Dispersive features of electrostatic waves in bounded quantum plasma under the effect of ionization

Abstract

Bounded plasma occur in waveguide of nanodevices with dielectric boundaries and the dimension of nanodevices control the frequency of oscillation and particle acceleration. A system with cylindrical bounded quantum plasma is used to study the electrostatic wave instability in the presence of magnetic field. Bohm potential, exchange-correlation potential and Fermi pressure significantly affect the characteristic frequency of oscillation of particle in bounded plasma. Using quantum hydrodynamic model, basic equations of cylindrical bounded quantum plasma are constructed and linearized under the effect of ionization rate. Dispersion relation for growing waves is obtained, which shows dependence on ionization rate, magnetic field, number density, wave vector and geometry of cylindrical waveguide. We investigated that growth rate increases with magnetic field, ionization rate, number density and poles of Bessel’s function, whereas it decreases with wave vector and radius of waveguide. The present investigation is performed on the basis of numerical parameters of astrophysical plasma.

Appendix

The coefficients used in Equation (11) are given below:

$$\begin{aligned}{ A}_0&=i\omega _{ce}\omega _{pe}^2\mu ^2, \end{aligned}$$

(12)

$$\begin{aligned} B_0&=(-\alpha ^4\mu ^2-V_f^4k^4\mu ^2-\alpha \omega _{ce}\omega _{pe}^2 \mu ^2-k_\Vert ^2\alpha ^4-k_\Vert ^2V_f^4k^4 \nonumber \\&\quad +2\alpha ^2V_f^2k^2k_{\Vert }^2-k^4V_f^2\omega _{pe}^2 +\alpha ^2k^2\omega _{pe}^2+2\alpha ^2V_f^2k^2\mu ^2), \end{aligned}$$

(13)

$$\begin{aligned}C_0&=\mu ^2(-2i\alpha ^5-2i\alpha V_f^4k^4+2i\alpha ^3V_f^2k^2-i\omega _{ci}^2 \omega _{ce}\omega _{pe}^2 \nonumber \\&\quad -2i\alpha ^2\omega _{ce}\omega _{pe}^2+4i\alpha ^2\omega _{ce}\omega _{pe}^2 -k_{\Vert }^2(2i\alpha ^5+2i\alpha V_f^4k^4 \nonumber \\&\quad -4i\alpha ^3V_f^2k^2)-k^2(2i\alpha V_f^2k^2\omega _{pe}^2-2i\alpha ^3\omega _{pe}^2), \end{aligned}$$

(14)

$$\begin{aligned}D_0=\mu ^2(\omega _{ci}^2\alpha ^4+\omega _{ci}^2V_f^4k^4 - 2\omega _{ci}^2\alpha ^2V_f^2k^2-\alpha \omega _{ci}^2\omega _{ce}\omega _{pe}^2\nonumber \\\quad -2\alpha ^3\omega _{ce}\omega _{pe}^2+\alpha ^4\omega _{pi}^2 +V_f^4k^4\omega _{pi}^2-2\alpha ^2V_f^2k^2\omega _{pi}^2)\nonumber \\\quad -k_{\Vert }^2(-\alpha ^4\omega _{pi}^2-\omega _{pi}^2V_f^4k^4+2\alpha ^2V_f^2k^2\omega _{pi}^2-\alpha ^4\omega _{ci}^2\nonumber \\\quad -V_f^4k^4\omega _{ci}^2+2\alpha ^2V_f^2k^2\omega _{ci}^2)-k^2(\alpha ^2\omega _{ci}^2\omega _{pe}^2\nonumber \\\quad -\omega _{ci}^2\omega _{pe}^2V_f^2k^2), \end{aligned}$$

(15)

$$\begin{aligned}E_0&=\mu ^2i(-2\alpha ^7-2\alpha ^3V_f^4k^4+2\alpha ^5V_f^2k^2-\alpha ^2\omega _{ci}^2\omega _{ce}\omega _{pe}^2\nonumber \\&\quad +\alpha ^4\omega _{ce}\omega _{e}^2+2\alpha ^5\omega _{pi}^2+2\alpha V_f^4k^4\omega _{pi}^2-4\alpha ^3V_f^2k^2\omega _{pi}^2\nonumber \\&\quad +\omega _{ci}\alpha ^4\omega _{pi}^2+\omega _{ci}V_f^4k^4\omega _{pi}^2-2\alpha ^2\omega _{ci}\omega _{pi}^2V_f^2k^2)\nonumber \\&\quad -k_{\Vert }^2i(-2\alpha ^5\omega _{pi}^2-2\alpha V_f^4k^4\omega _{pi}^2+4\alpha ^3V_f^2k^2\omega _{pi}^2+2\alpha ^7\nonumber \\&\quad +2\alpha ^3V_f^4k^4-4\alpha ^5V_f^2k^2)-k^2i(-2\alpha ^5\omega _{pe}^2\nonumber \\&\quad +2\alpha ^3V_f^2k^2\omega _{pe}^2), \end{aligned}$$

(16)

$$\begin{aligned}F_0&=\mu ^2(-\omega _{ci}^2\alpha ^6-\alpha ^2V_f^4k^4\omega _{ci}^2-2\alpha ^4\omega _{ci}^2V_f^2k^2+\alpha ^8\nonumber \\&\quad +\alpha ^4V_f^4k^4-2\alpha ^6V_f^2k^2-\alpha ^3\omega _{ci}^2\omega _{ce}\omega _{pe}^2-\alpha ^5\omega _{ce}\omega _{pe}^2\nonumber \\&\quad -\alpha ^6\omega _{pi}^2\nonumber -\alpha ^2V_f^4k^4\omega _{pi}^2+2\alpha ^4V_f^2k^2\omega _{pi}^2-\alpha ^5\omega _{ci}\omega _{pi}^2\nonumber \\&\quad -\alpha \omega _{ci}V_f^4k^4\omega _{pi}^2+2\alpha ^3\omega _{ci}V_f^2k^2\omega _{pi}^2)-k_{\Vert }^2{(\alpha }^4\omega _{pi}^2\omega _{ci}^2\nonumber \\&\quad +\alpha ^6\omega _{pi}^2+\omega _{ci}^2\omega _{pi}^2V_f^4k^4 +{ \alpha }^2V_f^4k^4\omega _{pi}^2-\alpha ^6\omega _{ci}^2\nonumber \\&\quad -2\alpha ^2V_f^2k^2\omega _{ci}^2\omega _{pi}^2-2\alpha ^4V_f^2k^2\omega _{pi}^2-\alpha ^2V_f^4k^4\omega _{ci}^2\nonumber \\&\quad + 2\alpha ^4V_f^2k^2\omega _{ci}^2-\alpha ^8-\alpha ^4V_f^4k^4+2\alpha ^6V_f^2k^2)\nonumber \\&\quad -k^2(\alpha ^4\omega _{ci}^2\omega _{pe}^2-\alpha ^2V_f^2k^2\omega _{ci}^2\omega _{pe}^2+ \alpha ^6\omega _{pe}^2\nonumber \\&\quad -\alpha ^4V_f^2k^2\omega _{pe}^2). \end{aligned}$$

(17)