The principal objective of this work was to derive a linear dispersion relation considering density, momentum, and energy conservation equations for electrons where we can assess thermal effects along with momentum exchange collisions and explore if this type of elastic collisions could act as an instability mechanism. So, we derived a fourth-degree dispersion relation from a two-fluid plasma of ions (cold, non-collisional, unmagnetized, and singly charged) and electrons (hot, collisional, and magnetized with inertia). Then, to validate the resulting mathematical expression, we applied some simplifying conditions made in other studies, confirming that our 4th-degree dispersion relation is a generalization of some linear models proposed earlier. Thus, we evaluated our model with SPT-100 Hall Thruster data, finding two azimuthal unstable modes with two instability regions each. The first mode, dominant near the anode extending to the beginning of the acceleration region with a frequency spectrum from similar to 3 KHz to similar to 100 KHz, corresponded to an instability related to momentum exchange collisions. The second one, although dominant in the plume region with a frequency spectrum from similar to 0.1 KHz to similar to 0.23 KHz, corresponded to a rotating spoke instability where elastic collisions act as a damping factor, especially inside the discharge chamber where we had the maximum rate of electron-neutral collisions. Furthermore, when we compared our results with those of other dispersion relations, we saw that elastic collisions could represent a source of instability or damping, depending on the nature of the unstable modes. Finally, we also assessed the relative importance of electron inertia over the development of instabilities, finding that when we have a model with temperature perturbations and collisional effects aside from plasma non-uniformities, density, and electrostatic fluctuations, inertia seems to be of minor significance, nevertheless, in simplified models without energy conservation equation and thermal effects, inertia has a relevant impact over the instability.
