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By combination of two independent approaches, nuclear resonant inelasticX-ray scattering and first-principles calculations in the framework of densityfunctional theory, we determine the element-resolved vibrational density ofstates in the ferromagnetic low temperature and paramagnetic high temperaturephase of LaFe$_{13-x}$Si$_x$. This allows us to derive the lattice andelectronic contribution to the entropy change at the first-order phasetransformation, which are both of considerable magnitude. The change in latticeentropy is dominated by magneto-elastic softening, which originates from theitinerant electron metamagnetism associated with Fe. This counteracts the largevolume change at the transition and leads to an unexpected, cooperativebehavior of magnetic, vibrational and electronic entropy change, which isresponsible for the large magneto- and barocaloric effect observed for thismaterial.

Element-resolved thermodynamics of magnetocaloric LaFe$_{13-x}$Si$_x$