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We report a $^{23}$Na and $^{75}$As nuclear magnetic resonance (NMR)investigation of Na$_{x}$FeAs series ($x=1$, 0.9, 0.8) exhibiting aspin-density wave (SDW) order below $T_{\rm SDW}=45$, 50 and 43 K for $x=1$,0.9, 0.8, respectively, and a bulk superconductivity below $T_c\approx 12$ Kfor x=0.9. Below $T_{\rm SDW}$, a spin-lattice relaxation reveals the presenceof gapless particle-hole excitations in the whole $x$ range, meaning that aportion of the Fermi surface remains gapless. The superconducting fraction asdeduced from the bulk susceptibility scales with this portion, while the SDWorder parameter as deduced from the NMR linewidth scales inversely with it. TheNMR lineshape can only be reproduced assuming an incommensurate (IC) SDW. Thesefindings qualitatively correspond to the mean-field models of competinginterband magnetism and intraband superconductivity, which lead to an IC SDWorder coexisting with superconductivity in part of the phase diagram.

Incommensurate spin-density wave and multiband superconductivity in Na$_{x}$FeAs as revealed by nuclear magnetic resonance