Figure 1

(a) The $\beta$-Mn crystal structure (space group $P{4}_{1}32$), showing magnetic Mn atoms ($12d$ site) only. Frustrated nearest-neighbor connections are shown as grey or burgundy lines. While all connections shown are of identical length to within 1% (see the Supplemental Material [15]), only connections shown the in same color can be brought into coincidence by symmetry. (b) Local Mn environment. The six nearest neighbors of each Mn are arranged in a “windmill” geometry, with each triangular “blade” perpendicular to one of the $⟨111⟩$ axes. The local $x$, $y$, and $z$ (out of the page) axes are defined for each Mn atom relative to the burgundy-colored triangle. This choice of coordinates aligns the $x$ axis with the twofold rotation axis, which is the only point symmetry element of the $12d$ site.Reuse & Permissions

Figure 2

Magnetic diffuse scattering patterns for $\beta \mathrm{\text{−}}{\mathrm{Mn}}_{0.8}{\mathrm{Co}}_{0.2}$ in four reciprocal-space planes, clockwise from top left $(001{)}^{*}$, $(110{)}^{*}$, $(11\overline{2}{)}^{*}$, and $(111{)}^{*}$. In each plane, the top left panel shows $T=1.5\mathrm{K}$ experimental neutron scattering data, the top right panel shows the RMC fit to data, the bottom right panel shows the scattering for the $J_1\mathrm{\text{−}}{J}_5$ model described in the text, and the bottom left panel shows the scattering for antiferromagnetic $J_1$ interactions only. Intensity scales are chosen such that the maximum value of $I(\mathbf{Q})$ in each plane is equal to 1, and $m\overline{3}m$ Laue symmetry has been applied to experimental data and RMC fits. The $J_1$ and $J_1\mathrm{\text{−}}{J}_5$ models are calculated using a mean-field theory described in the text. Whereas the $J_1$-only model is a very poor description of the data, both the RMC fit and the $J_1\mathrm{\text{−}}{J}_5$ model give good agreement with experiment.Reuse & Permissions

Figure 3

(a) Radial spin correlation function $⟨\mathbf{S}(0)·\mathbf{S}(r)⟩$ calculated from RMC spin configurations. (b) Variation of the sum of squared residuals ${\chi }^2$ as a function of the number of interaction parameters $J_n$ allowed to vary when fitting a Heisenberg model [Eq. (3)] to the data. The ${\chi }^2$ minimum for $J_1+J_n$ (red line) at $n=5$ indicates a good fit to data, which is not significantly improved by allowing a third parameter to vary (green line) or by including all $J_n$ up to $n=12$ (blue line).Reuse & Permissions

Figure 4

(a) Ferromagnetic $J_5$ interactions couple spins along helical chains, shown as thin red lines. The direction of propagation of each chain is shown as a thick red rod. The complete crystal structure can be described as four rod sublattices, each of which describes a triangular rod lattice. For clarity, only a single rod sublattice is shown. (b) Spin correlations $⟨\mathbf{S}(0)·\mathbf{S}(r_z)⟩$ along $J_5$ rods are always ferromagnetic and decay exponentially with distance $r_z$ along the rod. (c) Antiferromagnetic (AFM) $J_1$ interactions, shown as thin lines, couple spins from adjacent rods. An arbitrary central Mn atom, shown as a large red sphere, is coupled to two atoms within the same rod sublattice (colored red); its four other nearest neighbors belong to the other three rod sublattices (colored green, blue, and yellow). (d) The right panel shows spin correlations $⟨\mathbf{\sigma }(\mathbf{0})·\mathbf{\sigma }({\mathbf{r}}_{xy})⟩$ perpendicular to $J_5$ rods. Values shown represent a sum along each rod, as described in the text, and are normalized by $⟨|\mathbf{\sigma }(\mathbf{0}){|}^2⟩$. The spin correlations alternate between antiferromagnetic and ferromagnetic throughout the triangular rod lattice. The left panel shows spin correlations $⟨\mathbf{\sigma }(\mathbf{0})·\mathbf{\sigma }({\mathbf{r}}_{xy})⟩$ for a frustrated antiferromagnetic Heisenberg model on the triangular lattice (see the Supplemental Material [15]).Reuse & Permissions