Transition from the adiabatic to the sudden limit in core-electron photoemission

Experimental results for core-electron photoemission $J_{\mathbf{k}}\left(\omega \right)$ are often compared with the one-electron spectral function $A_c\left({\epsilon }_k-\omega \right),$ where $\omega$ is the photon energy, ${\epsilon }_k$ is the photoelectron energy, and the optical transition matrix elements are taken as constant. Since $J_{\mathbf{k}}\left(\omega \right)$ is nonzero only for ${\epsilon }_k>0,$ we must actually compare it with $A_c\left({\epsilon }_k-\omega \right)\theta \left({\epsilon }_k\right).\mathrm{}$ For metals $A_c\left(\omega \right)$ is known to have a quasiparticle (QP) peak with an asymmetric power-law [theories of Mahan, Nozières, de Dominicis, Langreth, and others (MND)] singularity due to low-energy particle-hole excitations. The QP peak starts at the core-electron energy ${\epsilon }_c,$ and is followed by an extended satellite (shakeup) structure at smaller $\omega .$ For photon energies $\omega$ just above threshold, ${\omega }_{\mathrm{th}}=-{\epsilon }_c,\hspace{0.5em}{A}_c\left({\epsilon }_k-\omega \right)\theta \left({\epsilon }_k\right)$ as a function of ${\epsilon }_k\hspace{0.5em}(\omega$ constant) is cut just behind the quasiparticle peak, and neither the tail of the MND line nor the plasmon satellites are present. The sudden (high-energy) limit is given by a convolution of $A_c\left(\omega \right)$ and a loss function, i.e., by the Berglund-Spicer two-step expression. Thus $A_c\left(\omega \right)$ alone does not give the correct photoelectron spectrum, neither at low nor at high energies. We present an extension of the quantum-mechanical (QM) models developed earlier by Inglesfield, and by Bardyszewski and Hedin to calculate $J_{\mathbf{k}}\left(\omega \right).\mathrm{}$ It includes recoil and damping, as well as shakeup effects and extrinsic losses, is exact in the high-energy limit, and allows calculations of $J_{\mathbf{k}}\left(\omega \right)$ including the MND line and multiple plasmon losses. The model, which involves electrons coupled to quasibosons, is motivated by detailed arguments. As an illustration we have made quantitative calculations for a semi-infinite jellium with the density of aluminum metal and an embedded atom. The coupling functions (fluctuation potentials) between the electron and the quasibosons are related to the random-phase-approximation dielectric function, and different levels of approximations are evaluated numerically. The differences in the predictions for the photoemission spectra are found small. We confirm the finding by Langreth that the BS limit is reached only in the keV range. At no photon energy are the plasmon satellites close to being either purely intrinsic or extrinsic. For photoelectron energies larger than a few times the plasmon energy, a semiclassical approximation gives results very close to our QM model. At lower energies the QM model gives a large peak in the ratio between the total intensity in the first plasmon satellite and the main peak, which is not reproduced by the SC expression. This maximum has a simple physical explanation in terms of different dampings of the electrons in the QP peak and in the satellite. For the MND peak $J_{\mathbf{k}}\left(\omega \right)$ and $A_c\left({\epsilon }_k-\omega \right)$ agree well for a range of a few eV, and experimental data can thus be used to extract the MND singularity index. For an embedded atom at a small distance from the surface there are, however, substantial deviations from the large-distance limit. Our model is simple enough to perform quantitative calculations allowing for band-structure and surface details.