Magnetosonic shock waves in magnetized quantum plasma with the evolution of spin-up and spin-down electrons

The quantum hydrodynamic model is used to study the linear and nonlinear properties of small amplitude magnetosonic shock waves in dissipative plasma with degenerate inertialess spin-up and spin-down electrons and inertial classical ions. Spin effects are considered via spin pressure and macroscopic spin magnetization current. A linear dispersion relation is derived analytically and plotted numerically for different plasma parameters such as spin density, polarization ratio, plasma beta, quantum diffraction, spin magnetization energy, and magnetic diffusivity. Employing the standard reductive perturbation technique, a Korteweg–de Vries–Burgers-type equation is derived for small amplitude waves and studied numerically. We have observed that an oscillatory and monotonic shock waves are generated depending upon the plasma configurations. The phase portraits of both oscillatory and monotonic shock waves are also presented. Interestingly, different plasma parameters are found to play a significant role in the transition of oscillatory to monotonic shock waves or vice versa. Most importantly it is found that, the magnetosonic excitations obtained with spin-up and spin-down electrons are significantly different from the usual electron ion quantum plasma. The work presented is related to magnetosonic waves in dense astrophysical environments such as a pulsar magnetosphere and neutron stars.

Figure 1

(a) Real part of the dispersion relation ${\omega }_r$ against $k$ for different values of spin polarization ratio $\kappa$. (b) Imaginary part of the dispersion relation ${\omega }_i$ against $k$ for different values of $\kappa$, where $H_e=0.1$, $\beta =0.5$, ${\epsilon }_0=0.2$, and $\gamma =0.4.$

Figure 2

Real and imaginary parts of the plasma normalized angular frequency of dispersion relation (19) against $k$ for different values of dimensionless plasma variables such as diffusion ($\gamma$), quantum diffraction ($H_e$), plasma beta ($\beta$), and Zeeman energy (${\epsilon }_0$). (a), (b) Plots for different values of $\gamma$ for $H_e=0.1$, $\beta =0.1$, and ${\epsilon }_0=0.5$; (c), (d) plots for different values of $H_e$ where $\beta =0.1$, ${\varepsilon }_0=0.2$, and $\gamma =0.03$; (e), (f) plots for different values of ${\varepsilon }_0$ where $H_e=0.1$, $\beta =0.1$, and $\gamma =0.07$; and (g), (h) plots for different values of $\beta$ where $H_e=0.1$, ${\varepsilon }_0=0.1$, and $\gamma =0.07.$

Figure 3

Phase velocity ($V_0$) defined by Eq. (23) against spin polarization density ratio $\kappa$ for $\beta =0.3$, ${\epsilon }_0=0.5$.

Figure 4

Shock wave represented by Eq. (30) for different values of $\beta$ where $H_e=0.1$, ${\varepsilon }_0=0.6$, ${\gamma }_0=0.01$, $U_0=0.95$.

Figure 5

Shock profile represented by Eq. (30) for different values of spin magnetic energy ${\varepsilon }_0$, where $H_e=0.1$, $\beta =0.15$, ${\gamma }_0=0.01$, $U_0=0.95$.

Figure 6

Shock profile represented by Eq. (30) for different values of quantum diffraction ($H_e$) where ${\varepsilon }_0=0.88$, $\beta =0.25$, ${\gamma }_0=0.01$, $U_0=0.95$.

Figure 7

Shock profile represented by Eq. (30) for different values of plasma diffusivity (${\gamma }_0$) where ${\varepsilon }_0=0.88$, $H_e=0.1$, $\beta =0.25$, $U_0=0.95$.

Figure 8

The phase portraits corresponding to Fig. 4.

Figure 9

The phase portraits corresponding to Fig. 5.

Figure 10

The phase portraits corresponding to Fig. 6.

Figure 11

The phase portraits corresponding to Fig. 7.