The elementary excitation spectra of the one-dimensional spin-1/2 XY model in the ferrimagnetic diamond chain at low temperature are calculated by using invariant eigenvector in this paper. And the elementary excitation energies in different cases are discussed. Therefore, analytic solutions of the three critical magnetic field intensities HC1, HC2 and Hpeak in the system are given. It is found that the analytic solutions of three critical magnetic field intensities are correct from the law of the magnetization changing with temperature under different external magnetic fields, and it is explained by the contributions of the three elementary excitations to the magnetization. The external magnetic field dependent magnetization presents a 1/3 magnetized plateau at low temperature. The variation of magnetic susceptibility either with temperature or with external magnetic field shows a double peak structure, this phenomenon indicates that the double peak structure originates from the competition among the ferromagnetic exchange interaction energy of intramolecular electronic spin parallel arrangement in dimer, the antiferromagnetic exchange interaction energy of intermolecular electronic spin antiparallel arrangement in dimer-monomer, the thermal disorder energy and the spin magnetic moment potential energy related to external magnetic field.