Nonrelativistic Charged Particle Research Articles

On grazing scattering of a charged particle by a flat solid surface

The sliding motion of a quantum nonrelativistic charged particle near a flat surface of a solid body is considered within the framework of the effective potential approach, which consists of the interaction of the particle with surface atoms and excitations, i.e. within the framework of the so-called surface channeling formalism. We have demonstrated that the imaginary part of this potential is amenable to classical interpretation. A detailed analysis of the angular scattering of a positively charged particle is presented, including surface channeling requirements.

On charged particle dynamics near flat solid surface

The motion of a non-relativistic charged particle near flat solid surface at small sliding angles is examined within the formalism of the effective potential composed by interaction of a particle with surface atoms and excited plasmons. As known, the presence of curvature in a solid surface can result in forming a set of discrete quantum states for a particle in the potential well of interaction with the surface.In this work we demonstrate that, at the glancing to flat-surface motion of fast charged particles, small channeling oscillations can be also observed for some special conditions. It is shown that the effective interaction potential represents a shallow well capable to hold a particle in a bound state. We succeeded to analytically determine the energy levels of a particle in such a potential well, as well as the critical angle for a particle to be trapped, in terms of known critical angle of channeling in crystals and a newly proposed function for simplification.

Evolution of an accelerated charged vortex particle in an inhomogeneous magnetic lens

We present a detailed analysis of the capture and acceleration of a nonrelativistic charged vortex particle (electron, positron, proton, etc.) with an orbital angular momentum in the field of an axisymmetric electromagnetic lens, typical for a linear accelerator. We account for the acceleration and the inhomogeneity of both electric and magnetic fields that may arise from real-life imperfections. We establish the conditions for this wave packet to be captured and successfully transported through the lens. We describe the transition process and explain how a free Laguerre-Gaussian packet can be captured into the Landau state of the lens and how its structure is preserved for all moments in time. We illustrate the developed formalism with several representative examples.

A product picture for quantum electrodynamics

We present a short account of our work to provide quantum electrodynamics (QED) with a product picture. We aim to complement the longer exposition in a recent paper in Foundations of Physics and to help to make that work more accessible. The product picture is a formulation of QED, equivalent to standard Coulomb gauge QED, in which the Hilbert space arises as (a certain physical subspace of) a product of a Hilbert space for the electromagnetic field and a Hilbert space for charged matter (i.e., the Dirac field) and the Hamiltonian arises as the sum of an electromagnetic Hamiltonian, a charged matter Hamiltonian, and an interaction term. (The Coulomb gauge formulation of QED is not a product picture because, in it, the longitudinal part of the electromagnetic field is made out of charged matter operators.) We also recall a “Contradictory Commutator Theorem” for QED, which exposes flaws in previous attempts at temporal gauge quantization of QED, and we explain how our product picture appears to offer a way to overcome those flaws. Additionally, we discuss the extent to which that theorem may be generalized to Yang–Mills fields. We also develop a product picture for nonrelativistic charged particles in interaction with the electromagnetic field and point out how this leads to a novel way of thinking about the theory of many nonrelativistic electrically charged particles with Coulomb interactions. In an afterword, we explain how the provision of a product picture for QED gives hope that one will be able likewise to have a product picture for (Yang Mills and for) quantum gravity—the latter being needed to make sense of the author's matter-gravity entanglement hypothesis. Also, we briefly discuss some similarities and differences between that hypothesis and its predictions and ideas of Roger Penrose related to a possible role of gravity in quantum state reduction and related to cosmological entropy.

Evolution of energy and magnetic moment of a quantum charged particle in power-decaying magnetic fields

We consider a quantum spinless nonrelativistic charged particle moving in the xy plane under the action of a homogeneous time-dependent magnetic field B(t)=B0(1+t/t0)−1−g, directed along the z-axis and described by means of the vector potential A(t)=B(t)[−y,x]/2. Assuming that the particle was initially in the thermal equilibrium state with a negligible coupling to a reservoir, we obtain exact formulas for the time-dependent mean values of the energy and magnetic moment in terms of the Bessel functions. Different regimes of the evolution are discovered and illustrated in several figures. The energy goes asymptotically to a finite value if g>0 (“fast” decay), while it goes asymptotically to zero if g≤0 (“slow” decay). The dependence on parameter t0 practically disappears when 1+g is close to zero value (“superslow” decay). The mean magnetic moment goes to zero for g>1, while it grows unlimitedly if g<1.

Pauli oscillator in noncommutative space

In this study, we investigate the Pauli oscillator in a noncommutative space. In other words, we derive wave function and energy spectrum of a spin half non-relativistic charged particle that is moving under a constant magnetic field with an oscillator potential in noncommutative space. We obtain critical values of the deformation parameter and the magnetic field, which they counteract the normal and anomalous Zeeman effects. Moreover, we find that the deformation parameter has to be smaller than [Formula: see text]. Then, we derive the Helmholtz free energy, internal energy, specific heat and entropy functions of the Pauli oscillator in the non commutative space. With graphical methods, at first, we compare these functions with the ordinary ones, and then, we demonstrate the effects of magnetic field on these thermodynamic functions in the commutative and noncommutative space, respectively.

Energy and Magnetic Moment of a Quantum Charged Particle in Time-Dependent Magnetic and Electric Fields of Circular and Plane Solenoids.

We consider a quantum spinless nonrelativistic charged particle moving in the plane under the action of a time-dependent magnetic field, described by means of the linear vector potential , with two fixed values of the gauge parameter : (the circular gauge) and (the Landau gauge). While the magnetic field is the same in all the cases, the systems with different values of the gauge parameter are not equivalent for nonstationary magnetic fields due to different structures of induced electric fields, whose lines of force are circles for and straight lines for . We derive general formulas for the time-dependent mean values of the energy and magnetic moment, as well as for their variances, for an arbitrary function . They are expressed in terms of solutions to the classical equation of motion , with . Explicit results are found in the cases of the sudden jump of magnetic field, the parametric resonance, the adiabatic evolution, and for several specific functions , when solutions can be expressed in terms of elementary or hypergeometric functions. These examples show that the evolution of the mentioned mean values can be rather different for the two gauges, if the evolution is not adiabatic. It appears that the adiabatic approximation fails when the magnetic field goes to zero. Moreover, the sudden jump approximation can fail in this case as well. The case of a slowly varying field changing its sign seems especially interesting. In all the cases, fluctuations of the magnetic moment are very strong, frequently exceeding the square of the mean value.

Remarks on fermions in a dipole magnetic field

This work is a continuation of our recent study of non-relativistic charged particles, confined to a sphere enclosing a magnetic dipole at its center [1]. In this sequel, we extend our computations in two significant ways. The first is to a relativistic spin- frac{1}{2} fermion and the second concerns the interpretation of the physics. Whereas in [1] we speculated on the possibility of observing such condensed matter systems in the astrophysics of extreme magnetic sources such as neutron stars, the physical systems in this study are more down-to-earth objects such as a C60 fullerine enclosing a current loop. We unpack some of the details of our previous analysis for the spinless fermion on the dipole sphere and adapt it to solve the eigenvalue problem for the single-particle Dirac Hamiltonian. In the strong-field/small-radius limit, the spectrum of the spin- frac{1}{2} Hamiltonian, like the spinless case, exhibits a Landau level structure in the |m| ≪ Q regime. It features a new, additional (approximately) zero-energy lowest Landau level which persists into the |m| < Q regime. As in the spinless system, the spectrum exhibits level-crossing as the strength of the magnetic field increases, with the wavefunctions localising at the poles in the strong-field/small-radius limit.

Drift Motion of Charged Particles in Inhomogeneous Magnetic and Strong Electric Fields

Distinctive features of the drift motion of a nonrelativistic charged particle in slowly varying magnetic and strong electric fields, for which the assumption that the electric drift velocity is small as compared to the total particle velocity is non-applicable, are studied. The variational principles of the drift motion are extended to the case of the strong electric field. The generalized Littlejohn’s Lagrangian is obtained and the extended set of drift equations is derived. The possibility of particle acceleration due to the drift motion along the strong electric field is demonstrated.

Observation of the Relativistic Reversal of the Ponderomotive Potential.

The secular dynamics of a nonrelativistic charged particle in an electromagnetic wave can be described by the ponderomotive potential. Although ponderomotive electron-laser interactions at relativistic velocities are important for emerging technologies from laser-based particle accelerators to laser-enhanced electron microscopy, the effects of special relativity on the interaction have only been studied theoretically. Here, we use a transmission electron microscope to measure the position-dependent phase shift imparted to a relativistic electron wave function when it traverses a standing laser wave. The kinetic energy of the electrons is varied between 80 and 300keV, and the laser standing wave has a continuous-wave intensity of 175 GW/cm^{2}. In contrast to the nonrelativistic case, we demonstrate that the phase shift depends on both the electron velocity and the wave polarization, confirming the predictions of a quasiclassical theory of the interaction. Remarkably, if the electron's speed is greater than 1/sqrt[2] of the speed of light, the phase shift at the electric field nodes of the wave can exceed that at the antinodes. In this case there exists a polarization such that the phase shifts at the nodes and antinodes are equal, and the electron does not experience Kapitza-Dirac diffraction. Our results thus provide new capabilities for coherent electron beam manipulation.

A pure non-neutral plasma under an external harmonic field: equilibrium thermodynamics and chaos

Motivated by the precedent study of Ordenes-Huanca and Velazquez (2016 J. Stat. Mech. 093303), we address the study of a simple model of a pure non-neutral plasma: a system of identical non-relativistic charged particles confined under an external harmonic field with frequency . We perform the equilibrium thermo-statistical analysis in the framework of continuum approximation. This study reveals the existence of two asymptotic limits: the known Brillouin steady state at zero temperature, and the gas of harmonic oscillators in the limit of high temperatures. The non-extensive character of this model is evidenced by the associated thermodynamic limit, , which coincides with the thermodynamic limit of a self-gravitating system of non-relativistic point particles in presence of Newtonian gravitation. Afterwards, the dynamics of this model is analyzed through numerical simulations. It is verified the agreement of thermo-statistical estimations and the temporal expectation values of the same macroscopic observables. The system chaoticity is addressed via numerical computation of Lyapunov exponents in the framework of the known tangent dynamics. The temperature dependence of Lyapunov exponent approaches to zero in the two asymptotic limits of this model, reaching its maximum during the transit between them. The chaos of the present model is very strong, since its rate is faster than the characteristic timescale of the microscopic dynamics . A qualitative analysis suggests that such a strong chaoticity cannot be explained in terms of collision events because of their respective characteristic timescales are quite different, and .

Optimizing the Longitudinal Dynamics of a Non-Relativistic Charged Particle Bunch in the Field of a Traveling Wave

Using the Lagrangian equation for the envelopes of the longitudinal motion of a non-relativistic charged particle bunch in the field of a traveling wave, we obtained solutions to the problem of optimizing this motion. The pronounced extrema of the bunch size and its spectrum were found and confirmed by test calculations. The technique developed and described in the paper can be of high interest for solving the problems of optimizing the dynamics of charged particle beams in resonant linear accelerators for ions at low energies.

Rising and lowering operator approach to the problem of a charged particle in a uniform magnetic field

In this paper, we analyse the quantization of cyclotron orbits for a 2D non-relativistic and spinless charged particle under the influence of a static and uniform magnetic field, the Landau problem. We follow the algebraic approach of rising and lowering operators. The Hamiltonian is expressed in terms of the usual rising and lowering operators of an isotropic quantum oscillator. Through a Bogoliubov transformation, we show that the Hamiltonian may be reduced to the one of a quantum oscillator that accounts for the formation of infinitely degenerate Landau levels. The explicit form of the position-space wave functions of Fock–Darwin states for a vanishing parabolic confinement potential is worked out by applying the techniques of ladder operators and the induction method. To explore the internal dynamics of the electron states, we compute the Fock–Darwin states’ density currents expressing them in terms of the ladder operators. In the supplementary material, we provide the reader with Mathematica algorithm that can help perform some of the calculations exposed in the paper. The charged particle in a magnetic field is used to explore some of the fundamental concepts of quantum mechanics as commuting operators, simultaneous eigenstates, degeneracy, rising and lowering operators, etc. This approach can be used as supplementary material in undergraduate final-year courses like quantum mechanics or solid state physics.

The oscillations of charged particle beam in crossed magnetic fields

In the present paper we are going to study the decrease of the velocity and density oscillations of the beam in the non-relativistic charged particle beams in the chosen configuration of the crossed field. This paper examines using a hydrodynamic approach and the Maxwell’s processes inside the cold beam. Results show that in the framework of the present model one can obtain physical conditions when it is possible to observe the cooling effect for the charged particle beam.

Particle dynamics in a magnetic field: symplectic reduction and classification of singular trajectories

The paper studies the dynamics of a classical non-relativistic charged particle moving in a superposition of force-free and constant magnetic fields. Using the first integrals of the problem, a symplectic reduction of the corresponding Newton-Lorentz equations to an auxiliary two-dimensional Hamiltonian system is performed. By the linearisation method, the types of singular points of the reduced Hamiltonian system are classified. The results obtained are applied to the problem of studying the trajectories of the original Hamiltonian system that are close to singular trajectories.

Quantum mechanics of bending of a nonrelativistic charged particle beam by a dipole magnet

Quantum mechanics of bending of a nonrelativistic monoenergetic charged particle beam by a dipole magnet is studied in the paraxial approximation. The transfer map for the position and momentum components of a particle of the beam between two transverse planes at different points on the curved optic axis of the system is derived starting with the nonrelativistic Schrödinger equation. It is found that the quantum transfer map contains the classical transfer map as the main part and there are tiny quantum correction terms. The negligibly small quantum corrections explain the remarkable success of classical mechanics in charged particle beam optics.

Magnetic Schrödinger operators as the quasi-classical limit of Pauli–Fierz-type models

We study the quasi-classical limit of the Pauli–Fierz model: the system is composed of finitelymany non-relativistic charged particles interactingwith a bosonic radiation field. We trace out the degrees of freedom of the field, and consider the classical limit of the latter. We prove that the partial trace of the full Hamiltonian converges, in resolvent sense, to an effective Schrödinger operator with magnetic field and a corrective electric potential that depends on the field configuration. Furthermore, we prove the convergence of the ground state energy of the microscopic systemto the infimum over all possible classical field configurations of the ground state energy of the effective Schrödinger operator.

Polarization of Diffraction Radiation on a Conducting Sphere and a Hemispheric Bulge in a Conducting Plane

The polarization characteristics of diffraction radiation appearing during the uniform motion of a nonrelativistic charged particle near an ideally conducting sphere and also a hemispherical bulge in a conducting plane are calculated. Our previously developed approach based on the method of images, which is known in electrostatics, is used. It is shown that, in the low-frequency range, radiation at a hemisphere is always linearly polarized, while radiation at a sphere is generally characterized by elliptical polarization.

Asymptotic Formulas in the Theory of Diffraction and Transition Radiation on a Conducting Sphere

The diffraction and transition radiation of a nonrelativistic charged particle on a perfectly conducting sphere is described using the method of image charges known from electrostatics. Asymptotic formulas that describe the characteristics of the radiation in the low- and high-frequency regions are constructed. It is shown that this approach in the high-frequency limit can be extended to the case of a relativistic incident particle.

Further Improvement of Reflection Efficiency of a Magnetic Mirror and Replenishment against Loss of Escaping Deuteron Ions

We reported previously the idea to improve reflection-ability of a magnetic mirror by installing a cyclotron resonance space in the front part of the mirror. However, since the previous analysis was insufficient from the examination after that, we complement the following two things in this work: 1) A simpler procedure of design to make a supplemental magnetic mirror with the simplest magnetic configuration, compared with the procedure reported previously. 2) A peculiar characteristic arising only in reflection of a nonrelativistic charged particle (a deuteron ion).