Angular Momentum Sensitive Two-Center Interference

In quantum mechanics the Young-type double-slit experiment can be performed with electrons either traveling through a double slit or being coherently emitted from two inversion symmetric molecular sites. In the latter one the valence photoionization cross sections of homonuclear diatomic molecules were predicted to oscillate over kinetic energy almost 50 years ago. Beyond the direct proof of the oscillatory behavior of these photoionization cross sections $\sigma$, we show that the angular distribution of the emitted electrons reveals hitherto unexplored information on the relative phase shift between the corresponding partial waves through two-center interference patterns.

Figure 1

Schematic representation of the coherent emission of photoelectrons from two inversion symmetric atomic sites—the molecular double-slit experiment. The left side of the figure shows the superposition of the photoelectrons from the left and the right atomic site leading to oscillations in the photoelectron partial cross sections with an oscillation frequency proportional to the inverse bond length. The right side of the figure shows the corresponding oscillation of the angular distribution anisotropy parameter $\beta$.

Figure 2

Schematic description of the experimental setup and the angular distribution anisotropy parameter $\beta$. The apparatus consists of 16 independently operating time-of-flight electron spectrometers placed on a circle. The gas inlet introduces the molecules into the interaction volume of the incident photon beam with the molecular target being in the center of that circle. The photons are generated in an undulator structure of $e^{+}$ and $e^{-}$ storage rings, in the case of the present experiments DORIS III at DESY in Hamburg and BESSY II in Berlin, respectively. The photons are monochromatized by plane grating monochromators. The angular distribution anisotropy parameter $\beta$ is visualized at bottom left.

Figure 3

Normalized experimental (symbols) and theoretical (dashed lines) partial cross sections $\sigma$ and angular distribution anisotropy parameters $\beta$ for the valence photoionization of ${\mathrm{N}}_{2}3{\sigma }_g^{-1}$, ${\mathrm{N}}_{2}1{\pi }_u^{-1}$, ${\mathrm{O}}_{2}1{\pi }_g^{-1}$, and ${\mathrm{O}}_{2}1{\pi }_u^{-1}$ displayed over the kinetic energy in electron volts (upper $x$ axis) and $k$ ($R_{\mathrm{eff}}^{-1}$) (lower $x$ axis). For the ${\mathrm{O}}_{2}1{\pi }_u^{-1}$ only the $a^4{\mathrm{\Pi }}_u$ state is included. These are the outer valences for both molecules with different parity. Panels (a) and (d) show the $\pi$-shifted $\sigma$ behavior for the two-center interference oscillations for the $g$ and $u$ orbitals of $N_2$ and $O_2$ respectively. Panels (b) and (e) show the corresponding $\beta$ oscillations for $N_2$ and $O_2$ where the respective gerade and ungerade states are consistently shifted by $\pi$. This phase shift is imprinted due to the inversion symmetry of the molecules and the corresponding sign change in the creation of ungerade orbitals. The $K$-shell $\beta$ data of Ref. [3] are depicted in panel (c). The asterisk in panel (a) marks a typical error bar for the high kinetic energy region of $N_2$ and $O_2$.