributions, because the degree of alignment in this experiment is not sufficient, θ = . ± . cos 061 0 03 m 2 , which means that about 60% of all the molecules have their axis located within a cone of 40°.
For further quantitative discussion on the photoelectron angular distributions measured by the momentum images in Fig. 3, polar plots are provided in Fig. 4. Data processing for the polar plots are written in the Methods section. Although the differences in Fig. 4(a-c) are small, one can recognize them in the polar plots. The reference frame of Fig. 4(a) is the laboratory coordinate system, whose z axis is the direction of XFEL's polarization vector. In this frame, the angular distribution is expressed by Eq. (1) in the Methods section. Thus we fitted Eq. (1) to the experimental data and calculated the asymmetry parameter β to be 0.65 ± 0.12. On the other hand, the photoelectron angular distribution in Fig. 4(b) is presented in the molecular reference frame, whose z axis is the molecular axis parallel to the direction of the polarization vector of the alignment laser (see Fig. 3). In the molecular frame, the photoelectron angular distribution is determined by Eq. (2) in the Methods section. Such angular distribution is equivalent to the XPD pattern, so that hereafter it is referred to as XPD. Even if the molecular axes are not fully aligned along the z axis (parallel to the polarization vector), the functional form of Eq. (2) is applicable to reproduce the experimental results. Therefore, we fitted Eq. (2) with up to L = 6 to the experimental data. With the help of this theoretical analysis of the numerical data, it is highlighted that the XPD pattern in Fig. 4(b) clearly exhibits a different shape from that for the photoelectron angular distribution in Fig. 4(a). Besides such a visual check, we examined the difference between the shapes shown in Fig. 4(a,b) quantitatively: Eq. (1) was fitted to the data in Fig. 4(b) and Eq. (2) to the data in Fig. 4(a). A weighted sum of squared errors for the each least squares fitting is summarized in Table 1. From this, one can see that Eq. (1) gives the smaller minimum weighted sum for Fig. 4(a) than that for Fig. 4(b) and that Eq. (2) gives the smaller minimum weighted sum for Fig. 4(b) than that for Fig. 4(a). These results are the firm evidence that the shape of the angular distribution shown in Fig. 4(b) differs from that in Fig. 4(a). Here we emphasize that the comparison of the minimum weighted sums for Eq.
(2) about one data ( Fig. 4(a) or Fig. 4(b)) does not make sense because the mathematical forms of them are different from each other. The comparison of the diffraction pattern with theoretical XPD results will be discussed later.
In the molecular frame, the XPD pattern in Fig. 4(c) is fitted with Eq. (3) in the Methods section. In this case, the molecular axis is aligned along the y axis in the figure (see Fig. 3) perpendicular to the xz plane of the paper, so that the polar plot gives the azimuthal-angle distribution of the XPD pattern. As can be seen from Eq. (3), the characteristic features of the azimuthal-angle distribution are restricted by conservation of angular momentum component on the molecular axis, so that the XPD pattern observed in this perpendicular polarization geometry provides less information about the molecular geometry than that in the parallel polarization.
Simulated XPD patterns. We now discuss how the multiple-scattering XPD theory helps us to interpret the observed XPD results (detailed in the Methods section). The 2p state is triply degenerate, so we consider the photoemission from the 2p z orbital to be aligned along the molecular axis and from the 2p x (2p y ) orbital to be aligned along the x axis (y axis) orthogonal to the molecular axis. The theoretical results for the XPD, which were calculated for light polarization along the molecular axis, are depicted as polar plots on the xz plane in Fig. 5 [i.e., XPD from the 2p z orbital: Fig. 5(b), that from the 2p x orbital: Fig. 5(c), and their sum: Fig. 5(a)]. Here, we take incoherent superposition of XPD from both the left and right I atoms. As can be seen in Fig. 5(a), the difference between XPD by full multiple-scattering calculation and that by single-scattering calculation is minor. This implies that at the photoelectron energy of 140 eV, the single-scattering effect predominates in XPD, as reported in the literature 13,[24][25][26] .
To elucidate the interference effect in XPD, the results of the computational experiment for the single-scattering approximation, Fig. 5(d,e) (see Methods). For the polarization geometry corresponding to light polarized along the molecular axis, the photoionization of the 2p z orbital creates both s-and d z 2-partial waves in the local region of the emitter's atomic site, owing to the dipole selection rule. In Fig. 5(d), however, the primary photoemission amplitude, Z 0 2 , exhibits the specific shape of the angular function of where Y 20 is a spherical harmonic, because the minor component of the s-partial wave contributes negligibly. Since the  Fig. 4(a) Data in Fig. 4

(b)
Eq. (1)  , see Fig. 5(b,d). On the other hand, the photoionization of the 2p x orbital produces a d xz -partial wave at the emitter's atomic site. This is made obvious by Fig. 5( Computational and experimental results. Here, we build on the two preceding subsections by comparing the observed and theoretical XPD patterns. The interference fine structure of the XPD shown in Fig. 5(a) is not observed in the XPD shown in Fig. 4(b). As mentioned earlier, this occurs because the molecular axes of I 2 are not fully aligned in our experiments. Thus, the axis distribution expressed by θ = . ± . cos 061 0 03 m 2 was taken into account in the XPD calculations. The acceptance angle, when we made the polar plot in Fig. 4(b), was also taken into account in the calculations. The computational results by the multiple-scattering XPD theory are shown in Fig. 6, along with the experimental results. In these calculations, we used an equilibrium internuclear distance of 2.666 Å for the ground-state I 2 molecule, which is the sole geometrical parameter in this practical application. As seen in Fig. 6, the theoretical results taking the axis distribution into account reproduce the experimental ones quite well. This demonstrates that the multiple-scattering XPD theory is a promising computational means for deriving molecular structures from experimental XPD patterns, which are more or less influenced by axis distributions of sample molecules.

Discussion
The present XPD pattern is strongly affected by the degree of alignment for the sample molecules, as described above. Thus, a question arises: how sensitive to the molecular structure is the XPD pattern, when averaged by the molecular-axis distribution? To answer this question, computational experiments have been performed: we calculated the XPD patterns by changing internuclear distances for both partially and fully aligned I 2 molecules. These results are shown in Fig. 7. On one hand, the XPD patterns averaged by the axis distribution expressed by θ = . ± . cos 061 0 03 m 2 are not especially sensitive to changes in internuclear distance of ± 0.5 Å [see Fig. 7(a)]. On the other hand, the XPD patterns from the fully aligned molecules are sensitive to such small changes in the internuclear distance [see Fig. 7(b)]. From this, one can conclude that to definitively determine a molecular structure from a measured XPD pattern, a higher degree of alignment of sample molecules is necessary. In other words, the XPD patterns exhibiting interference profiles, which would be measurable for highly aligned molecules, are essential for the application of our molecular-structure-determination methodology, see Ref. 13. Namely, this is a criterion for the necessary experimental data to derive molecular structures from them. It should be noted that to achieve higher degrees of alignment, for example, an electrostatic-molecular-deflector for selecting quantum states of sample molecules may be a suitable device 17,29,30 . Further, the most advanced molecular alignment or orientation technique allows us to align or orient state-selected asymmetric top molecules even in the field-free condition 31 .
In conclusion, we have successfully measured 2p XPD patterns from laser-aligned I 2 molecules using XFEL pulses, which were in strong agreement with the multiple-scattering XPD theory calculations. In light of this, we have proposed the criterion for applying our molecular-structure-determination methodology to experimental XPD data. Thus, the present work is a step toward ultra-fast photoelectron diffraction, which may enable the capture of ultra-fast molecular movies, for example, of photochemical reactions.

Methods
Experimental details. XFEL pulses with a duration of ~10 fs (FWHM) were triggered at a repetition rate of 30 Hz 27,28 . A focal spot size of ~1 μ m in diameter was created with a pair of Kirkpatrick-Baez (KB) mirrors 32 . During our experiment, the pulse energy of the XFEL was 300-450 μ J. The Nd:YAG laser pulses, with a duration of ~10 ns (FWHM), a pulse energy of ~800 mJ, and a repetition rate of 30 Hz, were used to align the I 2 molecules. The Nd:YAG laser was focused to a spot size of 80 μ m in diameter at the interaction point, resulting in an intensity of ~10 12 Wcm −2 . A spatial overlap between the XFEL and Nd:YAG laser was confirmed by monitoring their spot images on a Ce:YAG screen installed at the interaction region. A temporal overlap between the XFEL and YAG laser pulses was introduced by adjusting the time delay of the Nd:YAG laser pulses to the XFEL pulses, which were measured by a pin photo-diode for the former and a home-made photo-diode for the latter and placed equidistant from the interaction point. z x Figure 6. Comparison of computational and experimental I 2p XPD patterns. Bold solid curve: full multiple-scattering calculation taking the molecular axis distribution into account. Filled circles with error bars: experimental data. Thin solid curve: fitted result. The experimental data are the same as in Fig. 4(b). Each result was normalized by the area of the polar plots. . Red curves: equilibrium internuclear distance of 2.666 Å; blue curves: internuclear distance of (2.666 + 0.5) Å; and green curves: internuclear distance of (2.666-0.5) Å. The red curve in (a) is the same as the bold solid curve in Fig. 6.
A pulsed supersonic beam of rotationally cold I 2 molecules was formed by expanding a mixture of I 2 and He carrier gas from an Even-Lavie valve 33 with a nozzle diameter of 150 μ m, which was then collimated by a skimmer with a diameter of 3 mm. The valve was operated with a duration of 20.7 μ s for a driving pulse. The stagnation pressure of He was 35 bar, and the sample pressure of I 2 gas was ~0.006 bar, which was evaporated from solid I 2 by heating the valve containing it to 60 °C. Operating the pulse valve at the half of the XFEL repetition rate results in alternating measurements of electron and ion images with and without the sample gas.
The static electric fields in the extraction regions and the drift regions of the VMI were adjusted to optimize the velocity focusing to I 2p photoelectrons with a kinetic energy of 140 eV. The VMI was equipped with a micro-channel plate (MCP) backed by a phosphor screen with an active diameter of 75 mm. Electron images were recorded with a scientific complementary metal-oxide-semiconductor (sCMOS) camera mounted to the upper VMI, while ion images were recorded with a CCD camera mounted to the lower VMI. We accumulated momentum-image data for 740,000 XFEL pulses without Nd:YAG laser pulses, for 1,150,000 XFEL pulses with horizontally polarized Nd:YAG laser pulses, and for 550,000 XFEL pulses with vertically polarized Nd:YAG pulses, which correspond to 7 hours, 11 hours, and 5 hours of total time, respectively. Fragmentation dynamics upon 2p ionization of I 2 . Fragment ions are mainly produced in the following scenario: highly charged molecular ions are created via Auger cascades after I 2p photoionization by XFEL pulses, these create ion pairs of I m+ and I n+ , which then dissociate back-to-back due to a Coulomb explosion between them. Figure 8 shows a covariance map of fragment ions for 2,500 XFEL pulses with the horizontally polarized Nd:YAG laser pulses. A small portion of the I + ions is created by the Nd:YAG laser. This figure indicates that the main contribution to the prominent ring between r = 7.5 mm and 10.5 mm in Fig. 3(e) is due to the fragment ions with charge states of 4+ to 6+ . For the fragment-ion pairs with the charge-separation combinations of (4 -4), (4 -5), (4 -6), (5 -5), (5 -6), and (6 -6), which produce images between r = 7.5 mm and 10.5 mm, a kinetic energy release was estimated using SIMION 34 . Here, the kinetic energy release is 56 eV for I 4+ -I 4+ pairs with an internuclear distance of 4.1 Å when the Coulomb explosion occurs after the cascade Auger decays and 130 eV for I 6+ -I 6+ pairs with a distance of 3.9 Å.
Data processing for polar plots. A three-dimensional representation of the 2D photoelectron image in Fig. 3(b) is shown in Fig. 9. As seen in this figure, the central image, because of low-energy electrons, has a sharp and strong peak at the center and a slightly asymmetric long tail that extends to the photoelectron outer ring. Accordingly, we subtracted the extrapolated asymmetric tail contribution from the signals in the range of r = 27.5 mm to r = 35 mm, sector-by-sector on the xz plane. Then, we integrated the signals over Δ r = (27.5-35) mm, which corresponds to an azimuthal acceptance angle of 60°, and over a polar angle of 30° on the xz plane. Finally, the left-side and right-side numerical data with respect to the z = 0 plane were averaged, and the upper and lower data with respect to the x = 0 plane were averaged, considering the symmetry restriction of the XPD pattern imposed by the experimental geometry, It should be noted that the strong Nd:YAG laser field may give rise to a sideband structure in the 2p photoelectron spectrum as the result of above-threshold ionization 35 , in which photoelectrons interact with the laser field through the absorption or emission of a number of laser photons. In our experimental conditions, the number of exchanged laser photons is estimated to be ≤ 10 (Ref. 36). Thus, the sidebands may be included in our photoelectron images. However, these are not visible since they are smeared out by the photon-energy shot-by-shot fluctuation of Δ ≈ E 24 eV. where σ is the integrated cross section, β is the asymmetry parameter, and θ e is measured from the electric vector of the XFEL 37,38 . θ ( ) P cos e 2 denotes the Legendre polynomial of the second order. If, instead, the molecules have a definite orientation, then the angular distribution is described by an alternative form from Eq. (1). For example, when the molecular axis is parallel to the electric vector of the XFEL (see Fig. 3), the angular distribution in the xz plane can be expressed by where the polar angle Θ is measured from the molecular z axis 39,40 . Θ ( ) P cos L denotes the Legendre polynomial of the Lth order, and the A L coefficients are calculated from the relevant dipole-matrix elements. Due to the parity selection rule, the summation over L is restricted to even integers for molecules having inversion symmetry (like the I 2 molecule). When the molecular axis is perpendicular to the electric vector of the XFEL (see Fig. 3 where the azimuthal angle Φ is measured from the molecular x axis 39,40 . Namely, the azimuthal-angle distribution is restricted by the conservation of the angular momentum component on the molecular axis for linear molecules. In contrast to this, the polar-angle distribution is determined by intramolecular photoelectron diffraction. Parameterizations of Eqs (2) and (3) for the photoelectron angular distributions in the molecular frame, in other words the XPD, are derived from molecular photoionization theory. Numerical results Figure 9. Three-dimensional representation of the electron image shown in Fig. 3(b). Compared to the outer-ring intensity for 2p photoelectrons, the intensity in the central region is extremely strong. The tail of the central region extends to the outer ring.