  Proton-neutron, neutron-neutron and proton-proton momentum correlation
functions ($C_{pn}$, $C_{nn}$, $C_{pp}$) are systematically investigated for
$^{15}$C and other C isotopes induced collisions at different entrance channel
conditions within the framework of the isospin-dependent quantum molecular
dynamics (IDQMD) model complemented by the CRAB (correlation after burner)
computation code. $^{15}$C is a prime exotic nucleus candidate due to the
weakly bound valence neutron coupling with closed-neutron shell nucleus
$^{14}$C. In order to study density dependence of correlation function by
removing the isospin effect, the initialized $^{15}$C projectiles are sampled
from two kinds of density distribution from RMF model, in which the valence
neutron of $^{15}$C is populated on both 1$d$5/2 and 2$s$1/2 states,
respectively. The results show that the density distributions of valence
neutron significantly influence nucleon-nucleon momentum correlation function
at large impact parameter and high incident energy. The extended density
distribution of valence neutron largely weakens the strength of correlation
function. The size of emission source is extracted by fitting correlation
function using Gaussian source method. The emission source size as well as the
size of final state phase space is larger for projectiles sampling from more
extended density distribution of valence neutron corresponding 2$s$1/2 state in
RMF model. Therefore momentum correlation function can be considered as a
potential valuable tool to diagnose the exotic nuclear structure such as skin
and halo.
