  We identify an accretion disk atmosphere and corona from the high resolution
X-ray spectrum of Hercules X-1, and we determine its detailed physical
properties. More than two dozen recombination emission lines (from Fe XXVI at
1.78 A to N VI at 29.08 A) and Fe K-alpha, K-beta fluorescence lines were
detected in a 50 ks observation with the Chandra High-Energy Transmission
Grating Spectrometer (HETGS). They allow us to measure the density,
temperature, spatial distribution, elemental composition, and kinematics of the
plasma. We exclude HZ Her as the source of the recombination emission. We
compare accretion disk model atmospheres with the observed spectrum in order to
constrain the stratification of density and ionization, disk atmosphere area,
elemental composition, and energetics. The atmospheric spectrum observed during
the low state is photoionized by the main-on X-ray continuum, indicating that
the disk is observed edge-on during the low state. We infer the mean number of
scatterings N of Ly-alpha and Ly-beta line photons from H-like ions. We derive
N < 69 for O VIII Ly_alpha_1, which rules out the presence of a mechanism
modeled by Sako (2003) to enhance N VII emission via a line overlap with O
VIII. The line optical depth diagnostics are consistent with a flattened
atmosphere. Our spectral analysis, the disk atmosphere model, and the presence
of intense N VII and N VI lines (plus N V in the UV), confirm the
over-abundance of nitrogen relative to other metals, which was shown to be
indicative of CNO cycle processing in a massive progenitor.
