Isolated unstable Weibel modes in unmagnetized plasmas with tunable asymmetry

In this letter, an initially unmagnetized pair plasma with asymmetric velocity distributions is investigated where any unstable Weibel mode must be isolated, with discrete values for the growth rates and the unstable wavenumbers. For both a non-relativistic distribution with thermal spread and a high-relativistic two-stream distribution it is shown that isolated modes are excited and that, as the asymmetry tends to zero, the growth rate remains finite, as long as the distribution function is not precisely symmetric.Comment: Comments: references adde

Non-linear Weibel-type Soliton Modes

Discussion is given of non-linear soliton behavior including coupling between electrostatic and electromagnetic potentials for non-relativistic, weakly relativistic, and fully relativistic plasmas. For plasma distribution functions that are independent of the canonical momenta perpendicular to the soliton spatial structure direction there are, in fact, no soliton behaviors allowed because transverse currents are zero. Dependence on the transverse canonical momenta is necessary. When such is the case, it is shown that the presence or absence of a soliton is intimately connected to the functional form assumed for the particle distribution functions. Except for simple situations, the coupled non-linear equations for the electrostatic and electromagnetic potentials would seem to require numerical solution procedures. Examples are given to illustrate all of these points for non-relativistic, weakly relativistic, and fully relativistic plasmas.Comment: Accepted for publication at Journal of Physics A: Mathematical and Theoretica

Pitch-angle scattering in magnetostatic turbulence. II. Analytical considerations and pitch-angle isotropization

Aims. The process of pitch-angle isotropization is important for many applications ranging from diffusive shock acceleration to large-scale cosmic-ray transport. Here, the basic analytical description is revisited on the basis of recent simulation results. Methods. Both an analytical and a numerical investigation were undertaken of the Fokker-Planck equation for pitch-angle scattering. Additional test-particle simulations obtained with the help of a Monte-Carlo code were used to verify the conclusions. Results. It is shown that the usual definition of the pitch-angle Fokker-Planck coefficient via the mean-square displacement is flawed. The reason can be traced back to the assumption of homogeneity in time which does not hold for pitch-angle scattering. Conclusions. Calculating the mean free path via the Fokker-Planck coefficient has often proven to give an accurate description. For numerical purposes, accordingly, it is the definition that has to be exchanged in favor of the pitch-angle correlation function.Comment: 5 pages, 5 figures, accepted for publication in Astron. Astrophy

Cosmic-ray acceleration at collisionless astrophysical shocks using Monte-Carlo simulations

Context. The diffusive shock acceleration mechanism has been widely accepted as the acceleration mechanism for galactic cosmic rays. While self-consistent hybrid simulations have shown how power-law spectra are produced, detailed information on the interplay of diffusive particle motion and the turbulent electromagnetic fields responsible for repeated shock crossings are still elusive. Aims. The framework of test-particle theory is applied to investigate the effect of diffusive shock acceleration by inspecting the obtained cosmic-ray energy spectra. The resulting energy spectra can be obtained this way from the particle motion and, depending on the prescribed turbulence model, the influence of stochastic acceleration through plasma waves can be studied. Methods. A numerical Monte-Carlo simulation code is extended to include collisionless shock waves. This allows one to trace the trajectories of test particle while they are being accelerated. In addition, the diffusion coefficients can be obtained directly from the particle motion, which allows for a detailed understanding of the acceleration process. Results. The classic result of an energy spectrum with $E^{-2}$ is only reproduced for parallel shocks, while, for all other cases, the energy spectral index is reduced depending on the shock obliqueness. Qualitatively, this can be explained in terms of the diffusion coefficients in the directions that are parallel and perpendicular to the shock front.Comment: 12 pages, 15 figures, accepted for publication in A&

The Theory of Scanning Quantum Dot Microscopy

Electrostatic forces are among the most common interactions in nature and omnipresent at the nanoscale. Scanning probe methods represent a formidable approach to study these interactions locally. The lateral resolution of such images is, however, often limited as they are based on measuring the force (gradient) due to the entire tip interacting with the entire surface. Recently, we developed scanning quantum dot microscopy (SQDM), a new technique for the imaging and quantification of surface potentials which is based on the gating of a nanometer-size tip-attached quantum dot by the local surface potential and the detection of charge state changes via non-contact atomic force microscopy. Here, we present a rigorous formalism in the framework of which SQDM can be understood and interpreted quantitatively. In particular, we present a general theory of SQDM based on the classical boundary value problem of electrostatics, which is applicable to the full range of sample properties (conductive vs insulating, nanostructured vs homogeneously covered). We elaborate the general theory into a formalism suited for the quantitative analysis of images of nanostructured but predominantly flat and conductive samples