  The proton fraction in ${\sl \beta}$-stable neutron stars is investigated
within the framework of the Skyrme-Hartree-Fock theory using the extended
Skyrme effective interaction for the first time. The calculated results show
that the proton fraction disappears at high density, which implies that the
pure neutron matter may exist in the interior of neutron stars. The
incompressibility of the nuclear equation of state is shown to be more
important to determine the proton fraction. Meanwhile, it is indicated that the
addition of muons in neutron stars will change the proton fraction. It is also
found that the higher-order terms of the nuclear symmetry energy have obvious
effects on the proton fraction and the parabolic law of the nuclear symmetry
energy is not enough to determine the proton fraction.
