Landau-Lifshitz Equation Of Motion Research Articles

Effect of radiation-reaction on charged particle dynamics in a focused electromagnetic wave

The effect of radiation-reaction force on the dynamics of a charged particle in an intense focused light wave is investigated using the physically appealing Hartemann-Luhmann equation of motion. It is found that, irrespective of the choice of initial conditions, radiation reaction force causes the charged particle to cross the focal region, provided the particle is driven into regions where the radiation reaction force dominates over the Lorentz force, thus enhancing the forward energy gained by the particle from the intense light wave. This result is in sharp contrast to the well known result, derived in the absence of radiation reaction forces, where for certain initial conditions the particle reflects from the high intensity region of the focused light wave, thereby losing forward energy. From the perspective of energy gain, our studies clearly show that the parameter space for forward energy gain which is reduced by ponderomotive effects is compensated by radiation reaction effects. These results, which are of relevance to the present day direct laser acceleration schemes of charged particle, also agrees with that obtained using the well known Landau-Lifshitz equation of motion.

Radiation reaction enhancement in flying focus pulses

Radiation reaction (RR) is the oldest still-unsolved problem in electrodynamics. In addition to conceptual difficulties in its theoretical formulation, the requirement of exceedingly large charge accelerations has thus far prevented its unambiguous experimental identification. Here, we show how measurable RR effects in a laser-electron interaction can be achieved through the use of flying focus pulses (FFPs). By allowing the focus to counterpropagate with respect to the pulse phase velocity, a FFP overcomes the intrinsic limitation of a conventional laser Gaussian pulse (GP) that limits its focus to a Rayleigh range. For an electron initially also counterpropagating with respect to the pulse phase velocity, an extended interaction length with the laser peak intensity is achieved in a FFP. As a result, the same RR deceleration factors are obtained, but at FFP laser powers orders of magnitude lower than for ultrashort GPs with the same energy. This renders the proposed setup much more stable than those using GPs and allows for more accurate \emph{in situ} diagnostics. Using the Landau-Lifshitz equation of motion, we show numerically and analytically that the capability of emerging laser systems to deliver focused FFPs will allow for a clear experimental identification of RR.

Lorentz-Abraham-Dirac and Landau-Lifshitz equations of motion and the solution to a relativistic electron in a counterpropagating laser beam

Beginning with a critical examination of the Lorentz-Abraham (LA) classical equation of motion for an extended charge and the closely related Lorentz-Abraham-Dirac (LAD) equation of motion for a mass-renormalized point-charge, the Landau-Lifshitz (LL) approximate solution to the LAD equation of motion is determined for an electron subject to a counterpropagating linearly or circularly polarized plane-wave pulse with an arbitrarily shaped envelope. A convenient three-vector formulation of the LL equation is used to derive closed-form expressions for the velocities and associated powers of the electron directly in terms of the time in the laboratory frame. The three-vector formulation also reveals definitive criteria for the LL solution to be an accurate approximation to the LAD equation of motion and for the LL solution to reduce to the solution of the Lorentz force equation of motion that ignores radiation reaction. Semiclassical analyses are used to obtain simple conditions for determining the regimes where the quantum effects of either Compton electron scattering by the incident photons or electron recoil produced by the emitted photons is significant. It is proven that the LL approximation becomes an inaccurate solution to the LAD equation of motion only for large enough electron velocities and plane-wave intensities that quantum recoil effects on the electron can greatly alter the classical solution. Comparisons are made with previously published analytical and numerical solutions to the LL equation of motion for the velocity of an electron in a counterpropagating plane wave.10 MoreReceived 18 August 2021Accepted 1 November 2021DOI:https://doi.org/10.1103/PhysRevAccelBeams.24.114002Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasElectromagnetismElectron beams & opticsLaser driven electron accelerationQuantum theorySingle-particle dynamicsAccelerators & BeamsGeneral PhysicsParticles & FieldsNonlinear DynamicsCondensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Experimental verification of the Landau–Lifshitz equation

The Landau–Lifshitz (LL) equation has been proposed as the classical equation to describe the dynamics of a charged particle in a strong electromagnetic field when influenced by radiation reaction. Until recently, there has been no clear experimental verification. However, aligned crystals have remedied the situation: here, as in Nielsen et al CERN NA63 Collaboration (2020 Phys. Rev. D 102 052004), we report on a quantitative experimental test of the LL equation by measuring the emission spectra of electrons and positrons penetrating aligned single crystals. The recorded spectra are in remarkable agreement with simulations based on the LL equation of motion with moderate quantum corrections for recoil and, in the case of electrons in axially aligned crystals, spin and reduced radiation intensity.

Radiation reaction near the classical limit in aligned crystals

An accelerated charged particle emits electromagnetic radiation. If the driving force is sufficiently strong, the radiated energy becomes comparable to the kinetic energy of the particle and the back-action of the emitted radiation (radiation reaction) significantly alters the dynamics of the particle. The Landau-Lifshitz (LL) equation has been proposed as the classical equation to describe the dynamics of a charged particle in a strong electromagnetic field when the effects of radiation reaction are taken into account. Hitherto, the experimental problem in validating the LL equation has been to achieve sufficiently strong fields for radiation reaction to be important without quantum effects being prominent. Notwithstanding, here we provide a quantitative experimental test of the LL equation by measuring the emission spectrum for a wide range of settings for 50 GeV positrons crossing aligned silicon single crystals near the $(110)$ planar channeling regime as well as 40 GeV and 80 GeV electrons traversing aligned diamond single crystals near the $\langle100\rangle$ axial channeling regime. The experimental spectra are in remarkable agreement with predictions based on the LL equation of motion with small quantum corrections for recoil and, in case of electrons, spin and reduced radiation emission, as well as with a more elaborate quantum mechanical model. Our experiment clearly shows the inadequacy of the Lorentz force as the sole agent of force on the particles in the classical limit, due to its absence of radiative energy loss in describing the dynamics of high-energy charged particles in strong electromagnetic fields like those in aligned single crystals.

Radiation reaction of a charged particle with the external rectangular force

We considered the equation of motion of an accelerating point charged particle which produced the radiation reaction force that described by the Abraham-Lorentz equation. Then we considered another equation of motion from a different approach so-called Landau-Lifshitz equation which came from a limit in the case of a finite spherical shell charged particle. This latter equation shows that an acceleration of charged particle depends on an external force and its derivative. We showed the solution of the acceleration from the external rectangular force of these equations. So we solved the velocity and position of the charged particle. As a result, we compared the solutions from the Abraham-Lorentz and Landau-Lifshitz equations of motion.

Harmonic transition-state theory of thermal spin transitions

A rate theory for thermally activated transitions in spin systems is presented. It is based on a transition-state approximation derived from Landau-Lifshitz equations of motion and quadratic expansion of the energy surface at minima and first order saddle points. While the flux out of the initial state vanishes at first order saddle points, the integrated flux over the hyperplanar transition state is nonzero and gives a rate estimate in good agreement with direct dynamical simulations of test systems over a range in damping constant. The preexponential factor obtained for transitions in model systems representing nanoclusters with 3 to 139 transition metal adatoms is on the order of ${10}^{11}$ to ${10}^{13}\phantom{\rule{0.28em}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$, similar to that of atomic rearrangements.

Coercivity control in finite arrays of magnetic particles

Micromagnetic simulation has been performed for two-dimensional arrays of single-domain magnetic particles using the Landau-Lifshitz equation of motion and an energy minimization method. Effects of array anisotropy and spin positional disorder on the hysteresis loop and coercivity of the particle systems are investigated. Simulation results show that the hysteresis loop can be largely modified by breaking geometric symmetry of square arrays, and coercivity in general is found to increase with array disorder. Magnetic hysteresis is strongly affected by disorder when the array contains only a few particles.

Hard-axis magnetization behavior and the surface spin-flop transition in antiferromagneticFe∕Cr(100)superlattices

We investigated the hard-axis magnetization behavior of a biaxial antiferromagnetic $\mathrm{Fe}∕\mathrm{Cr}(100)$ superlattice. We discovered a surface spin-flop transition that separates the field-induced nonsymmetric state at low fields and the symmetric twisted state at higher fields. We studied the transition via realistic model calculations using the Landau-Lifshitz equations of motion, via stray field observations using magneto-optic indicator film imaging, and via polarized-neutron reflectivity measurements.

Theory and calculation of giant magneto-impedance effect in amorphous alloy ribbons and films

The detailed expressions of orientation-related relative permeability in amorphous alloy ribbons and films with in-plane nearly transverse uniaxial anisotropy under applied driving ac current flowing through the ferromagnetic materials and dc external magnetic field are derived by simultaneously solving the Maxwell's equations and the Landau-Lifshitz equation of motion and establishing the magnetic domain structure model. The calculation formula of mean relative permeability averaging over 90°orientation angle and the magneto-impedance are obtained. The relation between permeability and the component of tensor susceptibility is deduced. The dependences of the real part and imaginary part of the relative permeability on frequency at several values of the applied external field are calculated, and the corresponding magnetic permeability spectra are plotted. The calculation formula of permeability which were inferred by Panina et al and Kraus can be unified by our theoretical result which is in good agreement with the experimental data under lower external magnetic field for both amorphous ribbons and films.

High-Field FMR Investigation of Epitaxially Grown Fe[sub 4]N Films on a MgO(001) Substrate

Epitaxially grown Fe 4 N thin film on a MgO(001) substrate has been examined by high-field ferromagnetic resonance (FMR) spectroscopy. When the magnetic field was applied parallel (along the in-plane direction) and perpendicular to the substrate, the resonance appeared at 2.4 and 4.9 T at 94 GHz, respectively. The g-value and the effective magnetization were estimated to be 2.13 and 191 emu g - 1 , respectively, using the relation derived from the Landau-Lifshitz equation of motion. Superconducting quantum interference device (SQUID) measurements also showed that the Fe 4 N epitaxial film has a large magnetization anisotropy, supporting that the magnetic moments lie along the in-plane direction.

Spin excitations of magnetic vortices in ferromagnetic nanodots

Spin excitations of the magnetic vortex state in ferromagnetic nanodots are measured using Brillouin light scattering. Arrays of permalloy dots with 800-nm diameter and 60-nm thickness were fabricated by means of electron beam lithography and lift-off procedures. Two excitation modes are observed experimentally in the vortex state. One mode, at \ensuremath{\sim}12 GHz, decreases slightly in frequency to 11 GHz as an in-plane magnetic field is applied. The lower mode, at \ensuremath{\sim}8 GHz, is almost independent of applied field strength. Numerical and analytical calculations of the dynamic magnetization based on the Landau-Lifshitz equation of motion allows us to identify the higher and lower frequency modes as corresponding to dipole-dominated spin excitations localized inside the dot and at the dot edges, respectively.

Dynamic constitutive relations for polarization and magnetization.

In this paper we develop constitutive relations for materials where the magnetization and polarization may depend on both the electric and magnetic fields. The approach is general, and is based on a previously developed statistical-mechanical theory. We include the quadrupole-moment density as well as the dipole-moment density in the microscopic displacement field. This yields an electric gradient term in the constitutive equations. This leads to origin invariance in the multipole moments from which Maxwell's equations are defined. We present generalizations of Debye and Landau-Lifshitz equations of motion which are valid for nonequilibrium and contain memory. The reversible and relaxation terms in the polarization and magnetization evolution equations include the possibility of magnetoelectric coupling. Using constitutive relationship, we derive evolution equations for the displacement and induction fields from a Hamiltonian approach.

Dynamic equations for individual spins in materials with magnetic order: The ideal ferromagnet

The time and temperature dependence of individual spins is studied in an ideal ferromagnet. Starting from the Heisenberg Hamiltonian in a magnetic field and, building on linear-response theory, we derive the linearized microscopic analogy to the macroscopic Landau-Lifshitz equations of motion for magnetization. The dynamical equations take into account the influence of spin correlations in addition to the molecular field. In order to illustrate the validity of our approach, we calculate the temperature-dependent spin-wave spectrum and Landau damping by employing a finite-temperature perturbation theory for spin operators. The latter is adapted to suit higher-order spin-correlation functions which can be described by simple quasiparticle interactions with the spin system. Our results agree with those established on the thermodynamics of the Heisenberg model. In addition, we provide a method of how to deduce macroscopic equations of the Landau-Lifshitz type from first principles, including temperature dependence and damping.

Angular dependence of giant magnetoimpedance in an amorphous Co-Fe-Si-B ribbon

The field response of impedance is studied in a stress-annealed amorphous ribbon as a function of the angle of application of the external magnetic field in order to verify the role of induced anisotropies (and their distribution) and demagnetizing factors in the giant magnetoimpedance (GMI) phenomenon which occurs in soft magnetic materials. The experimental results are well explained by a theoretical model, based on the simultaneous solution of Maxwell equations and the Landau-Lifshitz equation of motion. Demagnetizing effects are properly taken into account in the case of ribbons or thin films. The physical parameters necessary to test the theory were obtained through complementary measurements of the ferromagnetic resonance and temperature dependence of magnetization. The results clearly indicate the enormous influence of the distribution of anisotropies on the GMI effect. Also, an experimental procedure for determining the easy-axis distribution function is proposed.

Exchange coupling energy determined by ferromagnetic resonance in 80NiFe/Cu multilayer films

Ferromagnetic resonance (FMR) experiments were carried out for [ 80NiFe(60 A ̊ )/Cu (d Cu A ̊ )] N , ( d Cu = 10–12, N = 2–5) multilayer films fabricated by a sputtering method. The main resonance peak had a broad shoulder located on the higher magnetic field side. Namely, the exchange energy between the magnetic layers was antiferromagnetic. In order to estimate the exchange energy, the FMR absorption spectrum was calculated using the Landau-Lifshitz equation of motion. The calculation model was improved so that we were able to analyze the spectrum for a multilayer with N magnetic layers. The exchange energy obtained from FMR analysis was compared with that estimated from the saturation magnetic field in the magnetoresistance (MR) curve. The positions and the intensity ratio of the resonance peaks were dependent on merely the exchange energy but not on the number of layers. The MR ratio was roughly proportional to the exchange energy.

Effect of interlayer exchange coupling on spin-wave spectra in magnetic double layers: Theory and experiment.

The spin-wave spectra of dipolar and exchange coupled double-layer systems are investigated both theoretically and experimentally. The theoretical formalism is based on a macroscopic description using the Landau-Lifshitz equation of motion and Rado-Weertman and Hoffmann boundary conditions. The experimental spin-wave spectra were obtained for ${\mathrm{Ni}}_{0.8}$${\mathrm{Fe}}_{0.2}$/Pd/${\mathrm{Ni}}_{0.8}$${\mathrm{Fe}}_{0.2}$ and ${\mathrm{Ni}}_{0.8}$${\mathrm{Fe}}_{0.2}$/Cr/Co systems by means of Brillouin light scattering. From comparison of the theoretical calculations and experimental data, the dependence of the interlayer exchange constant on the interlayer thickness is determined. In addition, we analyzed recent microwave absorption data for ${\mathrm{Ni}}_{0.8}$${\mathrm{Fe}}_{0.2}$/Cu/${\mathrm{Ni}}_{0.8}$${\mathrm{e}}_{0.2}$ taken from the literature. In all systems the interlayer exchange coupling was found to be ferromagnetic. It vanishes for interlayer thicknesses of the order of 10 A\r{} for Cr and 20--30 A\r{} for Cu and Pd. The full coupling limit is also treated in detail.

Recording and transition noise simulations in thin film media

Micromagnetic simulations of the recording of a single transition in thin-film metallic media are presented. The film is modeled as a planar array of hexagonal single-domain particles with long-range magnetostatic as well as possible nearest-neighbor exchange interactions. Magnetization configurations are determined by following the Landau-Lifshitz equations of motion with finite damping. The media parameters used here approximate Co films. The recording geometry resembles a close flying head with a small gap. Transition fluctuations, a source of transition noise, are also suited for both non-exchange-coupled and exchange-coupled media. It is shown that intergranular exchange coupling can significantly enhance transition noise and that films with well defined nonmagnetic grain boundaries exhibit better signal-to-noise ratios. >

Analog simulation of flux reversal in thin magnetic films by uniform rotation

The results of an analog computer study of flux reversal in thin magnetic films are presented. The computer diagram is given for the simulation of the Landau-Lifshitz equation of motion of the magnetization vector. Solutions of these equations were obtained for pulse fields applied parallel and perpendicular to the easy axis of a film with uniaxial anisotropy. Switching curves are given which show the effect of varying the relaxation frequency λ. The switching time varies by a factor of 3 as λ is varied from 108to 5 × 108Hz. The various definitions of switching time which have been reported in the literature are also compared. Nonlinear resonance curves are presented for λ, values of 108and 1.5 × 108Hz. The curves represent solutions for the angular velocity of the magnetization when a sinusoidal field is applied perpendicular to the easy axis. For applied fields as large as half the anisotropy field the resonant frequency and linewidths are significantly different (20 percent or more) from the small signal values which are useful for determining the anisotropy field H K and the relaxation frequency λ. It is concluded that the analog technique is convenient for studying magnetic switching problems when the problems can be formulated in terms of ordinary differential equations. Other problems suitable for this approach are studies of flux reversal with various waveforms for the excitation field and of the interaction between coupled films or particles.