Adiabatic Focusing Research Articles

EFFECT OF ADIABATIC FOCUSING ON THE DIFFUSION OF ENERGETIC CHARGED PARTICLES

ABSTRACT An approximate analytic description of a diffusion coefficient, including the effect of adiabatic focusing, has been developed. This description is formulated with the aid of stochastic differential equations and the steady perturbation solution of the Fokker–Plank transport equation. The analytical formula is based on three important assumptions. First, the pitch-angle diffusion coefficient is set to be separable from the spatial coordinate and the pitch-angle cosine. Second, the spatial dependence of the ratio between the mean-free path and focusing length is assumed to be weak. Third, the pitch-angle distribution relaxes quickly to a steady state. The new analytic formula could be applied to calculate the spatial diffusion coefficient in the interplanetary and interstellar space.

SIMULATION OF ENERGETIC PARTICLE TRANSPORT AND ACCELERATION AT SHOCK WAVES IN A FOCUSED TRANSPORT MODEL: IMPLICATIONS FOR MIXED SOLAR PARTICLE EVENTS

ABSTRACT We use numerical solutions of the focused transport equation obtained by an implicit stochastic differential equation scheme to study the evolution of the pitch-angle dependent distribution function of protons in the vicinity of shock waves. For a planar stationary parallel shock, the effects of anisotropic distribution functions, pitch-angle dependent spatial diffusion, and first-order Fermi acceleration at the shock are examined, including the timescales on which the energy spectrum approaches the predictions of diffusive shock acceleration theory. We then consider the case that a flare-accelerated population of ions is released close to the Sun simultaneously with a traveling interplanetary shock for which we assume a simplified geometry. We investigate the consequences of adiabatic focusing in the diverging magnetic field on the particle transport at the shock, and of the competing effects of acceleration at the shock and adiabatic energy losses in the expanding solar wind. We analyze the resulting intensities, anisotropies, and energy spectra as a function of time and find that our simulations can naturally reproduce the morphologies of so-called mixed particle events in which sometimes the prompt and sometimes the shock component is more prominent, by assuming parameter values which are typically observed for scattering mean free paths of ions in the inner heliosphere and energy spectra of the flare particles which are injected simultaneously with the release of the shock.

Modeling interstellar pickup ion distributions in corotating interaction regions inside 1 AU

AbstractWe present a modeling study of interstellar pickup ion (PUI) distributions in corotating interaction regions (CIRs). We consider gradual compressions associated with CIRs formed when fast speed streams overtake slower streams in the inner heliosphere. For the analysis, we adopt a simplified magnetohydrodynamic model of a CIR. The Energetic Particle Radiation Environment Module, a parallelized particle numerical kinetic code, is used to model PUI distributions using the focused transport equation, including adiabatic cooling/heating, adiabatic focusing, and parallel and perpendicular diffusion. The continuous injection of PUIs is handled as a source term with a ring distribution in velocity space that is produced from the local neutral density obtained from a hot model of the interstellar neutral gas. The simulated distributions exhibit a harder spectrum in the compression region and a softer spectrum in the rarefaction region than that in undisturbed solar wind. As an additional result, a v−5 power law tail distribution above the PUI cutoff speed (a knee in the distribution) emerges for a particular velocity gradient in the CIR. The tail above the PUI cutoff is sensitive to the CIR velocity gradient, and in one observational case studied, this relationship adequately explains the observed spectrum from 2 to 4 times the solar wind speed. This suggests that the velocity gradient associated with the CIR formation can efficiently create a seed population of PUIs before a shock forms even without stochastic acceleration. Thus, local CIR compressions without shocks may play a significant role in the acceleration process as suggested previously.

EFFECTS OF LATITUDINALLY DEPENDENT SOLAR WIND SPEED ON DIFFUSION COEFFICIENTS OF COSMIC RAYS IN THE PRESENCE OF ADIABATIC FOCUSING

The solar wind is observed to display high speeds in high heliolatitude coronal holes and low speeds near the ecliptic plane. The heliospheric magnetic field associated with the solar wind plays a very important role in the transport and modulation of charged energetic particles, including galactic cosmic rays (GCRs) and solar energetic particles (SEPs), in the three-dimensional heliosphere. In previous studies, a constant solar wind speed, which is independent of heliolatitude, was assumed and commonly used in modulation modeling of cosmic rays. In this work, we investigate the realistic latitudinally dependent solar wind speed and utilize the theoretical models in hyperbolic and piecewise polynomial forms to explore the important effects on the heliospheric magnetic field and the diffusion coefficients (parallel, perpendicular, and drift scale) of cosmic rays in the presence of adiabatic focusing. Comparisons of the diffusion coefficients derived from standard Parker field and modified magnetic fields are presented. Since the structures and properties of different solar wind sources (coronal streamer belt, polar coronal hole, and transition region between them) differ from each other in essence, we suggest that the latitudinally dependent solar wind speed and the corresponding heliospheric magnetic field and diffusion coefficients with adiabatic focusing should be employed in the global modeling studies of GCRs and SEPs in the heliosphere.

MODIFICATION OF THE PARALLEL SCATTERING MEAN FREE PATH OF COSMIC RAYS IN THE PRESENCE OF ADIABATIC FOCUSING

The cosmic ray mean free path in a large-scale nonuniform guide magnetic field with superposed magnetostatic turbulence is calculated to clarify some conflicting results in the literature. A new, exact integro-differential equation for the cosmic-ray anisotropy is derived from the Fokker-Planck transport equation. A perturbation analysis of this integro-differential equation leads to an analytical expression for the cosmic ray anisotropy and the focused transport equation for the isotropic part of the cosmic ray distribution function. The derived parallel spatial diffusion coefficient and the associated cosmic ray mean free path include the effect of adiabatic focusing and reduce to the standard forms in the limit of a uniform guide magnetic field. For the illustrative case of isotropic pitch angle scattering, the derived mean free path agrees with the earlier expressions of Beeck & Wibberenz, Bieber & Burger, Kota, and Litvinenko, but disagrees with the result of Shalchi. The disagreement with the expression of Shalchi is particularly strong in the limit of strong adiabatic focusing.

THE DIFFUSION APPROXIMATION VERSUS THE TELEGRAPH EQUATION FOR MODELING SOLAR ENERGETIC PARTICLE TRANSPORT WITH ADIABATIC FOCUSING. I. ISOTROPIC PITCH-ANGLE SCATTERING

The diffusion approximation to the Fokker-Planck equation is commonly used to model the transport of solar energetic particles in interplanetary space. In this study, we present exact analytical predictions of a higher order telegraph approximation for particle transport and compare them with the corresponding predictions of the diffusion approximation and numerical solutions of the full Fokker-Planck equation. We specifically investigate the role of the adiabatic focusing effect of a spatially varying magnetic field on an evolving particle distribution. Comparison of the analytical and numerical results shows that the telegraph approximation reproduces the particle intensity profiles much more accurately than does the diffusion approximation, especially when the focusing is strong. However, the telegraph approximation appears to offer no significant advantage over the diffusion approximation for calculating the particle anisotropy. The telegraph approximation can be a useful tool for describing both diffusive and wavelike aspects of the cosmic-ray transport.

Hot-electron nanoscopy using adiabatic compression of surface plasmons

Surface plasmon polaritons are a central concept in nanoplasmonics and have been exploited to develop ultrasensitive chemical detection platforms, as well as imaging and spectroscopic techniques at the nanoscale. Surface plasmons can decay to form highly energetic (or hot) electrons in a process that is usually thought to be parasitic for applications, because it limits the lifetime and propagation length of surface plasmons and therefore has an adverse influence on the functionality of nanoplasmonic devices. Recently, however, it has been shown that hot electrons produced by surface plasmon decay can be harnessed to produce useful work in photodetection, catalysis and solar energy conversion. Nevertheless, the surface-plasmon-to-hot-electron conversion efficiency has been below 1% in all cases. Here we show that adiabatic focusing of surface plasmons on a Schottky diode-terminated tapered tip of nanoscale dimensions allows for a plasmon-to-hot-electron conversion efficiency of ∼30%. We further demonstrate that, with such high efficiency, hot electrons can be used for a new nanoscopy technique based on an atomic force microscopy set-up. We show that this hot-electron nanoscopy preserves the chemical sensitivity of the scanned surface and has a spatial resolution below 50nm, with margins for improvement.

Quantitative Imaging of Rapidly Decaying Evanescent Fields Using Plasmonic Near-Field Scanning Optical Microscopy

Non-propagating evanescent fields play an important role in the development of nano-photonic devices. While detecting the evanescent fields in far-field can be accomplished by coupling it to the propagating waves, in practice they are measured in the presence of unwanted propagating background components. It leads to a poor signal-to-noise ratio and thus to errors in quantitative analysis of the local evanescent fields. Here we report on a plasmonic near-field scanning optical microscopy (p-NSOM) technique that incorporates a nanofocusing probe for adiabatic focusing of propagating surface plasmon polaritons at the probe apex, and for enhanced coupling of evanescent waves to the far-field. In addition, a harmonic demodulation technique is employed to suppress the contribution of the background. Our experimental results show strong evidence of background free near-field imaging using the new p-NSOM technique. Furthermore, we present measurements of surface plasmon cavity modes, and quantify their contributing sources using an analytical model.

NUMERICAL ANALYSIS OF THE FOKKER-PLANCK EQUATION WITH ADIABATIC FOCUSING: ISOTROPIC PITCH-ANGLE SCATTERING

We present numerical solutions to both the standard and modified two-dimensional Fokker-Planck equations with adiabatic focusing and isotropic pitch-angle scattering. With the numerical solution of the particle distribution function, we then discuss the related numerical issues, calculate the parallel diffusion coefficient using several different methods, and compare our numerical solutions for the parallel diffusion coefficient to the analytical forms derived earlier. We find the numerical solution to the diffusion coefficient for both the standard and modified Fokker-Planck equations agrees with that of Shalchi for the mean squared displacement method of computing the diffusion coefficient. However, we also show the numerical solution agrees with that of Litvinenko and Shalchi & Danos when calculating the diffusion coefficient via the velocity correlation function.

Bimodal fluxes of near‐relativistic electrons during the onset of solar particle events

We report for several solar energetic particle events (SEPs) intensity and anisotropy measurements of energetic electrons in the energy range ∼27 to ∼500 keV as observed with the Wind and ACE spacecraft in June 2000. The observations onboard Wind show bimodal pitch angle distributions (PADs), whereas ACE shows PADs with one peak, as usually observed for impulsive injection of electrons at the Sun. During the time of observation Wind was located upstream of the Earth's bow shock, in the dawn‐noon sector, at distances of ∼40 to ∼70REfrom the Earth, and magnetically well connected to the quasi‐parallel bow shock, whereas ACE, located at the libration point L1, was not connected to the bow shock. The electron intensity‐time profiles and energy spectra show that the backstreaming electrons observed at Wind are not of magnetospheric origin. The observations rather suggest that the bimodal electron PADs are due to reflection or scattering at an obstacle located at a distance of less than ∼150RE in the antisunward direction, compatible with the bow shock or magnetosheath of the magnetosphere of the Earth. For a modeling of the observations, we have performed transport simulations which include the effects of pitch angle diffusion, adiabatic focusing, and reflection at a boundary close to the point of observation. The results of the simulations demonstrate that the bimodal PADs are compatible with the reflection of electrons at a nearby boundary, at distances of ∼70RE. This finding is supported by the orbital configuration and the magnetic field direction: Whereas ACE is not connected, Wind is well connected to the magnetosphere of the Earth.

The telegraph equation for cosmic-ray transport with weak adiabatic focusing

Time-dependent solutions of a spatial diffusion equation are often used to describe the transport of solar energetic particles, accelerated in large solar flares. Approximate analytical solutions of the diffusion approximation can complement and guide detailed numerical solutions of the Fokker-Planck equation for the particle distribution function. The accuracy of the diffusion approximation is limited, however, because the signal propagation speed is infinite in the diffusion limit. An improved description of cosmic-ray transport is provided by the telegraph equation, characterised by a finite signal propagation speed. We derive the telegraph equation for the particle density, taking into account adiabatic focusing in a large-scale interplanetary magnetic field in a weak focusing limit. As an illustration, we calculate a propagating pulse solution of the telegraph equation, determine the rise time when the maximum particle intensity is reached at a given distance from the Sun, and compare the results with those obtained in the diffusion approximation. In comparison with the diffusion equation, the telegraph equation predicts an asymmetrical shape of the pulse and a shorter rise time. These potentially significant differences suggest that the more accurate telegraph equation should be used in analysis of the solar energetic particle data, at least to quantify the accuracy of the focused diffusion model.

Broadband absorbers and selective emitters based on plasmonic Brewster metasurfaces

We discuss the possibility of realizing utlrabroadband omnidirectional absorbers and angularly selective coherent thermal emitters based on properly patterned plasmonic metastructures. Instead of relying on resonant concentration effects that inherently limit the bandwidth, we base our design on the combination of two inherently nonresonant effects: plasmonic Brewster funneling and adiabatic plasmonic focusing. With this approach, we demonstrate compact, broadband absorption and emission spanning terahertz, infrared and optical frequencies, ideal for various energy and defense applications.

Physics of unbounded, broadband absorption/gain efficiency in plasmonic nanoparticles

Anomalous resonances in properly shaped plasmonic nanostructures can in principle lead to infinite absorption/gain efficiencies over broad bandwidths. By developing a closed-form analytical solution for the fields scattered by conjoined semicircles, we outline the fundamental physics behind these phenomena, associated with broadband adiabatic focusing of surface plasmons at the nanoscale. We are able to justify the apparent paradox of finite absorption/gain in the limit of infinitesimally small material loss/gain, and we explore the potential of these phenomena in nonlinear optics, spasing, energy-harvesting and sensing.

ON THE DIFFERENT ANALYTICAL RESULTS OBTAINED FOR THE PARALLEL DIFFUSION COEFFICIENT OF COSMIC PARTICLES WITH ADIABATIC FOCUSING

A spatially varying mean magnetic field gives rise to so-called adiabatic focusing of energetic particles propagating through the universe. In the past, different analytical approaches have been proposed to calculate the particle diffusion coefficient along the mean field with focusing. In the present paper, we show how these different results are related to each other. New results for the parallel diffusion coefficient that are more general than previous results are also presented.

A DIRECT APPROACH FOR DETERMINING THE PERPENDICULAR MEAN FREE PATH OF SOLAR ENERGETIC PARTICLES IN A TURBULENT AND SPATIALLY VARYING MAGNETIC FIELD

A direct approach for explicitly determining the perpendicular mean free path of solar energetic particles (SEPs) influenced by parallel diffusion and composite dynamical turbulence in a spatially varying magnetic field is presented. As theoretical applications of the direct approach, we investigate the inherent relations between the perpendicular mean free path and various parameters concerning physical properties of SEPs as well as those of interplanetary conditions such as the solar wind and the turbulent magnetic field. Comparisons of the perpendicular mean free paths with and without adiabatic focusing are also presented. The direct method shows encouraging agreement with spacecraft observations, suggesting it is a reliable and useful tool for use in theoretical investigations and space weather forecasting.

Simulating cosmic-ray transport with adiabatic focusing

Context. Magnetized space plasmas such as the heliosphere contain a spatially variable guide magnetic field that is superposed by turbulence. Understanding the transport of energetic particles in such an environment is important for many applications.Aims. The modifications of the scattering mean free path caused by the effect of adiabatic focusing have so far been studied only analytically. Here, the exact analytical conditions are reproduced in a numerical simulation to allow for a detailed comparison.Methods. A Monte-Carlo simulation code was modified to include a guide magnetic field with a constant focusing length. The turbulence strength was kept constant either throughout the simulation space or relative to the variable background field strength.Results. In contrast to both quasi-linear and nonlinear analytical predictions, an increased parallel mean free path is found for almost all cases. Particle diffusion is found to be asymmetric depending on the initial particle velocity relative to the guide field. Conclusions. The increased parallel mean free path can be explained by a coherent particle motion toward lower magnetic field strengths. The possible reduction of the diffusion parameter is consequently obscured.

EFFECTS OF NON-ISOTROPIC SCATTERING, MAGNETIC HELICITY, AND ADIABATIC FOCUSING ON DIFFUSIVE TRANSPORT OF SOLAR ENERGETIC PARTICLES

Transport of solar energetic particles in interplanetary space is analyzed. A new systematic derivation of the diffusion approximation is given, which incorporates the effects of non-isotropic scattering, magnetic helicity, and adiabatic focusing in a non-uniform large-scale magnetic field. The derivation is based on a system of stochastic differential equations, equivalent to the Fokker-Planck equation, and the new method is a generalization of the Smoluchowski approximation in the theory of the Brownian motion. Simple, physically transparent expressions for the transport coefficients are derived. Different results of earlier treatments of the problem are related to the assumptions regarding the evolving particle distribution.

A DIRECT METHOD TO DETERMINE THE PARALLEL MEAN FREE PATH OF SOLAR ENERGETIC PARTICLES WITH ADIABATIC FOCUSING

The parallel mean free path of solar energetic particles (SEPs), which is determined by physical properties of SEPs as well as those of solar wind, is a very important parameter in space physics to study the transport of charged energetic particles in the heliosphere, especially for space weather forecasting. In space weather practice, it is necessary to find a quick approach to obtain the parallel mean free path of SEPs for a solar event. In addition, the adiabatic focusing effect caused by a spatially varying mean magnetic field in the solar system is important to the transport processes of SEPs. Recently, Shalchi presented an analytical description of the parallel diffusion coefficient with adiabatic focusing. Based on Shalchi's results, in this paper we provide a direct analytical formula as a function of parameters concerning the physical properties of SEPs and solar wind to directly and quickly determine the parallel mean free path of SEPs with adiabatic focusing. Since all of the quantities in the analytical formula can be directly observed by spacecraft, this direct method would be a very useful tool in space weather research. As applications of the direct method, we investigate the inherent relations between the parallel mean free path and various parameters concerning physical properties of SEPs and solar wind. Comparisons of parallel mean free paths with and without adiabatic focusing are also presented.

Ultrafast Plasmon Propagation in Nanowires Characterized by Far-Field Spectral Interferometry

Spectral interferometry is employed to fully characterize amplitude and phase of propagating plasmons that are transmitted through silver nanowires in the form of ultrashort pulses. For nanowire diameters below 100 nm, the plasmon group velocity is found to decrease drastically in accordance with the theory of adiabatic focusing. Furthermore, the dependence of the plasmon group velocity on the local nanowire environment is demonstrated. Our findings are of relevance for the design and implementation of nanoplasmonic signal processing and in view of coherent control applications.

A diffusive description of the focused transport of solar energetic particles

The transport of solar energetic charged particles along the interplanetary magnetic field in the ecliptic plane of the sun can be described roughly by a one-dimensional diffusion equation. Large-scale spatial variations of the guide magnetic field can be taken into account by adding an additional term to the diffusion equation that includes the effect of adiabatic focusing. We solve this equation analytically by assuming a point-like particle injection in time and space and a spatial power-law dependence for the focusing length and the spatial diffusion coefficient. We infer the intensity- and anisotropy-time profiles of solar energetic particles from this solution. Through these the influence of different assumptions for the diffusion parameters can be seen in a mathematically closed form. The comparison of calculated and measured intensity- and anisotropy-time profiles, which are a powerful diagnostic tool for interplanetary particle transport, gives information about the large-scale spatial dependence of the focusing length and the diffusion coefficient. For an exceptionally large solar energetic particle event, which did occur on 2001 April 15, we fit the 27−512 keV electron intensities and anisotropies observed by the Wind spacecraft using the theoretically derived profiles. We find a linear spatial dependence of the mean free path along the guiding magnetic field. We also find the mean free path to be energy independent, which supports the theory of “velocity-dependent diffusion”. This means that the intensity profiles for the discussed energies exhibit the same shape if they are plotted against the traveled distance and not against the time. In this case the profiles differ only in their maximum values and we can determine the energy spectra of the solar flare electrons out of the scaling factor we need to fit the data. The derived spectra exhibits a power-law dependence ∝E −3 kin in an energy range from ∼50 keV to ∼500 keV.