Thermal conductivity (kappa) of a distorted spin diamond-chain system, Cu_3(CO_3)_2(OH)_2, is studied at low temperatures down to 0.3 K and in magnetic fields up to 14 T. In zero field, the kappa(T) curve with heat current along the chain direction has very small magnitudes and shows a pronounced three-peak structure. The magnetic fields along and perpendicular to the chains change the kappa strongly in a way having good correspondence to the changes of magnetic specific heat in fields. The data analysis based on the Debye model for phononic thermal conductivity indicates that the heat transport is due to phonons and the three-peak structure is caused by two resonant scattering processes by the magnetic excitations. In particular, the spin excitations of the chain subsystem are strongly scattering phonons rather than transporting heat.
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