The magnetic structure of the zigzag chain family Na$_{x}$Ca$_{1-x}$V$_2$O$_4$ determined by muon-spin rotation

We present a muon spin rotation measurements on the complete zig-zag chain Na$_x$Ca$_{1-x}$V$_2$O$_4$ family. In this family we explore the magnetic properties from the metallic NaV$_2$O$_4$ to the insulating CaV$_2$O$_4$. We find a critical $x_c$ which separates the low and high $x$ transition temperature and its magnetic ground state. For $xx_c$ multiple subphases are revealed. Based on the muon data, dipolar field simulation indicate a helical incommensurate structure in $x=1$ with a $\sim4.88$ periodicity.