The crux of understanding the superconducting mechanism in pressurized Ruddlesden-Popper nickelates hinges on elucidating their structural phases. Under ambient conditions, the trilayer nickelate La4Ni3O10 stabilizes in a twinned monoclinic structure with space group P21/c. Upon heating, it undergoes a structural transition to the tetragonal I4/mmm phase at Ts ~ 1030 K, while a second transition associated with the onset of density-weave (DW) ordering emerges upon cooling below TDW ~ 135 K. Here from pressure-temperature x-ray diffraction on high quality flux-grown single crystals we unequivocally demonstrate a direct tetragonal-to-monoclinic transition with no trace of intermediate orthorhombic Bmab phase. Ab initio density-functional theory calculations as a function of pressure fully corroborate the experimental observations. The transition unfolds as a 2-fold superstructure due to the emergence of commensurate superlattice reflections and can be progressively suppressed from 1030 K down to 20 K under 14 GPa. Notably, from XRD we establish the first observation of weak incommensurate satellite reflections associated with the DW ordering in fluxgrown samples, as previous findings were confined to only to crystals grown by the floating-zone technique. This is further reinforced by Raman spectroscopy that reveal the emergence of additional phonon modes below 130 K, implying a breaking of monoclinic P21/c symmetry.

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