QN3D: A three-dimensional quasi-neutral hybrid particle-in-cell code with applications to the tilt mode instability in field reversed configurations. The tilt mode instability of a field reversed configuration (FRC) is discussed. Previous numerical models have not adequately explained the behavior of this mode. A particle-in-cell (PIC) model in Cartesian coordinates is introduced with an explanation as to why it represents the physics of FRC’s more closely. The PIC model is implemented in a FORTRAN code, QN3D. The major elements of this code are presented including the many techniques required for its optimization. We discuss many of the major factors in optimization that are dependent upon features of the Cray-2 multiprocessor. Testing of the code is presented in three phases. First, single particle motion is analyzed. Next the normal modes are calculated and simulated. Finally, QN3D is applied to the rigid rotor problem. These tests indicate that the code is suited to model plasma phenomena in the parameter regimes of interest. Lastly, two cases of the tilt mode problem are treated. The results match current experiments and confirm our initial hypothesis as to why other models are not adequate.