We present a systematic numerical study of core-level x-ray photoemission intensity in metals, with emphasis upon studying the deviations from behavior asymptotically close to threshold. For a model with a contact potential and linear conduction-electron dispersion, we have evaluated the photoemission intensity for core-hole phase shifts δ(0) between 0.05π and 0.5π. We find the following results. (i) The asymptotic regime extends out to 0.01 to 0.1 times the conduction bandwidth (D) from threshold. The range of the asymptotic regime decreases with increasing (absolute) phase shift. (ii) The linear relation between the integrated photoemission intensity and the asymptotic form holds for all phase shifts above ∼0.1D. Due to our normalization procedure we cannot say whether it holds below this value. (iii) Discrepancies exist between numerical estimates of the deviations from asymptotic behavior and approximate analytic estimates. (iv) A definition of the frequency-dependent threshold singularity exponent α(ω) in terms of a moment of the photoemission intensity is stable out to the conduction-band edge and may prove useful to experimentalists attempting to extract exponents from their data.

• Received 7 March 1985

DOI:https://doi.org/10.1103/PhysRevB.32.555