The strong magnetic fields (B ~ 10^{12} - 10^{13} G) characteristic of neutron stars make all the properties of an atom strongly dependent on the transverse component K_perp of its generalized momentum. In particular, the photoionization process is modified substantially: (i) threshold energies are decreased as compared with those for an atom at rest, (ii) cross section values are changed significantly, and (iii) selection rules valid for atoms at rest are violated by the motion so that new photoionization channels become allowed. To calculate the photoionization cross sections, we, for the first time, employ exact numerical treatment of both initial and final atomic states. This enables us to take into account the quasi-bound (autoionizing) atomic states as well as coupling of different ionization channels. We extend the previous consideration, restricted to the so-called centered states corresponding to relatively small values of K_perp, to arbitrary states of atomic motion. We fold the cross sections with the thermal distribution of atoms over K. For typical temperatures of neutron star atmospheres, the averaged cross sections differ substantially from those of atoms at rest. In particular, the photoionization edges are strongly broadened by the thermal motion of atoms; this magnetic broadening exceeds the usual Doppler broadening by orders of magnitude. The decentered states of the atoms give rise to the low-energy component of the photoionization cross section. This new component grows significantly with increasing temperature above 10^{5.5} K and decreasing density below 1 g/cm^3, i.e., for the conditions expected in atmospheres of middle-aged neutron stars.
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