Generalized non-ideal treatment and growth rates analysis of drift waves instabilities in a collisions-free magnetized dusty plasma

Dust acoustic wave propagation in weak magnetized homogeneous dusty plasmas has been extensively and successfully studied by different authors. Here, it is analyzed the behavior of electrostatic drift waves of very low frequency, appearing as due to the motion of plasma particles across the static magnetic field and the inhomogeneity introduced by the distribution of the particles in the dusty plasma generating the well-known diamagnetic effect. To study this situation, the kinetic Vlasov model in plasmas will be used. However, these models tend to behave unstably driven by the quasi-equilibrium of the diamagnetic current arising by the heavier species of dusty plasma particles. While other authors have used a standard state equation, such as the Van der Waals one, here, a more accurate model has been chosen as the equation of Ree and Hoover, a state equation that can be considered as a new form of the state equation. Together with the multipole approximation of the dispersion function, Z53 [P. Martin et al., J. Math. Phys. 21, 280-283 (1980)] allows a new derivation to obtain the simplest form of the dispersion relation in the form of a polynomial with complex coefficients. This mathematical approach was realized successfully and allowed straightforward calculus of the growth rate due to the drift unstable mode. This problem has been extensively analyzed in this work, including the non-ideal effect of dusty plasmas. These unstable modes are well discriminated and plotted as a function of the density of the grains in dusty plasma and their radius. Temperature gradients and charge fluctuation have been ignored.