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Geometrical frustration arised in spin ices leads to fascinating emergent physical properties. Nowadays there is a wide diversity of the artificial structures, mimicking spin ice at the nanoscale and demonstrating some new effects. Most of the nanoscaled spin ices are two dimensional. Ferromagnetic inverse opal-like structures (IOLS) are among inspiring examples of the three-dimensional system exhibiting spin ice behaviour. However, a detailed examination of its properties is not straightforward. An experimental technique which is able to unambiguously recover magnetization distribution in 3D mesoscaled structures is lacking. In this work, we used an approach based on complementary exploiting of small-angle neutron diffraction technique and micromagnetic simulations. An external magnetic field was applied along three main directions of the IOLS mesostructure. Comparison of the calculated and measured data allowed us to determine IOLS magnetic state. The results are in good agreement with the spin ice model. Moreover influence of the demagnetizing field and vortex states on the magnetizing process were revealed. Additionally, we speculate that this approach can be also applied to other 3D magnetic mesostructures.

The magnetic properties of an inverse opal-like cobalt-based structure having the symmetry of the face-centered cubic lattice are studied. The magnetization reversal of the structure in a magnetic field applied along the [ $$1\overline 1 0$$ ] axis is described using a phenomenological model, which uses the ice rule for the local magnetization of nanostructure elements. This description predicts the absence of a long-range magnetic order in two directions that are normal to the magnetic field. The magnetic structure is analyzed by smallangle neutron diffraction. Neutron diffraction patterns are measured in an external magnetic field varying from −1.2 to 1.2 T and applied along the crystallographic [ $$1\overline 1 0$$ ] direction. A magnetic contribution to the neutron scattering intensity is extracted. It is shown that this contribution is close to zero over the entire applied magnetic field range for all scattering planes that are normal to the field, which agrees well with the proposed model.

Magnetization hysteresis loops of a three-dimensional nanoscale analog of spin ice based on the nickel inverse opal-like structure (IOLS) have been studied at room temperature. The samples are produced by filling nickel into the voids of artificial opal-like films. The spin ice behavior is induced by tetrahedral elements within the IOLS, which have the same arrangement of magnetic moments as a spin ice. The thickness of the films vary from a two-dimensional, i.e., single-layered, antidot array to a three-dimensional, i.e., multilayered, structure. The coercive force, the saturation, and the irreversibility field have been measured in dependence of the thickness of the IOLS for in-plane and out-of-plane applied fields. The irreversibility and saturation fields change abruptly from the antidot array to the three-dimensional IOLS and remain constant upon further increase of the number of layers $n$. The coercive force ${H}_{c}$ seems to increase logarithmically with increasing $n$ as ${H}_{c}={H}_{c0}+\ensuremath{\alpha}ln(n+1)$. The logarithmic law implies the avalanchelike remagnetization of anisotropic structural elements connecting tetrahedral and cubic nodes in the IOLS. We conclude that the ``ice rule'' is the base of mechanism regulating this process.