  Precision measurements of slow neutron cross sections with atoms have several
scientific applications. In particular the n-$^{4}$He s-wave scattering length
is important to know both for helping to constrain the nuclear three-body
interaction and for the proper interpretation of several ongoing slow neutron
experiments searching for other types of neutron-atom interactions. We present
new measurements of the ratios of the neutron differential scattering cross
sections for natural isotopic-abundance mixtures of the noble gases He, Ar, Kr,
and Xe to natural isotopic abundance Ne. These measurements were performed
using a recently developed neutron scattering apparatus for gas samples located
on a pulsed slow neutron beamline which was designed to search for possible
exotic neutron-atom interactions and employs both neutron time of flight
information and a position-sensitive neutron detector for scattering event
reconstruction. We found agreement with the literature values of scattering
cross sections inferred from Ar/Ne, Kr/Ne and Xe/Ne differential cross section
ratios over the $q$ range of $1 - 7$ nm$^{-1}$. However for the case of He/Ne
we find that the cross section inferred differs by 11.3% (7.6 $\sigma$) from
previously-reported values inferred from neutron phase shift measurements, but
is in reasonable agreement with values from other measurements. The very large
discrepancy in the He/Ne ratio calls for a new precision measurement of the
n-$^{4}$He scattering length using neutron interferometry.
