The European Physical Journal Plus

We study an intense space charge dominated beam in the realm of beam optics of particle accelerators. We assume an anisotropic beam with different external focussing constants and with different phase space emittances in the two transverse directions. We consider the root mean square moment equations of the beam with linear transverse coupling of particle motion ( $$x$$ and $$y$$ ). Magnetic focussing fields and the space charge Coulomb’s self-field forces are expressed in a matrix format and describe the governing equations for the beam outer envelope. By linearizing the equations, we investigate the collective response of ensemble of particles. This response is stated as coherent mode frequencies for the second-order even (envelope) and odd (tilting) oscillation modes of the beam cross section. We discuss the main features of the eigenfrequencies in detail. The approach known as Chernin’s matrix formalism is conceptually equivalent to the second-order moments of the full Vlasov–Poisson kinetic equation in four-dimensional phase space. We show that the relations obtained via matrix approach are in full agreement with the eigenfrequencies of transverse oscillations obtained by the linearized kinetic Vlasov–Poisson equations.

The Bohr Hamiltonian for the triaxial nuclei with Pöschl-Teller potential for the $\beta$ -part and a harmonic oscillator around $\gamma = \frac{\pi}{6}$ for the $\gamma$ -part is solved. An approximate separation of the variables occurs when the potential has the form $v(\beta,\gamma)=u(\beta)+v(\gamma)$ . The $\beta$ -part has been solved using the Nikiforov-Uvarov method. The total wave function has been derived and an expression for the total energy is represented. The electric quadrupole transition rates are evaluated. The spectra and B(E2) s are compared to the experimental data.

We compute the spectrum of quasinormal frequencies of regular black holes obtained in the presence of non-linear electrodynamics. In particular, we perturb the black hole with a minimally coupled massive scalar field, and we study the corresponding perturbations adopting the 6th-order WKB approximation. We analyze in detail the impact on the spectrum of the charge of the black hole, the quantum number of angular momentum and the overtone number. All modes are found to be stable. Finally, a comparison with other charged black holes is made, and an analytical expression for the quasinormal spectrum in the eikonal limit is provided.

In this contribution, we study the quasi-normal modes of a Generic-class of a regular black hole with a magnetic charge in nonlinear electrodynamics, considering scalar and electromagnetic perturbations. The Generic-class contains the Bardeen-class, Hayward-class, and New-class solutions, it can represent a black hole with two horizons or one horizon. First, we obtain the critical values of the magnetic charge and mass; Then, using the third–order WKB approximation, we can determine the dependence of the quasi-normal modes with the parameters of the Generic-class. Finally, the transmission and reflection coefficients of the scattered wave in the third–order WKB approximation are calculated.

This article is focused on the intermediate mass fragments emission induced by the Kr projectile at relativistic energy in nuclear emulsion. The study of correlation between intermediate mass fragments and the total charge confined in fragments for $$Z\ge $$ 2 with different targets (such as H, CNO and AgBr) shows that there is increase in mean multiplicity of intermediate mass fragments with increase in target mass. The observed anti-correlation between the mean multiplicities of particles emitted from the target and the bound system charge $$(Z_{\mathrm{b}})$$ reflects the relation of the $$Z_{\mathrm{b}}$$ to the centrality of the collision.

This paper presents the Ayon-Beato Garcia black hole (ABG) in AdS spacetime known as regular black hole solutions at origin ( $$r\rightarrow 0$$ ); it has anti-de Sitter and de Sitter form depending upon the charge-to-mass ratio $$ >1$$ and $$<1$$ , respectively. To investigate the solution, we compute the thermodynamic quantities mass, temperature, entropy and specific heat in extended phase space. The cosmological constant $$\Lambda $$ is expressed in terms of thermodynamical pressure P as $$P=-\Lambda /8\pi $$ . By observing the behaviour of the Gibbs free energy, we study the first- and second-order phase transition. The first-order phase transition is seen at below the critical point, and it terminates at the critical point where the phase transition becomes second order which is characterized by the critical exponents. We also explore the equation of state for black holes and their critical exponents and prove it to be the same those of Van der Waals fluid. In the case of regular black hole, the area law is violated. For the ABG black hole, a new thermodynamics quantity $$\phi _M$$ is defined which is required for the consistency of both first law of black hole thermodynamics and Smarr relation.

We investigate the nonlinear propagation of multidimensional magnetosonic shock waves (MSWs) in a dissipative quantum magnetoplasma. A macroscopic quantum magnetohydrodynamic (QMHD) model is used to include the quantum force associated with the Bohm potential, the pressure-like spin force, the exchange and correlation force of electrons, as well as the dissipative force due to the kinematic viscosity of ions and the magnetic diffusivity. The effects of these forces on the properties of arbitrary amplitude MSWs are examined numerically. It is found that the contribution from the exchange-correlation force appears to be dominant over those from the pressure gradient and the other similar quantum forces, and it results in a transition from monotonic to oscillatory shocks in the presence of either the ion kinematic viscosity or the magnetic diffusivity.

The superposition of different statistics defines the concept of superstatistics, which makes an equilibrium statistical mechanics incorporate fluctuations. Herein, we study the thermodynamical properties of the anharmonic oscillator for Dirac equation in commutative and noncommutative (NC) spaces using q-deformed superstatistics. By employing the biconfluent Heun functions, the energy spectrum and the wave functions of the considered system are expressed in explicit form. Moreover, the thermodynamical properties of the anharmonic oscillator ensemble in terms of the effective Boltzmann factor are derived, and in order to illustrate the results, some figures are plotted for the clarity of our results.