Cent. Eur. J. Phys. • 11(9) • 2013 • 1091-1098 DOI: 10.2478/s11534-012-0140-x Central European Journal of Physics Above-threshold ionization of hydrogen and hydrogen-like ions by X-ray pulses Research Article Henri Bachau 1* , Olimpia Budriga 2 , Mihai Dondera 3† , Viorica Florescu 3‡ 1 Centre des Lasers Intenses et Applications, Université Bordeaux I-CNRS-CEA, FR-33405 Talence, France 2 National Institute for Laser, Plasma and Radiation Physics, RO-077125 Bucharest-M˘ agurele, Romania 3 Faculty of Physics and Centre for Advanced Quantum Physics, University of Bucharest, RO-077125 Bucharest-M˘ agurele, Romania Received Apr 02, 2012; accepted Sep 21, 2012 Abstract: This paper adresses the problem of above-threshold ionization (ATI) of hydrogen interacting with an in- tense 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 one- electron 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. PACS (2008): 32.80.rm, 32.80.Fb, 32.30.Rj Keywords: Multiphoton ionization • X-ray • Above threshold ionization • Hydrogen• hydrogenic ions © Versita sp. z o.o. * E-mail: bachau@celia.u-bordeaux1.fr † E-mail: don@baratu.fizica.inibuc.ro ‡ E-mail: flor@baratu.fizica.inibuc.ro 1. Introduction The emergence of X-ray free-electron lasers which pro- duce radiation at increasing peak intensities [1, 2], moti- vates the theoretical interest for the study of the atomic ionization induced by multiphoton absorption in the ex- treme ultraviolet (EUV) and X-ray photon energy range. It is expected that new experimental results, like the re- 1091