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We study the magnetic excitations of the trimeric magnetic compoundsA$_3$Cu$_3$(PO$_4$)$_4$ (A = Ca, Sr, Pb). The spectra are analyzed in terms ofthe Heisenberg model and a generic spin Hamiltonian that accounts for thechanges in valence electrons distribution along the bonds among magnetic ions.The analytical results obtained in the framework of both Hamiltonians arecompared to each other and to the available experimental measurements. Theresults based on our model show better agreement with the experimental datathan those obtained with the aid of the Heisenberg model. For all trimers, ouranalysis reveals the existence of one thin energy band referring to theflatness of observed excitation peaks.

Magnetic excitations in the trimeric compounds A$_3$Cu$_3$(PO$_4$)$_4$ (A = Ca, Sr, Pb)