  Muon spin rotation and relaxation studies have been performed on a "111"
family of iron-based superconductors NaFe_1-xNi_xAs. Static magnetic order was
characterized by obtaining the temperature and doping dependences of the local
ordered magnetic moment size and the volume fraction of the magnetically
ordered regions. For x = 0 and 0.4 %, a transition to a nearly-homogeneous long
range magnetically ordered state is observed, while for higher x than 0.4 %
magnetic order becomes more disordered and is completely suppressed for x = 1.5
%. The magnetic volume fraction continuously decreases with increasing x. The
combination of magnetic and superconducting volumes implies that a
spatially-overlapping coexistence of magnetism and superconductivity spans a
large region of the T-x phase diagram for NaFe_1-xNi_xAs . A strong reduction
of both the ordered moment size and the volume fraction is observed below the
superconducting T_C for x = 0.6, 1.0, and 1.3 %, in contrast to other iron
pnictides in which one of these two parameters exhibits a reduction below TC,
but not both. The suppression of magnetic order is further enhanced with
increased Ni doping, leading to a reentrant non-magnetic state below T_C for x
= 1.3 %. The reentrant behavior indicates an interplay between
antiferromagnetism and superconductivity involving competition for the same
electrons. These observations are consistent with the sign-changing s-wave
superconducting state, which is expected to appear on the verge of microscopic
coexistence and phase separation with magnetism. We also present a universal
linear relationship between the local ordered moment size and the
antiferromagnetic ordering temperature TN across a variety of iron-based
superconductors. We argue that this linear relationship is consistent with an
itinerant-electron approach, in which Fermi surface nesting drives
antiferromagnetic ordering.
