A full phonon intensity cancellation is reported in a longitudinal polarized inelastic neutron scattering experiment performed on the magnetocaloric compound MnFe$_{4}$Si$_{3}$, a ferromagnet with $T_{Curie}$ $approx$ 305 K. The TA[100] phonon polarized along the $c$-axis measured from the Brillouin zone center $textbf{G}$=(0, 0, 2) is observed only in one ($sigma_{z}^{++}$) of the two non-spin-flip polarization channels and is absent in the other one ($sigma_{z}^{--}$) at low temperatures. This effect disappears at higher temperatures, in the vicinity of $T_{Curie}$, where the phonon is measured in both channels with nonetheless marked different intensities. The effect is understood as originating from nuclear-magnetic interference between the nuclear one-phonon and the magnetovibrational one-phonon scattering cross-sections. The total cancellation reported is accidental, i.e. does not correspond to a systematic effect, as established by measurements in different Brillouin zones.
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