For incident x-ray energies below the core level ionization threshold, one-photon absorption (OPA) leads to off-resonant x-ray excitations that are mediated by a virtual intermediate electronic state. In the linear regime of x-ray interaction, this intermediate state may decay radiatively and the maximum energy of the emitted photon is given by the initial and final atomic state as well as the energy of the incident photon. In the nonlinear excitation regime, however, this intermediate state

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... be depleted by a second x-ray absorption process as described in the sequential two-photon absorption (TPA) mechanism. Since the same intermediate state is involved in the OPA and TPA processes, the cross-section values for OPA and TPA are interconnected. We report on the determination of cross-section values for off-resonant excitation conditions, and show how the data can be used for estimating the TPA cross section in the sequential absorption mechanism. The determined off-resonant cross sections for cobalt exhibit a Lorentz-like energy-dependent behavior allowing for an energy-dependent description of the OPA and TPA cross sections with semiempirical models. The dependence of TPA cross sections on atomic number is discussed and different scaling laws are investigated. The effect of the lifetimes of a virtual intermediate state is also studied in relation to the incidence x-ray energy.