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Abstract:This paper adresses the problem of above-threshold ionization (ATI) of hydrogen interacting with an intense X-ray electromagnetic field. Two approaches have been used. In the first approach, we calculate generalized differential and total cross sections based on second-order perturbation theory for the electron interaction with a monochromatic plane wave, with the A 2 and A · P contributions from the nonrelativistic Hamiltonian (including retardation) treated exactly. In the second approach, we solve the time-dependent Schrödinger equation (TDSE) for a pulsed plane wave using a spectral approach with a basis of oneelectron orbitals, calculated with L 2 -integrable B-spline functions for the radial coordinate and spherical harmonics Y for the angular part. Retardation effects are included up to O(1/ ), they induce extra terms forcing the resolution of the TDSE in a three dimensional space. Relativistic effects [of O 1/ 2 ] are fully neglected. The isoelectronic series of hydrogen is explored in the range Z = 1 − 5 in both TDSE and perturbative approaches. Photoelectron angular distributions are obtained for photon energies of 1 keV and 3 keV for hydrogen, and photon energy of 25 keV for the hydrogenic ion B 4+ . Perturbative and TDSE calculations are compared. 32.80.rm, 32.80.Fb, 32.30.Rj PACS (2008):