The magnetic structure and fluctuations of tetragonal $\mathrm{Gd}\mathrm{Rh}{\mathrm{In}}_5$ were studied by resonant x-ray diffraction at the Gd $L_{\mathit{II}}$ and $L_{\mathit{III}}$ edges, followed by a renormalization group analysis for this and other related Gd-based compounds, namely ${\mathrm{Gd}}_2\mathrm{Ir}{\mathrm{In}}_8$ and $\mathrm{Gd}{\mathrm{In}}_3$. These compounds are spin-only analogs of the isostructural Ce-based heavy-fermion superconductors. The ground state of $\mathrm{Gd}\mathrm{Rh}{\mathrm{In}}_5$ shows a commensurate antiferromagnetic spin structure with propagation vector $\vec{\tau }=(0,\frac{1}{2},\frac{1}{2})$, corresponding to a parallel spin propagation along the $\vec{a}$ direction and antiparallel propagation along $\vec{b}$ and $\vec{c}$. The spin direction lies along $\vec{a}$. A comparison between this magnetic structure and those of other members of the $R_m{(\mathrm{Co},\mathrm{Rh},\mathrm{Ir})}_n{\mathrm{In}}_{3m+2n}$ family ($R=\text{rare}$ earth, $n=0,1$; $m=1,2$) indicates that, in general, $\vec{\tau }$ is determined by a competition between first- $\left(J_1\right)$ and second-neighbor $\left(J_2\right)$ antiferromagnetic (AFM) interactions. While a large $J_1∕J_2$ ratio favors an antiparallel alignment along the three directions (the $G$-AFM structure), a smaller ratio favors the magnetic structure of $\mathrm{Gd}\mathrm{Rh}{\mathrm{In}}_5$ ($C$-AFM). In particular, it is inferred that the heavy-fermion superconductor $\mathrm{Ce}\mathrm{Rh}{\mathrm{In}}_5$ is in the frontier between these two ground states, which may explain its noncollinear spiral magnetic structure. The critical behavior of $\mathrm{Gd}\mathrm{Rh}{\mathrm{In}}_5$ close to the paramagnetic transition at $T_N=39\mathrm{K}$ was also studied in detail. A typical second-order transition with the ordered magnetization critical parameter $\beta =0.35$ was experimentally found, and theoretically investigated by means of a renormalization group analysis. Although the Gd $4f^7$ electrons define a half-filled, spherically symmetrical shell, leading to a nearly isotropic spin system, it is argued that a significant spin anisotropy must be claimed to understand the second order of the paramagnetic transition of $\mathrm{Gd}\mathrm{Rh}{\mathrm{In}}_5$ and the related compound ${\mathrm{Gd}}_2\mathrm{Ir}{\mathrm{In}}_8$.

• Received 30 June 2006

DOI:https://doi.org/10.1103/PhysRevB.74.214428