  Two Chandra observations have been used to search for thermal X-ray emission
from within and around the Crab Nebula. Dead-time was minimized by excluding
the brightest part of the Nebula from the field of view. A dust-scattered halo
comprising 5% of the strength of the Crab is clearly detected with surface
brightness measured out to a radial distance of 18 arcminutes. Coverage is 100%
at 4 arcminutes, 50% at 12 arcminutes, and 25% at 18 arcminutes. The observed
halo is compared with predictions based on 3 different interstellar grain
models and one can be adjusted to fit the observation. This dust halo and
mirror scattering form a high background region which has been searched for
emission from shock-heated material in an outer shell. We find no evidence for
such emission. We can set upper limits a factor of 10-1000 less than the
surface brightness observed from outer shells around similar remnants. The
upper limit for X-ray luminosity of an outer shell is about 10e34 erg/s.
Although it is possible to reconcile our observation with an 8-13 solar mass
progenitor, we argue that this is unlikely.
