The Modulated Crystal Structure of K₂V₃O₈

Weak lattice distortions can tune exchange pathways and magnetic interactions in square-lattice quantum magnets. K₂V₃O₈ is a mixed valence (V⁴⁺/V⁵⁺) fresnoite oxide that exhibits strong spin-lattice coupling at low temperature. We combine single-crystal neutron diffraction (90 K) and laboratory X-ray diffraction (50 K) to solve the low-temperature structure as an orthorhombic (3 + 1)D incommensurately modulated phase in superspace group Cmm2(β,0,1/2)0s0 [No. 196]. What initially appeared as two independent modulation vectors, q₁ = 0.3132(6)[110] + 1/2c* and q₂ = 0.3132(6)[11̅0] + 1/2c*, are more naturally described as a single one-dimensional modulation wave q = 0.626(1)a* + 1/2c* in a C-centered orthorhombic lattice, related to the parent tetragonal cell by the transformation a + b, -a + b, c. Refinement with a 4-fold rotational twin inherited from the P4bm parent structure solves oxygen-dominated framework distortions and K⁺ displacements. A de Wolff section (t = 0.40) enables a symmetry-mode decomposition, identifying three dominant mm2 (C_2v) modes: GM3 for framework tilt, A5 for interlayer shear, and Z5 for c-axis breathing. The mode-resolved structure provides a unified, symmetry-based explanation for reported low-temperature Raman and IR anomalies and clarifies the structural origin of the spin-lattice coupling in the S = 1/2 two-dimensional quantum spin compound.