As is well known,copper is such an unbelievable element that it can affect the phase transition behaviors of binary TiNi alloy when it displaces Ni element up to near upon 25%.The martensitic transition behaviors of TiNi1-xCux alloys appear from high-temperature cubic B2 phase to intermediate B19 structure with orthorhombic system and then finally to low-temperature B19' phase with monoclinic system with x 10% on cooling,so called two-stage martensitic phase transformation.Whereas,it directly transforms into orthorhombic B19 phase withx 20% on cooling,so called one-stage martensitic phase transformation.The orthorhombic B19 phase becomes final low-temperature phase while monoclinic phase will be unstable on cooling.The electronic structures and the formation energies of various point defects, Mulliken bond orders,etc.are studied for TiNi1-xCuxx alloys,however,the phase transition pathway at an atomic level has not been described at all,and further,the difference in transition pathway between TiNi and Ti1Ni1-xCuxx has not been understood so far.In this work,we optimize the crystal structures of TiNi and Ti50Ni25Cu25 alloys with initial geometry from experimental data.In order to choose the proper positions of Cu atom,we calculate the total energy of each doping system and find the most stable configuration.To study the transformation mechanism of TiNi,we calculate the phonon-dispersion spectra of each phase with both frozen-phonon method and linear response method,and then find the atomic vibrations with the imaginary frequency.Finally,with the help of this atomic vibration direction with negative frequency,we find the intermediate structures by the linear interpolation method and calculate their total energies.The phase transformation of TiNi from cubic to orthorhombic phase is driven by the phonon softening at the M point (0.5,0.5,0) of Brillouin zone.For orthorhombic and monoclinic phase,TiNi has real phonon frequencies for all k points and modes.A barrier of 1.6 meV is calculated between orthorhombic and monoclinic phase while no barrier is found between cubic and orthorhombic phase of TiNi,so it is easy to transform from cubic to orthorhombic and then to monoclinic phase.There exists a potential energy barrier of 10.3 meV at least between orthorhombic and monoclinic phase for Ti50Ni25Cu25,which is too high for its transition to overcome the maximum value of potential energy which corresponds to =93.4.The difference in transition pathway between TiNi and Ti50Ni25Cu25 accords well with the experimental measurement,so that the copper concentration with 25% in binary TiNi alloy will offer a new transition path from cubic to orthorhombic phase.