Transition from the adiabatic to the sudden limit in core-level photoemission: A model study of a localized system

We consider core electron photoemission in a localized system, where there is a charge transfer excitation. Examples are transition metal and rare earth compounds, chemisorption systems, and high-${T}_{c}$ compounds. The system is modeled by three electron levels, one core level, and two outer levels. In the initital state the core level and one outer level is filled (a spinless two-electron problem). This model system is embedded in a solid state environment, and the implications of our model system results for solid state photoemission are discussed. When the core hole is created, the more localized outer level (d) is pulled below the less localized level $(L).$ The spectrum has a leading peak corresponding to a charge transfer between L and d (``shakedown''), and a satellite corresponding to no charge transfer. The model has a Coulomb interaction between these levels and the continuum states into which the core electron is emitted. The model is simple enough to allow an exact numerical solution, and with a separable potential an analytic solution. Analytic results are also obtained in lowest order perturbation theory, and in the high-energy limit of the semiclassical approximation. We calculate the ratio $r(\ensuremath{\omega})$ between the weights of the satellite and the main peak as a function of the photon energy $\ensuremath{\omega}.$ The transition from the adiabatic to the sudden limit is found to take place for quite small kinetic energies of the photoelectron. For such small energies, the variation of the dipole matrix elements is substantial and described by the energy scale ${E}_{d}.$ Without the coupling to the photoelectron, the corresponding ratio ${r}_{0}(\ensuremath{\omega})$ shows a smooth turn-on of the satellite intensity, due to the turn on of the dipole matrix element. The characteristic energy scales are ${E}_{d}$ and the satellite excitation energy $\ensuremath{\delta}E.$ When the interaction potential with the continuum states is introduced an energy scale ${E}_{s}=1/({2R}_{s}^{2})$ enters, where ${R}_{s}$ is a length scale of the interaction (scattering) potential. At threshold there is typically a (weak) constructive interference between intrinsic and extrinsic contributions, and the ratio $r(\ensuremath{\omega}{)/r}_{0}(\ensuremath{\omega})$ is larger than its limiting value for large $\ensuremath{\omega}.$ The interference becomes small or weakly destructive for photoelectron energies of the order ${E}_{s}.$ For larger photoelectron energies $r(\ensuremath{\omega}{)/r}_{0}(\ensuremath{\omega})$ therefore typically has a weak undershoot. If this undershoot is neglected, $r(\ensuremath{\omega}{)/r}_{0}(\ensuremath{\omega})$ reaches its limiting value on the energy scale ${E}_{s}$ for the parameter range considered here. In a ``shake-up'' scenario, where the two outer levels do not cross as the core hole is created, we instead find that $r(\ensuremath{\omega}{)/r}_{0}(\ensuremath{\omega})$ is typically reduced for small $\ensuremath{\omega}$ by interference effects, as in the case of plasmon excitation. Even for this shake-down case, however, the results are very different from those for a simple metal, where plasmons dominate the picture. In particular, the adiabatic to sudden transition takes place at much lower energies in the case of a localized excitation. The reasons for the differences are briefly discussed.