  We present a study of the narrow Fe K$\alpha$ line in seven bright, nearby
AGN that have been observed extensively with the Chandra High Energy
Transmission Grating (HETG). The HETG data reveal a wider Fe K$\alpha$ line in
the first order spectrum than in the second and third order spectra, which we
interpret as the result of spatially extended Fe K$\alpha$ emission. We utilize
these differences in narrow Fe K$\alpha$ line widths in the multi-order Chandra
HETG spectra to determine the spatial extent and intrinsic velocity width of
the emitting material in each object. We find that there is modest evidence for
spatially extended emission in each object, corresponding to extension of
$r\sim5-100$ pc. These distances are significantly larger than those inferred
from velocity widths assuming gravitational motions, which give $r\sim0.01-1$
pc. This implies that either the gas is emitting at a range of radii, with
smaller radii dominating the velocity width and larger radii dominating the
spatial extent, or that the gas is exhibiting non-gravitational motions, which
we suggest would be outflows due to slight excess redshift in the line and
velocities that exceed the freefall velocity. We also use the spatial extent
information to estimate the mass of the emitting gas by counting fluorescing
iron atoms, finding masses on the order of
$M_\mathrm{gas}\sim10^5-10^8\,M_\odot$. Future work with observatories like
XRISM will be able to extend this study to a larger number of AGN and decrease
uncertainties that arise due to the low signal-to-noise of the higher order
HETG data.
