  We study theoretically the di-neutron spatial correlations and the crossover
from superfluidity of neutron Cooper pairs in the $^1S_0$ pairing channel to
Bose-Einstein condensation (BEC) of di-neutron pairs for both symmetric and
neutron matter in the microscopic relativistic pairing theory. We take the bare
nucleon-nucleon interaction Bonn-B in the particle-particle channel and the
effective interaction PK1 of the relativistic mean-field approach in the
particle-hole channel. It is found that the spatial structure of neutron Cooper
pair wave function evolves continuously from BCS-type to BEC-type as density
decreases. We see a strong concentration of the probability density revealed
for the neutron pairs in the fairly small relative distance around $1.5 {\rm
fm}$ and the neutron Fermi momentum $k_{Fn}\in[0.6,1.0] {\rm fm^{-1}}$.
However, from the effective chemical potential and the quasiparticle excitation
spectrum, there is no evidence for the appearance of a true BEC state of
neutron pairs at any density. The most BEC-like state may appear at
$k_{Fn}\thicksim0.2 {\rm fm^{-1}}$ by examining the density correlation
function. From the coherence length and the probability distribution of neutron
Cooper pairs as well as the ratio between the neutron pairing gap and the
kinetic energy at the Fermi surface, some features of the BCS-BEC crossover are
seen in the density regions, $0.05 {\rm fm^{-1}}<k_{Fn}<0.7 (0.75) {\rm
fm^{-1}}$, for the symmetric nuclear (pure neutron) matter.
