Magnetic transition due to the inter-singlet spin-exchange interaction and elastic softening by the interplay of electric quadrupoles in the distorted kagome lattice antiferromagnet Tb₃Ru₄Al₁₂

The distorted kagome lattice antiferromagnet Tb₃Ru₄Al₁₂ with a hexagonal structure has the Néel temperature T_N = 22 K. To clarify the 4 f -electronic state and an influence of electric quadrupoles in Tb₃Ru₄Al₁₂, ultrasonic measurements on a single-crystalline sample at zero magnetic field and under fields were carried. A characteristic elastic softening of the transverse modulus C₆₆ originating from a quadrupole interaction was found. The crystal electric field parameters were determined to reproduce C₆₆, magnetic susceptibilities, and magnetization curves. The obtained level scheme is that the ground and first excited states are singlets, despite the existence of both the magnetic transition and the quadrupole interaction, indicating that Tb₃Ru₄Al₁₂ is a curious compound. The positive sign of the quadrupole-quadrupole coupling constant for C₆₆ indicates a ferroquadrupolar-type interaction of the electric quadrupole O_xy or O² . The anisotropic magnetic field dependencies of T_N in the field along [100] and [001] were also clarified.