content_image/375276.jpg content_image/375278.jpg content_image/375284.jpg
<img src='content_image/375276.jpg'>

<img src='content_image/375278.jpg'>

K − ( θ ) are unstable. The unstable domain defined by Eqs. (12–13) is graphically summarized in Figure 4 when one recognizes immediately the domain outlined in Figs. 1. 

## IV. GROWTH RATE ANALYSIS 

It turns out that the dispersion equation from Eq. (8) is biquadratic and can be solved exactly for any parameter set. In terms of Ω $^{2}$, the two roots are, 

<img src='content_image/375284.jpg'>

Unstable modes clearly arises from Ω 2 $_{1}$, since Ω 2 $_{2}$> 0 for any parameters. Of course, the sign analysis of Ω 2 1 yields the unstable domain already explained. 

Analytic results are acessible not only in the electrostatic approximation. Indeed, allowing for magnetic field perturbations and following the steps outlined in Section II, the exact 

