We model an accretion disk atmosphere and corona photoionized by a central X-ray continuum source. We calculate the opacity and radiation transfer for an array of disk radii, to obtain the two-dimensional structure of the disk and its X-ray recombination emission. The atmospheric structure is insensitive to the viscosity alpha. We find a feedback mechanism between the disk structure and the central illumination, which expands the disk and increases the solid angle subtended by the atmosphere. We model the disk of a neutron star X-ray binary. We map the temperature, density, and ionization structure of the disk, and we simulate the high resolution spectra observable with the Chandra and XMM-Newton grating spectrometers. The X-ray emission lines from the disk atmosphere are detectable, especially for high-inclination binary systems. The grating observations of two classes of X-ray binaries already reveal important spectral similarities with our models. The line spectrum is very sensitive to the structure of each atmospheric layer, and it probes the heating mechanisms in the disk. The model spectrum is dominated by double-peaked lines of H-like and He-like ions, plus weak Fe L. Species with a broad range of ionization levels coexist at each radius: from Fe XXVI in the hot corona, to C VI at the base of the atmosphere. The choice of stable solutions affects the spectrum, since a thermal instability is present in the regime where the X-ray recombination emission is most intense.
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