Incommensurate magnetic modulation in K-rich cryptomelane, K$_x$Mn$_8$O$_{16}$ ($x\approx1.46$)
Cryptomelane is a hollandite-like material consisting of K$^+$ cations in an $\alpha$-MnO$_2$ tunnel-like crystallographic motif. A sample with stoichiometry K$_{1.461(4)}$Mn$_8$O$_{16}$ has been synthesised and its magnetic properties investigated using variable-temperature magnetic susceptibility, heat capacity, and neutron powder diffraction. Three distinct magnetic transitions at $184$\,K, $54.5$\,K, and $24$\,K are assigned to K$_{1.461(4)}$Mn$_8$O$_{16}$. Magnetic Bragg peaks emerge below $54.5$\,K, and from their positions they indicate a modulated magnetic structure which is incommensurate with the crystallographic nuclear structure. The model consistent with the data is a dual-$\vec{k}_\mathrm{mag}$ structure with a canted ferrimagnetic $\vec{k}_\mathrm{mag}=0$ component and an incommensurate $\vec{k}_\mathrm{mag}=(0,0,k_z)$ [$k_z=0.36902(15)$] component, with the latter most likely to be of the helical type formerly proposed for this material. Below 24\,K, there is a magnetic transition, which gives rise to a different set of magnetic Bragg peaks indicative of a highly complex magnetic structure.