Pressure and oxygen-isotope substitution on density-wave transitions in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>La</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <mml:msub> <mml:mi>Ni</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:msub> <mml:mi mathvariant="normal">O</mml:mi> <mml:mn>10</mml:mn> </mml:msub> </mml:mrow> </mml:math>

Understanding the interplay between magnetism and superconductivity in nickelate systems is a key objective in condensed matter physics. Gaining microscopic insights into magnetism—particularly as it emerges near superconductivity—requires a synergistic approach that combines complementary experimental techniques with controlled tuning of external parameters. In this paper, we present a systematic investigation of the three-layer Ruddlesden-Popper (RP) nickelate La 4 Ni 3 O 10 using muon-spin rotation/relaxation ( μ SR ) and resistivity measurements. At ambient pressure, two incommensurate spin-density-wave (SDW) transitions are identified at T SDW ≃ 132 K and T * ≃ 80 – 90 K . Comparison of the observed internal magnetic fields with dipole-field calculations reveals a magnetic structure consistent with antiferromagnetically coupled SDW order on the outer two Ni-O layers, with smaller moments on the inner Ni-O layer. Above T * , the moments lie primarily in the a b plane, but below this temperature they undergo a subtle distortion and develop a c -axis component. The internal fields at the muon stopping sites appear abruptly at T SDW , suggesting a first-order-like nature of the SDW transition, which is closely linked to the charge-density wave (CDW) order occurring at the same temperature ( T SDW = T CDW ). Under applied pressure, all transition temperatures—including T SDW , T * , and T CDW —are suppressed at a nearly uniform rate of ≃ − 13 K/GPa. This behavior contrasts with that of the two-layer RP nickelate La 3 Ni 2 O 7 , where pressure enhances the separation between the SDW and CDW transitions. The oxygen-isotope substitution ( O 16 → O 18 ) reveals that the CDW transition temperature shifts to higher values in the O 18 -substituted samples. The isotope effect on T SDW and T * differs significantly. Specifically, when the CDW and SDW orders are intertwined, a notable isotope effect is observed on T SDW , leading to equal transition temperatures and nearly identical isotope shifts for both T CDW and T SDW . In contrast, at T * , where the SDW transition occurs independently of th