  We present Chandra ACIS-S sub-array observations of the quiescent neutron
star low-mass X-ray binaries X7 and X5 in the globular cluster 47 Tuc. The
large reduction in photon pile-up compared to previous deep exposures enables a
substantial improvement in the spectroscopic determination of the neutron star
radius and mass of these neutron stars. Modeling the thermal emission from the
neutron star surface with a non-magnetized hydrogen atmosphere and accounting
for numerous sources of uncertainties, we obtain for the neutron star in X7 a
radius of $R=11.1^{+0.8}_{-0.7}$ km for an assumed stellar mass of $M=1.4$
M$_{\odot}$ (68% C.L.). We argue, based on astrophysical grounds, that the
presence of a He atmosphere is unlikely for this source. Due to eclipses and
variable absorption, the quiescent low-mass X-ray binary X5 provides less
stringent constraints, leading to a radius of $R=9.6^{+0.9}_{-1.1}$ km,
assuming a hydrogen atmosphere and a mass of $M=1.4$ M$_{\odot}$. When combined
with all other existing spectroscopic radius measurements, these measurements
strongly favor radii in the 9.9-11.2 km range for a ~1.5 M$_{\odot}$ neutron
star and point to a dense matter equation of state that is somewhat softer than
the nucleonic ones that are consistent with laboratory experiments at low
densities.
