A multiple scattering theoretical approach to time delay in high energy core-level photoemission of heteronuclear diatomic molecules

We present analytical expressions of momentum-resolved core-level photoemission time delay in a molecular frame of a heteronuclear diatomic molecule upon photoionization by a linearly polarized soft x-rays attosecond pulse. For this purpose, we start to derive a general expression of photoemission time delay based on the first order time dependent perturbation theory within the one electron and single channel model in the fixed-in-space system (atoms, molecules and crystals) and apply it to the core-level photoemission within the electric dipole approximation. By using multiple scattering theory and applying series expansion, plane wave and muffin-tin approximations, the core-level photoemission time delay $t$ is divided into three components, $t_{\rm abs}$, $t_{\rm path}$ and $t_{\rm sc}$, which are atomic photoemission time delay, delays caused by the propagation of photoelectron among the surrounding atoms and the scattering of photoelectron by them, respectively. We applied single scattering approximation to $t_{\rm path}$ and obtained $t_{\rm path}^{(1)}(k,\theta)$ with polarization vector parallel to the molecular axis for a heteronuclear diatomic molecule, where $\theta$ is the angle of measured photoelectron from the molecular axis. $t$ is approximated well with this simplified expression $t_{\rm path}^{(1)}(k,\theta)$ in the high energy regime ($k\gtrsim 3.5\,\, {\rm a.u.}^{-1}$), and the validity of this estimated result is confirmed by comparing it with multiple scattering calculations for C 1$s$ core-level photoemission time delay of CO molecules. $t_{\rm path}^{(1)}(k,\theta)$ shows characteristic dependence on $\theta$, it becomes zero at $\theta=0$, exhibits EXAFS type oscillation with $2kR$ at $\theta=\pi$, where $R$ is the bondlength, and gives just the travelling time of photoelectron from the absorbing atom to the neighbouring atom at $\theta=\pi/2$.