Lattice Green function for electrons in magnetic field

An effective single-band Hamiltonian representing a crystal electron in a uniform magnetic field is constructed from the tight-binding form of a Bloch band by replacing $\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ by the operator $\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}\ensuremath{-}\frac{e\stackrel{\ensuremath{\rightarrow}}{A}}{c}$. The resultant Schr\"odinger equation becomes a finite-difference equation whose eigenvalues can be computed by a matrix method. The magnetic flux which passes through a lattice cell, divided by a flux quantum, yields a dimensionless parameter whose rationality or irrationality highly influences the nature of the computed spectrum. The graph of the spectrum over a wide range of "rational" fields is plotted. A recursive structure is discovered in the graph, which enables a number of theorems to be proven, bearing particularly on the question of continuity. The recursive structure is not unlike that predicted by Azbel', using a continued fraction for the dimensionless parameter. An iterative algorithm for deriving the clustering pattern of the magnetic subbands is given, which follows from the recursive structure. From this algorithm, the nature of the spectrum at an "irrational" field can be deduced; it is seen to be an uncountable but measure-zero set of points (a Cantor set). Despite these-features, it is shown that the graph is continuous as the magnetic field varies. It is also shown how a spectrum with simplified properties can be derived from the rigorously derived spectrum, by introducing a spread in the field values. This spectrum satisfies all the intuitively desirable properties of a spectrum. The spectrum here presented is shown to agree with that predicted by A. Rauh in a completely different model for crystal electrons in a magnetic field. A new type of magnetic "superlattice" is introduced, constructed so that its unit cell intercepts precisely one quantum of flux. It is shown that this cell represents the periodicity of solutions of the difference equation. It is also shown how this superlattice allows the determination of the wave function at nonlattice sites. Evidence is offered that the wave functions belonging to irrational fields are everywhere defined and are continuous in this model, whereas those belonging to rational fields are only defined on a discrete set of points. A method for investigating these predictions experimentally is sketched.

A two-mode cylindrical magnetic field is considered, the potential of which has a minimum. The object of this work is the study of the parameters of an electron beam when it moves in a solenoid field with the longitudinal trap formed by the magnetic field, and the construction of the computational model of the motion of an electron beam. The problem is posed of the stability of the motion of electrons in such solenoid magnetic field. The possibility of obtaining oscillatory modes of particle motion has been studied. It was found that for oscillations of particles with an energy of tens of kiloelectronvolts in the potential well in a well, the field with the amplitude of tens of thousands of Oersteds is required. For the solenoid magnetic field of the solenoid, the formation of electron beam with an energy of 55 keV in the longitudinal and radial directions during its transportation is studied. A section of a magnetron gun was used as the physical object. One possible direction is to combine the two matched magnetic systems of the gun to create the potential magnetic field well. It is shown that, for the chosen conditions, the motion of electrons can be associated with the model of three-dimensional oscillations. In this work, on the basis of the Hamiltonian formalism of the motion of electrons in a magnetic field and an algorithm for numerically finding solutions to the differential equations of dynamics, a software tool is constructed that allows one to obtain arrays of values of particle trajectories in the volume. The use of the software made it possible to simulate the main dependences of the motion of the electron beam in a given two-mode solenoid magnetic field. The results of numerical simulation of electron trajectories in the gradient magnetic field with the point secondary emission cathode located in the middle of the system are presented. The formation of the beam with energy of 55 keV in the radial and longitudinal directions during its transportation in a solenoid magnetic field with a large gradient is considered. For significant time intervals, the possibility of three-dimensional oscillations is shown and the operating modes of the magnetic system are obtained, in which the particle undergoes stable three-dimensional oscillations. The influence of the initial conditions during emission on the occurrence of the reciprocating oscillatory effect has been studied. It is shown that for a given electron energy and fixed magnetic field, the parameter that determines the reflection of a particle, is the polar angle of entry relative to the axis of the cylindrical magnetic field. The dependence of the formation of the final distribution of particles on the amplitude and gradient of the magnetic field along the axis of the system is investigated. The results of numerical simulation on the motion of the electron flow are presented. The characteristics of the resulting electron beam are considered on the basis of a model of electron flow motion. The obtained simulation results show that it is possible to establish the phenomenon of oscillatory-return longitudinal motion under experimental conditions. Keywords: electron beam, magnetron gun, three-dimensional oscillations, electron dynamics, gradient magnetic field, mathematical modeling.

Simultaneous measurements of the gamma ray differential energy spectra, electric field disturbances, and meteorological conditions provided by experimental facilities located at Mt. Aragats in Armenia allows to establish the model of particle acceleration and propagation in thunderstorm atmosphere. We present comparisons of measured and modeled thunderstorm ground enhancements (TGEs). The origin of the majority of TGEs is the MOS process —the modification of energy spectra of cosmic ray electrons in the atmospheric electric fields. The gamma ray differential energy spectra are well described by the power law function with indexes in the range for the electric field strengths 0.8−1.5 kV/cm at altitudes of 3400−5000 m a.s.l. The good agreement of the characteristics of experimental and simulated TGEs gives hope to estimate the intracloud electric field by the observed parameters of TGE gamma ray energy spectra.

A number of physical phenomena in crystals are determined by their electron energy spectrum. Some phenomena are associated with the motion of electrons in the periodic field of the lattice and with their scattering on lattice vibrations. The optical, electrical, magnetic, galvanomagnetic, and other properties of crystals of dielectrics, semiconductors, and metals are intimately connected with the nature of the electron energy spectrum and the geometry of the isoenergetic surfaces of the electrons in the crystal, the peculiarities of vibrations of the lattice atoms, and the dispersion of the frequencies of these vibrations. This chapter discusses the energy spectrum of the electrons in a crystal.

Intensity enhancements of the ring current electrons associated with the VLF emissions during the geomagnetic storms and substorms, which have been observed by the equatorially orbiting S³‐A satellite (Explorer 45), are so far limited only to electrons of energies below the order of 10 keV. Furthermore, onsets of the enhancement are detected in the lowest observable energy channel first, then shifting toward higher energies with time, a feature, which is opposite to the well‐known energy dependence expected from the drift motion of electrons in the earth's dipole magnetic field. In order to explain these findings on the ring current electron enhancements, trajectories of electrons injected into the nightside of the magnetosphere from the geomagnetic tail are calculated by modifying Ejiri's calculations which were used for the interpretation of the nose events of the ring current protons observed also by the S³‐A satellite. The results indicate that the electron intensity enhancements are limited to electrons below the order of 10 keV simply because of the location of the observations which are initially confined to the dusk‐midnight sector outside the plasmasphere. If the observations are made in the morning sector, where many VLF emissions such as chorus are observed, the enhancement should not be limited only to electrons below the order of 10 keV but should extend to electrons up to the order of 100 keV. The apparent inverse dispersion, i.e., the appearance of the enhancement at lower energies first, is also due to the location of the observations. Namely, if the observations are made in the morning sector outside the plasmasphere, the normal dispersion with the first enhancement by high‐energy electrons should be seen. Since the adiabatic energy increase of charged particles during cross‐magnetic field inward motion in the dipole field is largest for the 90° pitch angle particles, the energy at the source is lowest for 90° pitch angle electrons for the same energy at the observing location. The smaller enhancement of the small pitch angle electron intensity can therefore be explained by the energy spectra of source electrons in the geomagnetic tail.

We present here a review of two problems motivated by 2D models for high T, superconductivity. The first part concerns the energy spectrum of 2D Bloch electrons in a uniform magnetic field. A semiclassical analysis provides a qualitative as well as a quantitative understanding of this spectrum. In the second part we make the case for the application of “Quantum Chaos" to strongly correlated fermion systems. It is illustrated by the level spacing distribution for the t - J model in two dimensions.

The range of solid dielectrics available for use in the field of electrical and electronics engineering are wide and properties of these materials are equally diverse. Apart from physical and chemical properties there are other innumerable factors that contribute to the dielectric behaviour of insulating materials. The interaction and interdependence of these factors are so complex that even a century of research work has not rendered any unified approach to the study of process of ionization, conduction and breakdown mechanism in dielectric media. Solid dielectrics as used in the field of electrical and electronics are not only subjected to electric field, but also under the influence of both electric and magnetic fields. Considering this, an attempt has been made in this paper to explain the conductivity of solid dielectric under the influence of both electric and magnetic fields which has not been presented so far in the available literature. A generalised mathematical treatment of conduction current of dielectric materials in the presence of both electric and cross-magnetic field has been also presented.

For Bloch electrons in a magnetic field, explicit solutions are obtained at the center of the spectrum for the Bethe ansatz equations of Wiegmann and Zabrodin. When the magnetic flux per plaquette is $\frac{1}{Q}$ with $Q$ an odd integer, distribution of the roots of the Bethe ansatz equation is uniform except at two points on the unit circle in the complex plane. For the semiclassical limit $Q\ensuremath{\rightarrow}\ensuremath{\infty}$, the wave function is ${|\ensuremath{\psi}(x)|}^{2}=(\frac{2}{sin \ensuremath{\pi}x})$, which is critical and unnormalizable. For the golden-mean flux, the distribution of roots has exact self-similarity and the distribution function is nowhere differentiable. The corresponding wave function also shows a clear self-similar structure.

We present a new approach to the problem of Bloch electrons in magnetic field, by making explicit a natural relation between magnetic translations and the quantum group U9(sl2). The approach allows to express the spectrum and the Bloch function as solutions of the Bethe-Ansatz equations typical for completely integrable quantum systems

Need for the controllable finishing of materials such as semiconductors and ceramics, has remarkably increased in the fields of electronics and precision machinary. A new lapping method has been developed by utilizing functional magnetic fluid which has the ability to move in the magnetic field. The magnetic fluid is a suspension of fine magnetite Fe3O4(smaller than 150A) in liquid. This lapping method is based on the floatation of nonmagnetic grains suspended into the magnetic fluid in the magnetic field. Lapping experiments were conducted on soda lime glass. It was found from the experiments that the new lapping method is able to control the stock removal rate through varying the current to the electromagnet. The present method may be usefully employed in finishing such parts as aspherical lenses.

Electron and ion acceleration mechanisms during the coalescence process of two adjacent magnetic loops through cohelicity and counterhelicity magnetic reconnection are investigated by using a two-dimensional, electromagnetic, and relativistic particle-in-cell (PIC) simulation. Three types of acceleration mechanisms are found by the PIC simulations. (1) Electrons in a current sheet generated between two adjacent magnetic loops are accelerated by the electric field induced during the collisionless magnetic reconnection, and their energy spectrum is characterized by the power law. (2) Ions trapped in the front of fast magnetosonic shock waves generated during the coalescence process are promptly accelerated by the surfatron acceleration mechanism. (3) During the coalescence process through counterhelicity magnetic reconnection, ions outside the current sheet are accelerated by E×B drift, where the electrostatic fields E perpendicular to local magnetic field are generated by the collision between surrounding magnetic field barrier and electron-dominated jet from the current sheet. During the coalescence process through cohelicity magnetic reconnection, the electron energy spectrum in the current sheet is characterized by the power law whose index is about 5, while during the coalescence process through counterhelicity magnetic reconnection, the electron energy spectrum is characterized by the double power law whose indices are about 3.3 and 6. The simulation results obtained here are applied to the proton and electron acceleration during solar flares. The maximum energy of accelerated electrons reaches about 100 keV, while the maximum energy of accelerated protons by the surfatron acceleration mechanism is about 10 MeV for both cohelicity and counterhelicity case.

The double hollow cathode metal ion source produces metal elements by hollow cathode sputtering and has a high material utilization rate. However, at present, a high magnetic mirror field is generated by an electromagnet, which is huge in volume and the edge field of a high magnetic mirror field seriously affects ion extraction. Miniaturization of the discharge chamber is a feasible method to solve this problem. In this paper, a two-dimensional fluid model of the double hollow cathode discharge structure under magnetic mirror field constraints is developed to study the double hollow cathode discharge and ion sputtering. The results show that the sputtering depth flux is formed above the sputtering voltage of 400 V. The evolution of the sputtering depth profile results from the combined effect of the wall ion flux, spatial radial electric field, and ion collisions. The location of the maximum ion sputtering depth depends mainly on the energetic ions entering the hollow cathode from the anode. The magnetic field and thermal electron emission strongly correlate with the sputtering depth. A small discharge chamber structure is proposed with a hollow cathode length of 36 mm, which is 24 mm smaller than the ion source established by Peking University in 2017, using a magnetic mirror field of 1400 Gs at the center point, which can be realized in the form of a permanent magnet. The electron density in the center of the discharge chamber is 2.88 × 1021 m-3 and 99% of the electrons are confined within a radius of about 3.7 mm.

The layered nanostructures ferromagnet/superconductor (F/S) due to combination of incompatible in homogeneous materials properties are the most perspective materials for use in the new field of electronics – a superconducting spintronics. A new type of logical devices based on the layered F/S nanostructures and combining the advantages of the superconducting and magnetic recording channels in one sample is offered. Each channel can be separately controlled by weak magnetic field or current pulse and the switching time is of order of 10−10 – 10−11 s. The implementation of such devices on base of high-temperature superconductors will allow using nitrogen instead of expensive helium for cooling.

The BPT-4500 and method of tuning the acceleration by focusing a plasma lens is described. Test results show that overall thruster efficiency and plume shape is improved by adjusting the curvature of magnetic field lines in the acceleration zone. Thrust efficiency is improved to 50% from 21% at 150 V and from 58% to 62.5% at 350 V using this technique. At 125 V, the BPT-4500 operates at 45% thrust efficiency and 1030 s specific impulse yielding a thrust/power ratio only 10% less than a 600 s hydrazine arcjet. Introduction Multi-mode Hall thruster capability enables one thruster to serve constant power applications needing the highest possible thrust for orbit transfer and high specific impulse for station keeping. Towards this objective, we have evaluated a GENERAL DYNAMICS (formerly PRIMEX Aerospace Company) approach for tuning the BPT-4500 for the best possible performance over a range of accelerating potentials. This paper describes our initial exploratory efforts to improve thruster efficiency by actively changing magnetic field shape. Thruster Description The BPT-4500 is a 4.5 kW Hall field plasma accelerator based on an earlier 4000 watt device''. These thrusters employ a short discharge to minimize contact with the insulators and pole pieces to shape the curvature of the magnetic field in the ion creation zone. As is well known in Hall type accelerators, the magnetic field lines become nearly equipotential lines defining the acceleration vector of the ions. Also, the technique of adjusting electric potential shape by magnetized electrons in a magnetic field is commonly called a plasma lens. By virtue of the discharge properties in the BPT and shaping these fields, we have attempted to Copyright (c) 2001 by GENERAL DYNAMICS. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. create a plasma lens with adjustable focus. Plasma Lens Focusing Morozov in the Soviet Union has shown that using two sets of electromagnets to change the gradient of the radial magnetic field can have a significant effect on the behavior of ion acceleration in Hall type discharges. Gavryushin and Kim have shown that the axial gradient of the radial magnetic field regulated by varying the degree of screening of the accelerator channel. These accelerators are characterized by relatively long acceleration lengths in channels that typically have a length to width of 2:1. Insofar as these accelerators had a long discharge, the curvature of the field was not emphasized in the description of this early work. Instead, the researchers discuss primarily the axial gradient of Curvature of ion path changed by Anode focal length Anode potential contour Cathode potential contour Figure 1 Hall Acceleration with Plasma Lens Concept (c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. the magnetic field. The BPT discharge is configured to take place in a very short channel, typically with a length to width of 1:2. As such, physical evidence from a variety of geometry studies suggests that the BPT discharge creates ions in a thin, -1-2 mm layer, occurring approximately 3-4 mm downstream of the anode, schematically illustrated in Fig 1. As such, we hypothesize the BPT discharge embodies Hall acceleration within a plasma lens. Our innovation is to exploit the features of this short discharge by adjusting the focal length of a plasma lens using a trim or focusing coil. Variation of the curvature of the fields in this layer has the effect of changing the initial trajectory of ions. Changes in the shape downstream of the ionization zone are minimized by the fact that magnetic fields are manipulated inside the primary magnetic structure. The BPT-4500 thruster shown in Fig 2 is composed of a primary magnetic structure with four outer coils and an inner coil. The BPT-4500 has a 13 cm mid-diameter and a 2 cm annulus providing slightly more discharge area than the BPT-4000. The schematic cross section of the thruster is shown in Fig. 3. This shows the anode is surrounded by a secondary magnetic structure to add or subtract from the magnetic field radius of curvature in the annulus between insulator rings. Figure 2 BPT-4500 Laboratory Style Thruster Flight Version A flight capable version of the thruster is under development in the Lockheed Martin funded Hall Thruster Propulsion System (HTPS) program. This flight thruster is referred to as the BPT-4000. This thruster has a fixed magnetic field and plasma lens taken from the optimal configuration found here. GENERAL DYNAMICS is on track to deliver a flight qualified thruster, power processor and xenon flow control system in 2003.

Performance characteristics of the magnetic multipole plasma source depend strongly on the structure of the line-cusp magnetic fields and the behavior of the primary electrons. In order to design and improve the plasma source efficiently, we developed a computer code which can simulate primary electron orbits in the three-dimensional magnetic field generated in the source. Using this code, we compared the orbits of the primary electrons in the arc discharges with and without the mode-flip phenomenon. The result indicates that the trapping of the primary electrons in a local magnetic mirror fields causes the mode flip due to the reduction of the effective anode area. Trapping also degrades the spatial uniformity of the source plasma density, due to local arc discharge. The structure of the magnetic field can be optimized so that the primary electrons fill the center region of the source. The 3D simulation of the electron orbits turned out to be effective and useful in designing a multipole source.