Determination of the magnetic structures in the heavy fermion superconductor Ce${3}$PtIn${11}$

Heavy fermion superconductor Ce$_3$PtIn$_{11}$ ($ T_{\rm c} = 0.32 ~ {\rm K} $) belongs to the family of tetragonal indides Ce$_n M_m $In$_{3n+2m}$ ($M$ = Co, Rh, Ir, Pd, Pt) [1]. The compound undergoes two successive magnetic transitions at $T_{\rm N1} = 2.2 ~ {\rm K}$ and $T_{\rm N2} = 2.0 ~ {\rm K}$, clearly observed in thermodynamic quantities such as specific heat. However, the magnetic structures between $T_{\rm N1}$ and $T_{\rm N2}$ and below $T_{\rm N2}$ have not been revealed. Ce$_3$PtIn$_{11}$ has two inequivalent Ce sites in the crystallographic unit cell. The Ce(2) site carries a substantial magnetic moment, whereas a magnetic moment at the Ce(1) site is almost perfectly screened by Kondo effect [2].
To determine the magnetic structures of Ce$_3$PtIn$_{11}$, we performed a comprehensive analysis of the $^{115}$In-NQR spectra employing a magnetic dipolar model [4]. Based on this analysis, we confirmed that the magnetic moment at the Ce(1) site is very small ($\sim$ 4% of that at the Ce(2) site) but finite. Further, for $T_{\rm N2} < T < T_{\rm N1}$ and $T < T_{\rm N2}$, the magnetic moments at the Ce(1) and Ce(2) sites are parallel to the $c$-axis, and the in-plane propagation vectors in the two Ce atoms sublattices are ($q_a,~q_b$) = (1/2, 1/2). In the temperature interval $T_{\rm N2} < T < T_{\rm N1}$, both of them have a form of ${\bf q }$ = (1/2, 1/2, 0 or 1/2). For $T < T_{\rm N2}$, the magnetic structure in the Ce(2) sublattice is defined by the propagation vector ${\bf q}$ = (1/2, 1/2, 1/2 or 0), whereas the Ce(1) sublattice forms a six-fold period structure given by ${\bf q}$ = (1/2, 1/2, 1/6 or 1/3).