The free energy contained in electron drift, electron collision, and gradient of plasma density, temperature, magnetic can trigger the different frequency and wavelength instabilities in hall thrusters. The instabilities will destroy the stable discharge of plasma, affect the matching degree between the thruster and the power processing unit, and down the performance of the thruster. Based on this, the instabilities triggered by electron collision and gradient of plasma density and magnetic field in the hall thruster is studied by using dispersion relation derived from the fluid model. The results show that: 1) The instabilities can be excited at any axial position from the near anode region of the thruster to the plume region when the effect of electron inertia、electron collision with neutral atoms and electron drift are included in the model. The transition of the lower-hybrid mode excited by electron collision into the ion sound mode take place with the azimuthal wavenumber ky is increasing. Where k=2π/λ ，λ is the wave length. The real frequency ωr corresponding to the maximum growth rate γmax slightly decreases with collision frequency increasing for ky=10m-1. However, the maximum real frequency and real frequency ωr corresponding to the maximum growth rate γmax will not change with collision frequency varying for ky=300m-1. Independent on the size of ky, the growth rate of mode triggered by electron collision increases with collision frequency increasing. 2) The plasma density gradient effect plays the dominant role in driving instabilities when the electron inertia, electron-neutral collisions and plasma density gradient are simultaneously included in the model. The dynamic behavior of the model does not change with the increasing of ky, but the eigenvalue of the model increases with the ky increasing. Since the sign of anti-drift frequency induced by the plasma density gradient is changed, and the mode eigenvalue have the opposite change trend on both sides of point κN=0. When the sign of ωs and ωr are opposite, the density gradient effect has a stabilization effect on instability excitation (κN＞0). When the sign of ωs and ωr are same, the density gradient effect enhances the excitation of instability (κN＜0) .3) If the gradient of the plasma density and magnetic field, electron inertia and electron-neutral collisions are included in the dispersion, the mode eigenvalue relies on the size of electron drift frequency, and the diamagnetic drift frequency induced by the gradient of density and magnetic field. When the density gradient and magnetic gradient effect are considered, there is a stable window in the discharge channel. However，if the electron inertia and electron-neutral collisions are also included, the stable window will disappear.
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