Dynamical effects in x-ray spectra and the final-state rule

The dynamical theory of x-ray spectra due to Nozières and De Dominicis (ND) is evaluated here numerically for numerous model systems including cases where the corehole potential possesses a bound state. It is shown that the resulting emission spectra obey the final-state rule rather accurately. An approximate but analytical derivation of this rule is given which provides insight into the mechanisms leading to the final-state rule. The evaluations are performed with the use of two methods, one based on an integral equation together with a separable core-hole potential and the other based on determinantal wave functions for a finite number ($N$) of electrons in a box. The equivalence of the two methods is demonstrated both formally and numerically. By comparing them we prove the finite- $N$ approach to be accurate already for rather small $N$. We also show that a separable potential does not give rise to any spurious results but can actually be chosen to yield the same ND spectrum as a local potential. The ND theory of x-ray photoemission spectra is discussed and form calculations of the exponent function $\alpha \left(\omega \right)$ for several model systems closely corresponding to simple metals we conclude the equivalence of this theory and its asymptotic approximation as far as the extraction of asymmetry indices is concerned. Recent criticism of the major conclusions reached here and in previous work is refuted.