Nonlinear x-ray four-wave mixing experiments are becoming feasible due to rapid advances in high harmonic generation and synchrotron radiation coherent x-ray sources. By tuning the difference of two x-ray frequencies across the valence excitations, it is possible to probe the entire manifold of molecular electronic excitations. We show that the wave vector and frequency profiles of this x-ray analogue of coherent Raman spectroscopy provide an excellent real-space probe that carries most valuable structural and dynamical information, not available from spontaneous Raman techniques.

High-energy-resolution $\mathrm{C}1s$ photoelectron spectra of graphite were measured at excitation energies of 340, 870, 5950, and $7940\mathrm{eV}$ using synchrotron radiation. On increasing the excitation energy, i.e., increasing kinetic energy of the photoelectron, the bulk origin $\mathrm{C}1s$ peak position shifts to higher binding energies. This systematic shift is due to the kinetic-energy loss of the high-energy photoelectron by kicking the atom and is clear evidence of the recoil effect in photoelectron emission. It is also observed that the asymmetric broadening increases for the higher-energy photoelectrons. All these recoil effects can be quantified in the same manner as the Mössbauer effect for $\gamma$-ray emission from nuclei embedded in crystals.

A systematic description of coherent ultrafast x-ray spectroscopies in terms of nonlinear response functions and susceptibilities is developed. Correlation-function expressions of charge and current densities provide a unified treatment and classification of information content of the various possible techniques and connect them with their optical counterparts. Applications to pump-probe and four-wave mixing measurements are discussed.

High energy resolution photoelectron spectroscopy of conduction electrons in the vicinity of the Fermi edge in Al and Au at excitation energies of 880 and 7940 eV was carried out using synchrotron radiation. For the excitation energy of 7940 eV, the observed Fermi energy of Al shows a remarkable shift to higher binding energy as compared with that of Au, with accompanying broadening. This is due to the recoil effect of the emitted photoelectrons. The observed spectra are well reproduced by a simple model of Bloch electrons based on the isotropic Debye model.

Dissociation of the ${\mathrm{CF}}_4^{+}$ molecular ion has been investigated using monochromatized synchrotron radiation and the energy-selected electron–ion coincidence method. The ${2t}_2^{-1\mathrm{}}C$ ionic state produces fragment ions ${\mathrm{CF}}_2^{+}$ and ${\mathrm{CF}}_3^{+}$. The branching ratio ${\mathrm{CF}}_2^{+}$/ ${\mathrm{CF}}_3^{+}$ is increased by a factor of $\sim 6$ by the C $1s$ excitation to the lowest unoccupied antibonding orbital, from the ratio at the direct ionization of the ${2t}_2^{-1\mathrm{}}$ orbital. This increase is interpreted in the light of the nuclear motion in the core-excited state, with the help of theoretical calculations based on the vibronic model.

The charge transfer from an adatom to a semiconductor substrate of one-dimensional quantum dot array is evaluated theoretically. Due to the Van Hove singularity in the density of electron states at the band edges, the charge transfer decay rate is enhanced nonanalytically in terms of the coupling constant $g$ as $g^{4∕3}$. The optical absorption spectrum for the ionization of a core level electron of the adatom to the conduction band is also calculated. The reversible non-Markovian process and irreversible Markovian process in the time evolution of the adatom localized state manifest themselves in the absorption spectrum through the branch point and pole contributions, respectively.