Photoelectron diffraction from single oriented molecules: Towards ultrafast structure determination of molecules using x-ray free-electron lasers

We provide a molecular structure determination method, based on multiple-scattering x-ray photoelectron diffraction (XPD) calculations. This method is applied to our XPD data on several molecules having different equilibrium geometries. Then it is confirmed that, by our method, bond lengths and bond angles can be determined with a resolution of less than 0.1 Å and ${10}^{\circ }$, respectively. Differently from any other scenario of ultrafast structure determination, we measure the two- or three-dimensional XPD of aligned or oriented molecules in the energy range from 100 to 200 eV with a $4\pi$ detection velocity map imaging spectrometer. Thanks to the intense and ultrashort pulse properties of x-ray free-electron lasers, our approach exhibits the most probable method for obtaining ultrafast real-time structural information on small to medium-sized molecules consisting of light elements, i.e., a “molecular movie.”

Figure 1

Illustration of x-ray photoelectron diffraction. In a simulation, the interference between a $p$-type photoelectron wave emitted from the left atom and $s$-type waves scattered by the right atom makes a fringe pattern. The interference is caused by the phase difference between the two waves due to the path length difference.

Figure 2

(a) Calculated two-dimensional C $1s$ XPD image of CO molecules at ${\varepsilon }_p=$ 150 eV and (b) polar plot of the XPD. (c) Profiles of single ($|Z_1+Z_2{|}^2$), double ($|Z_1+Z_2+Z_3{|}^2$), and full multiple scattering. (d) Components of the single-scattering intensity. In (c), the results for double scattering and full multiple scattering almost perfectly overlap. In (d), positive and negative values for $2\mathrm{Re}\left(Z_1^{*}{Z}_2\right)$ are shown separately.

Figure 3

Polar plots of C $1s$ XPD for C${}_2$H${}_4$I${}_2$. XPD patterns for the anti form are shown in the left column and those for TS2 form are given in the right column. Positive and negative values of $2\mathrm{Re}\left(Z_1^{*}{Z}_2\right)$ are shown separately. The positive value of the interference term is so small that it is hardly observed in the patterns. Insets: Molecular orientations and polarization vectors.

Figure 4

Calculated two-dimensional C $1s$ XPD images of C${}_2$H${}_4$I${}_2$ molecule for the anti and TS2 forms at ${\varepsilon }_p=150$ eV and polar plots of them. XPD patterns were calculated with full multiple scattering. Insets: Molecular orientations and polarization vectors.

Figure 5

(a) $R$ factor versus C-O bond length of CO${}_2$: The solid vertical line indicates the C-O distance at the $R_{\min }$; the dashed vertical line, the equilibrium distance in the ground state; and the shaded (yellow) band, the range of the bond lengths with $R<R_{\min }+\mathrm{Var}\left(R_{\min }\right)$. (b) Calculated two-dimensional C $1s$ XPD image at the $R_{\min }$. (c) Polar plots of C $1s$ XPDs: Filled circles with error bars are the experimental data at ${\varepsilon }_p=150$ eV; the thin solid curve, the fit of it, the bold solid curve, the XPD calculation at the $R_{\min }$; and the dashed curve, the calculation at the maximum $R$. The calculated results are integrated over the relevant experimental acceptance angles (in plane, $\pm {10}^{\circ }$; out of plane, $\pm {20}^{\circ }$; for both photoelectrons and fragment ions) to compare them with the experimental ones. Inset: Molecular orientation and polarization vector.

Figure 6

Comparison between $R$ factors calculated in two ways: The dashed line shows $R$ calculated using the phase shift for the equilibrium geometry of CO${}_2$ in the ground state, whereas the solid line indicates $R$ calculated by phase shifts evaluated for each model structure.

Figure 7

(a) $R$ factor versus N-O bond length and O-N-O angle of NO${}_2$. (b) Calculated two-dimensional N $1s$ XPD image at the $R_{\min }$. (c) Polar plots of N $1s$ XPDs: Filled circles with error bars are the experimental data at ${\varepsilon }_p=90$ eV; the thin solid curve, the fit of it; the bold solid curve, the XPD calculation at the $R_{\min }$; and the dashed curve, the calculation at the maximum $R$. The calculated results are integrated over the relevant experimental acceptance angles for photoelectrons (in plane, $\pm 5^{\circ }$; out of plane, $\pm {30}^{\circ }$) and fragment ions (in plane, $\pm {10}^{\circ }$; out of plane, $\pm {20}^{\circ }$) to compare them with the experimental ones. Inset: Molecular orientation and polarization vector. In (a), $R$ takes values $<R_{\min }+\mathrm{Var}\left(R_{\min }\right)$ inside the bold contour at the bottom.

Figure 8

Same as Fig. 7 except for the perpendicular polarization geometry. Inset: Molecular orientation and polarization vector.

Figure 9

(a) $R$ factor versus B-F bond length and angle $\vartheta$ of BF${}_3$. (b) Two-dimensional B $1s$ XPD image at the $R_{\min }$. (c) Polar plots of B $1s$ XPDs: Filled circles with error bars are the experimental data at ${\varepsilon }_p=120$ eV; the thin solid curve, the fit of it; the bold solid curve, the XPD calculation at the $R_{\min }$; and the dashed curve, the calculation at the maximum $R$. The calculated results are integrated over the relevant experimental acceptance angles for photoelectrons (in-plane, $\pm 5^{\circ }$; out of plane, $\pm {30}^{\circ }$) and fragment ions (in plane, $\pm {10}^{\circ }$; out of plane, $\pm {20}^{\circ }$) to compare them with the experimental ones. Inset: Molecular orientation and polarization vector. In (a), $R$ takes values $<R_{\min }+\mathrm{Var}\left(R_{\min }\right)$ inside the bold contour at the bottom.

Figure 10

(a) $R$ factor versus C-F bond length and H-C-F angle of CH${}_3$F. (b) Two-dimensional F $1s$ XPD image at the $R_{\min }$. (c) Polar plots of F $1s$ XPDs: Filled circles with error bars are the experimental data at ${\varepsilon }_p=150$ eV; the thin solid curve, the fit of it; the bold solid curve, the XPD calculation at the $R_{\min }$; and the dashed line, the calculation at the maximum $R$. The calculated results are integrated over the relevant experimental acceptance angles for photoelectrons (in plane, $\pm 5^{\circ }$; out of plane, $\pm {30}^{\circ }$) and fragment ions (in plane, $\pm {10}^{\circ }$; out of plane, $\pm {20}^{\circ }$) to compare them with the experimental ones. Inset: Molecular orientation and polarization vector. In (a), $R$ takes values $<R_{\min }+\mathrm{Var}\left(R_{\min }\right)$ inside the bold contour at the bottom.

Figure 11

Schematic representation of the photoelectron diffractometer. Two beams propagating in a collinear arrangement intersect a molecular beam at the center of the diffractometer. An optical laser is used to align or orient the molecules that are probed by ionizing them using a time-delayed XFEL pulse. The XPD image is recorded by the upper VMI spectrometer. The degree of alignment or orientation is quantified using the 2D momentum distribution of the ionic fragments that is registered by the lower VMI spectrometer.

Figure 12

Detailed view of the VMIS for electron trajectory focusing. Two meshes, M${}_1$ and M${}_2$, delimit the extraction region (uniform extraction field) and the end of the drift tube of 80-mm diameter, respectively. The laser propagates along the $x$ direction, causing a line source of 3-mm length, from which three external points, $-1.5$, 0, and 1.5 mm on the $x$ axis, are chosen. From each point eight electrons with a 200-eV kinetic energy are ejected by the ${45}^{\circ }$ angle spacing in the $x$-$z$ plane. In the focusing plane on the MCP, electron trajectories of the same ejection angle but different positions come together. The trajectories focused at five points, from right to left, on the MCP correspond to ejection angles of $0^{\circ }$ (+$x$ direction), $\pm {45}^{\circ }$, $\pm {90}^{\circ }$ ($\pm z$ direction), $\pm {135}^{\circ }$, and ${180}^{\circ }$ ($-x$ direction), respectively. As the VMIS is cylindrically symmetric around the $z$ axis, the electron trajectories are retained under the rotation of those around the $z$ axis. This is indicated as circles in the upper figure. The potential condition of the electrodes for electron trajectory focusing is as follows: M${}_1$ (M${}_1^{\prime }$), 10 kV ($-10$ kV); L${}_1$, 10 kV; L${}_2$, 10 kV; and D, 4.6 kV. The potentials of C${}_1$ (C${}_1^{\prime }$) and C${}_2$ (C${}_2^{\prime }$) are tuned to keep a uniform extraction field of 1.25 kV/cm: C${}_1$ (C${}_1^{\prime }$), 1.9 kV ($-1.9$ kV); and C${}_2$ (C${}_2^{\prime }$), 5.9 kV ($-5.9$ kV).