Complex Magnetic Order in a Decorated Spin Chain System Rb$_2$Mn$_3$(MoO$_4$)$_3$(OH)$_2$

Macroscopic magnetic properties and microscopic magnetic structure of Rb$_2$Mn$_3$(MoO$_4$)$_3$(OH)$_2$ (space group $Pnma$) are investigated by magnetization, heat capacity and single-crystal neutron diffraction measurements. The compound's crystal structure contains bond-alternating [Mn$_3$O$_{11}$]$^{\infty}$ chains along the $b$-axis, formed by isosceles triangles of Mn ions occupying two crystallographically nonequivalent sites (Mn1 site on the base and Mn2 site on the vertex). These chains are only weakly linked to each other by nonmagnetic oxyanions. Both SQUID magnetometry and neutron diffraction experiments show two successive magnetic transitions as a function of temperature. On cooling, it transitions from a paramagnetic phase into an incommensurate phase below 4.5~K with a magnetic wavevector near ${\bf k}_{1} = (0,~0.46,~0)$. An additional commensurate antiferromagnetically ordered component arises with ${\bf k}_{2} = (0,~0,~0)$, forming a complex magnetic structure below 3.5~K with two different propagation vectors of different stars. On further cooling, the incommensurate wavevector undergoes a lock-in transition below 2.3~K. The experimental results suggest that the magnetic superspace group is $Pnma.1'(0b0)s0ss$ for the single-${\bf k}$ incommensurate phase and is $Pn'ma(0b0)00s$ for the 2-${\bf k}$ magnetic phase. We propose a simplified magnetic structure model taking into account the major ordered contributions, where the commensurate ${\bf k}_{2}$ defines the ordering of the $c$-axis component of Mn1 magnetic moment, while the incommensurate ${\bf k}_{1}$ describes the ordering of the $ab$-plane components of both Mn1 and Mn2 moments into elliptical cycloids