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In crystals lacking space inversion symmetry, antisymmetric exchange interactions can arise, which are generally expressed in the form of D • (S_i x S_j). In normal magnetic ordering by symmetric exchange interactions of the form J S_i•S_j, the magnetic moments are aligned parallel or antiparallel to each other into a collinear structure. However, when the antisymmetric exchange is active, the magnetic moments experience a twisting force, resulting in various types of spiral magnetic orderings. The twisting nature of exchange interactions has recently attracted considerable attention because it often generates nontrivial emergent magnetic structures such as chiral soliton lattices and magnetic skyrmion lattices. They are the crystallization of particle-like spin-swirling objects composed of spiral magnetic modulation waves with a single sense of rotation (helicity), which is considered to be the source of topological stability. Therefore, it is important to observe the helicity of the component waves.

The experimental observation of the magnetic helicity is challenging. Although polarized neutron diffraction is a powerful technique, it requires a delicate instrumental setup. In addition, it is unsuitable for samples that contain neutron-absorbing elements such as Eu and Gd. In this study, we employed resonant X-ray diffraction using a synchrotron radiation photon source. We used a diamond phase-retarder system inserted in the incident beam path to manipulate the incident polarization to left- and right-handed circularly polarized states, which have direct sensitivity to the magnetic helicity through different scattering cross-sections for the opposite sense of rotation. We applied a phase-retarder scan to observe the cycloidal ordering in the noncentrosymmetric magnet EuIrGe₃, with four mirror reflection planes including the four-fold c-axis.

The cycloidal ordering is characterized by a propagation vector (δ, δ, 0.8), where δ=0.012 at 2.0 K. The Fourier component for this structure is either (1, 1, i√2) or (1, 1, - i √2), where the sign is related to the cycloidal helicity. Furthermore, there arise four magnetic domains described by (δ, δ, 0.8), (-δ, δ, 0.8), (-δ, -δ, 0.8), and (δ, -δ, 0.8). The helicities of all the four domains were measured.

The helicities perfectly reflected the C_4v symmetry of the crystal structure. All four Fourier components were related by the 90° rotations and by the mirror reflections. This result clearly shows that the helicity of the cycloid is uniquely selected by the antisymmetric interactions. We believe that this method can be widely applied to the helicity measurements of various types of spiral magnetic systems.

(Written by T. Matsumura on behalf of all the authors.)