  We present a new study of the X-ray spectral properties of the Crab Pulsar.
The superb angular resolution of the Chandra X-ray Observatory enables
distinguishing the pulsar from the surrounding nebulosity. Analysis of the
spectrum as a function of pulse phase allows the least-biased measure of
interstellar X-ray extinction due primarily to photoelectric absorption and
secondarily to scattering by dust grains in the direction of the Crab Nebula.
We modify previous findings that the line-of-sight to the Crab is
under-abundant in oxygen and provide measurements with improved accuracy and
less bias. Using the abundances and cross sections from Wilms, Allen & McCray
(2000) we find [O/H] = $(5.28 \pm 0.28)\times10^{-4}$ ($4.9 \times10^{-4}$ is
solar abundance). We also measure for the first time the impact of scattering
of flux out of the image by interstellar grains. We find $\tau_{\rm scat} =
0.147 \pm 0.043$. Analysis of the spectrum as a function of pulse phase also
measures the X-ray spectral index even at pulse minimum --- albeit with
increasing statistical uncertainty. The spectral variations are, by and large,
consistent with a sinusoidal variation. The only significant variation from the
sinusoid occurs over the same phase range as some rather abrupt behavior in the
optical polarization magnitude and position angle. We compare these spectral
variations to those observed in Gamma-rays and conclude that our measurements
are both a challenge and a guide to future modeling and will thus eventually
help us understand pair cascade processes in pulsar magnetospheres. The data
were also used to set new, and less biased, upper limits to the surface
temperature of the neutron star for different models of the neutron star
atmosphere.
