Conditions and growth rate of Rayleigh instability in a Hall thruster under the effect of ionization

Rayleigh instability is investigated in a Hall thruster under the effect of ionization . The instability occurs only when the frequency of the oscillations $\omega $ falls within a frequency band, described by $\omega _{\min } <<\omega <<\omega _{\max } $ where $\omega _{\max } =\frac{\alpha k_y u_0 +\sqrt {(\alpha k_y u_0 )^2-4\alpha (\alpha +v)\left( {\alpha k_y u_0 +\frac{\omega _{pi} ^2\Omega ^2}{\omega _{pe} ^2}} \right)} }{2(\alpha +v)}$, $\omega _{\min } =\frac{\alpha }{2}\left( {1-\frac{v}{k_y u_0 }} \right)+\alpha \sqrt {\frac{1}{4}-\frac{v}{2k_y u_0 }-\frac{\omega _{pi} ^2V_{thE} ^2\left( {\frac{\partial ^2u_0 }{\partial x^2}+\frac{\Omega }{n_0 }\frac{\partial n_0 }{\partial x}} \right)}{\alpha ^2\omega _{pe} ^2u_0 }} $ ,$u_0 $ is the drift velocity of the electrons, $\Omega $ is their gyration frequency under the effect of magnetic field, $k_y $ is the wave propagation constant, $n_0 $ is the plasma density together with ${\partial n_0 } \mathord{\left/ {\vphantom {{\partial n_0 } {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}$ as the density gradient, and $V_{thE} $, $\alpha $, $v$ and $\omega_{p_i} (\omega_{p_e} )$ are the electron thermal velocity, ionization rate, collision frequency and plasma frequency of the ions (electrons), respectively. A relevant Rayleigh equation is derived and solved numerically using fourth-order Runge-Kutta method for investigating the perturbed potential under the effect of ionization rate and collision frequency. It is obtained that the growth rate of the instability gets reduced with the ionization rate and collision frequency.