Ion-beam–plasma interaction effects on electrostatic solitary wave propagation in ultradense relativistic quantum plasmas

Understanding the transport properties of charged particle beams is important not only from a fundamental point of view but also due to its relevance in a variety of applications. A theoretical model is established in this article, to model the interaction of a tenuous positively charged ion beam with an ultradense quantum electron-ion plasma, by employing a rigorous relativistic quantum-hydrodynamic (fluid plasma) electrostatic model proposed in McKerr et al. [M. McKerr, F. Haas, and I. Kourakis, Phys. Rev. E 90, 033112 (2014)]. A nonlinear analysis is carried out to elucidate the propagation characteristics and the existence conditions of large amplitude electrostatic solitary waves propagating in the plasma in the presence of the beam. Anticipating stationary profile excitations, a pseudomechanical energy balance formalism is adopted to reduce the fluid evolution equation to an ordinary differential equation. Exact solutions are thus obtained numerically, predicting localized excitations (pulses) for all of the plasma state variables, in response to an electrostatic potential disturbance. An ambipolar electric field form is also obtained. Thorough analysis of the reality conditions for all variables is undertaken in order to determine the range of allowed values for the solitonic pulse speed and how it varies as a function of the beam characteristics (beam velocity and density).

Figure 1

The Mach number $M$ is depicted versus (a) the beam velocity $V_{b0}$ (for $\delta =0.01$) and (b) the beam density $\delta$ (taking $V_{b0}=0.2$). We have considered different values of the (equilibrium) electron density $n_{e0}$ (as listed in the inset label) and ${\mu }_b=1$ ($H^{+}$ beam).

Figure 2

The soliton existence region, i.e., the interval of the permitted Mach number $M$ values, is depicted with $n_{e0}$ in units of ${10}^{11}{\mathrm{m}}^{-1}$ (a) for different values of $V_{b0}$ with $\delta =0.01$, (b) for different values of $V_{b0}$ with $\delta =0.2$, (c) for different values of $\delta$ with $V_{b0}=0.2$, and (d) for different values of $\delta$ with $V_{b0}=0.5$.

Figure 3

The pseudopotential $S\left(\varphi \right)$ is depicted in terms of the electrostatic potential $\varphi$ for different values of the unperturbed electron density $n_{e0}$ in two cases: (a) $V_{b0}=0$ and (b) $V_{b0}=0.2$. We have assumed $\delta =0.01, {\mu }_b=1$, and $M=1.2$ everywhere.

Figure 4

The plasma (fluid) state variables are shown in terms of the space variable $X$, for different values of the unperturbed (equilibrium) electron density $n_{e0}$. We have taken $M=1.2, V_{b0}=0.2, \delta =0.01$, and ${\mu }_b=1$.

Figure 5

(a) The pseudopotential function $S\left(\varphi \right)$ is shown in terms of the electrostatic potential $\varphi$, for different values of the beam velocity $V_{b0}$. (b) The maximum pulse amplitude ${\varphi }_m$ is depicted versus the Mach number $M$, for different values of the (equilibrium) beam velocity $V_{b0}$. We have taken $n_{e0}={10}^{11}{m}^{-1}$ (or ${\xi }_0=0.0603798$), ${\mu }_b=1, \delta =0.01$, and $M=1.4$ as indicative values.

Figure 6

The plasma state variables are shown in terms of the space variable $X$, for different values of the ion beam velocity $V_{b0}$, taking $n_0={10}^{11}{\mathrm{m}}^{-1}$ (or ${\xi }_0=0.0603798$), $M=1.4, \delta =0.01$, and ${\mu }_b=1$.

Figure 7

(a) The pseudopotential $S\left(\varphi \right)$ is shown in terms of the electrostatic potential $\varphi$ for different values of the unperturbed beam density $\delta$. (b) The maximum pulse amplitude ${\varphi }_m$ is shown versus the Mach number $M$. We have taken $V_{b0}=0.2, n_{e0}={10}^{11}{m}^{-1}$ (or ${\xi }_0=0.0603798$), ${\mu }_b=1$, and $M=1.2$ as indicative values.

Figure 8

The plasma state variables are shown in terms of the space variable $X$, for different values of the beam density $\delta$, with $n_{e0}={10}^{11}{m}^{-1}$ (or ${\xi }_0=0.0603798$), $M=1.2, V_{b0}=0.2$, and ${\mu }_b=1$.