  Using the DMRG method we calculate the surface spin correlation function,
$C_L(l)=< S_l^z S_{L+1-l}^z>$, in the spin $S=3/2$ antiferromagnetic Heisenberg
chain. For comparison we also investigate the $S=1/2$ chain with S=1 impurity
end spins and the S=1 chain. In the half-integer spin models the end-to-end
correlations are found to decay to zero logarithmically, $C_L(1)\sim (\log
L)^{-2d}$, with $d=0.13(2)$. We find no surface order, in clear contrast with
the behavior of the S=1 chain, where exponentially localized end spins induce
finite surface correlations. The lack of surface order implies that end spins
do not exist in the strict sense. However, the system possesses a
logarithmically weakly delocalizing boundary excitation, which, for any chain
lengths attainable numerically or even experimentally, creates the illusion of
an end spin. This mode is responsible for the first gap, which vanishes
asymptotically as $\Delta_1 \approx (\pi v_S d)/(L\ln L)$, where $v_S$ is the
sound velocity and $d$ is the logarithmic decay exponent. For the half-integer
spin models our results on the surface correlations and on the first gap
support universality. Those for the second gap are less conclusive, due to
strong higher-order corrections.
