Cross Section for Energy Transfer Between Two Moving Particles
Classical cross section for energy transfer in two-particle Coulomb collision
COULOMB COLLISION↗
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Stochastic acceleration of electrons. I - Effects of collisions in solar flares
Stochastic acceleration of thermal electrons to nonrelativistic energies is studied under solar flare conditions. We show that, in turbulent regions, electron-whistler wave interactions can result in the acceleration of electrons in times comparable to or shorter than the Coulomb collision time. The kinetic equation describing the evolution of the electron energy distribution including stochastic acceleration by whistlers and energy loss via Coulomb interactions is solved for an initial thermal electron energy spectrum. In general, the shape of the resulting electron distributions are characterized by the energy E(c) where systematic energy gain by turbulence equals energy loss due to Coulomb collisions. For energies less than E(c), the spectra are steep (quasi-thermal) whereas above E(c), the spectra are power laws. We find that hard X-ray spectra computed using the electron distributions obtained from our numerical simulations are able to explain the complex spectral shapes and variations observed in impulsive hard X-ray bursts. In particular, we show that the gradual steepening observed by Lin et al. (1981) could be due to a systematic increase in the density of the plasma (due to evaporation) and the increasing importance of collisions instead of the appearance of a superhot thermal component.
Hamilton, Russell J.↗
Establishing criteria for the transition from kinetic to fluid modeling in hollow cathode analysis
Hollow cathodes for plasma switch applications are investigated via 2D3V particle-in-cell simulations of the channel and plume region. The kinetic nature of the plasma within the channel is dependent on the thermalization rate of electrons, emitted from the insert. When Coulomb collisions occur at a much greater rate than ionization or excitation collisions, the electron energy distribution function rapidly relaxes to a Maxwellian and the plasma within the channel can be described accurately via a fluid model. In contrast, if inelastic processes are much faster than Coulomb collisions, then the electron energy distribution function in the channel exhibits a notable high-energy tail, and a kinetic treatment is required. This criterion is applied to hollow cathodes from the literature, revealing that a fluid approach is suitable for most electric propulsion applications, whereas a kinetic treatment can be more critical to accurate modeling of plasma switches.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
The behaviour of minor species in the solar wind
The flow of minor species in the solar wind is considered on the basis of a model in which all species are heated in an arbitrary manner as they leave the sun and the minor species interact with the background proton-electron plasma through the radial electric field associated with the latter and by means of Coulomb collisions. In order to produce satisfactory results in which the ions all move at more or less the same speed at the orbit of the earth, it is necessary to introduce heating functions such that each species is given energy in proportion to its mass. Coulomb collisions are found to be important as a means of removing energy from the heavier species close to the sun and bringing all species closer to thermal equilibrium at great distances from the sun. Substantial velocity differences can occur between species, especially close to the sun. Furthermore it is not difficult to construct solutions in which the bulk velocity of the helium ions exceeds that of the solar wind, as is often observed.
Ryan, J. M.↗
Gamma-ray emission and electron acceleration in solar flares
Recent observations have extended the spectra of the impulsive phase of flares to the GeV range. Such high-energy photons can be produced either by electron bremsstrahlung or by decay of pions produced by accelerated protons. In this paper we investigate the effects of processes which become important at high energies. We examine the effects of synchrotron losses during the transport of electrons as they travel from the acceleration region in the corona to the gamma-ray emission sites deep in the chromosphere and photosphere, and the effects of scattering and absorption of gamma rays on their way from the photosphere to space instruments. These results are compared with the spectra from so-called electron-dominated flares, observed by GRS on the Solar Maximum Mission, which show negligible or no detectable contribution from accelerated protons. The spectra of these flares show a distinct steepening at energies below 100 keV and a rapid falloff at energies above 50 MeV. Following our earlier results based on lower energy gamma-ray flare emission we have modeled these spectra. We show that neither the radiative transfer effects, which are expected to become important at higher energies, nor the transport effects (Coulomb collisions, synchrotron losses, or magnetic field convergence) can explain such sharp spectral deviations from a simple power law. These spectral deviations from a power law are therefore attributed to the acceleration process. In a stochastic acceleration model the low-energy steepening can be attributed to Coulomb collision and the rapid high-energy steepening can result from synchrotron losses during the acceleration process.
Petrosian, Vahe↗
Inner Magnetospheric Superthermal Electron Transport: Photoelectron and Plasma Sheet Electron Sources
Two time-dependent kinetic models of superthermal electron transport are combined to conduct global calculations of the nonthermal electron distribution function throughout the inner magnetosphere. It is shown that the energy range of validity for this combined model extends down to the superthermal-thermal intersection at a few eV, allowing for the calculation of the en- tire distribution function and thus an accurate heating rate to the thermal plasma. Because of the linearity of the formulas, the source terms are separated to calculate the distributions from the various populations, namely photoelectrons (PEs) and plasma sheet electrons (PSEs). These distributions are discussed in detail, examining the processes responsible for their formation in the various regions of the inner magnetosphere. It is shown that convection, corotation, and Coulomb collisions are the dominant processes in the formation of the PE distribution function and that PSEs are dominated by the interplay between the drift terms. Of note is that the PEs propagate around the nightside in a narrow channel at the edge of the plasmasphere as Coulomb collisions reduce the fluxes inside of this and convection compresses the flux tubes inward. These distributions are then recombined to show the development of the total superthermal electron distribution function in the inner magnetosphere and their influence on the thermal plasma. PEs usually dominate the dayside heating, with integral energy fluxes to the ionosphere reaching 10(exp 10) eV/sq cm/s in the plasmasphere, while heating from the PSEs typically does not exceed 10(exp 8) eV/sq cm/s. On the nightside, the inner plasmasphere is usually unheated by superthermal electrons. A feature of these combined spectra is that the distribution often has upward slopes with energy, particularly at the crossover from PE to PSE dominance, indicating that instabilities are possible.
Khazanov, G. V.↗
Inner Magnetospheric Superthermal Electron Transport: Photoelectron and Plasma Sheet Electron Sources
Two time-dependent kinetic models of superthermal electron transport are combined to conduct global calculations of the nonthermal electron distribution function throughout the inner magnetosphere. It is shown that the energy range of validity for this combined model extends down to the superthermal-thermal intersection at a few eV, allowing for the calculation of the entire distribution function and thus an accurate heating rate to the thermal plasma. Because of the linearity of the formulas, the source terms are separated to calculate the distributions from the various populations, namely photoelectrons (PEs) and plasma sheet electrons (PSEs). These distributions are discussed in detail, examining the processes responsible for their formation in the various regions of the inner magnetosphere. It is shown that convection, corotation, and Coulomb collisions are the dominant processes in the formation of the PE distribution function, and that PSEs are dominated by the interplay between the drift terms. Of note is that the PEs propagate around the nightside in a narrow channel at the edge of the plasmasphere as Coulomb collisions reduce the fluxes inside of this and convection compresses the flux tubes inward. These distributions are then recombined to show the development of the total superthermal electron distribution function in the inner magnetosphere and their influence on the thermal plasma. PEs usually dominate the dayside heating, with integral energy fluxes to the ionosphere reaching 10(exp 10) eV/sq cm/s in the plasmasphere, while heating from the PSEs typically does not exceed 10(exp 8)eV/sq cm/s. On the nightside, the inner plasmasphere is usually unheated by superthermal electrons. A feature of these combined spectra is that the distribution often has upward slopes with energy, particularly at the crossover from PE to PSE dominance, indicating that instabilities are possible.
Khazanov, G. V.↗
Guided Plasmaspheric Hiss Interactions with Superthermal Electrons: Resonance Curves and Timescales - Part 1
Under the proper conditions, guided plasmaspheric hiss is shown to be more efficient than Coulomb collisions at scattering electrons in the superthermal energy range of 50 to 500 eV. Broadband, whistler mode hiss becomes guided by plasma density gradients, intensifying the wave energy densities and focusing the wave normal angles. These waves are shown to interact through Cherenkov (Landau) resonance with electrons below 500 eV, and the presented equatorial plane timescales for pitch angle, energy, and mixed diffusion are shown to be faster than Coulomb collision timescales for typical values at the inner edge of the plasmapause and in detached plasma regions. In the latter case, energy diffusion timescales of less than 100 s for small pitch angle electrons between 250 and 500 eV indicate that these waves have the potential to dramatically change the distribution function.
Liemohn, M. W.↗
Guided Plasmaspheric Hiss Interactions with Superthermal Electrons: Resonance Curves and Timescales - 1
Under the proper conditions, guided plasmaspheric hiss is shown to be more efficient than Coulomb collisions at scattering electrons in the superthermal energy range of 50 to 500 eV. Broadband, whistler mode hiss becomes guided by plasma density gradients, intensifying the wave energy densities and focusing the wave normal angles. These waves are shown to interact through Cherenkov (Landau) resonance with electrons below 500 eV, and the presented equatorial plane timescales for pitch angle, energy, and mixed diffusion are shown to be faster than Coulomb collision timescales for typical values at the inner edge of the plasmapause and in detached plasma regions. In the latter case, energy diffusion timescales of less than 100 s for small pitch angle electrons between 250 and 500 eV indicate that these waves have the potential to dramatically change the distribution function.
Liemohn, M. W.↗
Energy-momentum-conserving stochastic differential equations and algorithms for the nonlinear Landau-Fokker-Planck equation
Coulomb collision is a fundamental diffusion process in plasmas that can be described by the Landau-Fokker-Planck (LFP) equation or the stochastic differential equation (SDE). While energy and momentum are conserved exactly in the LFP equation, they are conserved only on average by the conventional corresponding SDEs, suggesting that the underlying stochastic process may not be well defined by such SDEs. Here, in this study, we derive new SDEs with exact energy-momentum conservation for the Coulomb collision by factorizing the collective effect of field particles into individual particles and enforcing Newton's third law. These SDEs, when interpreted in the Stratonovich sense, have a particularly simple form that represents pure diffusion between particles without drag. To demonstrate that the new SDEs correspond to the LFP equation, we develop numerical algorithms that converge to the SDEs and preserve discrete conservation laws. Simulation results are presented in a benchmark of various relaxation processes.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Divergence of the total cross section for three body rearrangement collisions with coulomb interactions
Three charged particles 1, 2, 3 collide according to the reaction 1+(2+3) yields (1+3)+2, where (2+3) and (1+3) are hydrogenlike bound states. It is shown when (1+3) is in a highly excited state n, due to the repulsive potential, the cross section in the first Born approximation behaves as 1/n which makes the total cross section to diverge like ln n. The total cross sections in the higher orders of the Born approximation are similarly divergent logarithmically.
Omidvar, K.↗
A kinetic study of solar wind electrons in the transition region from collision dominated to collisionless flow
We have studied the evolution of the velocity distribution function of a test population of electrons in the solar corona and inner solar wind region, using a recently developed kinetic model. The model solves the time dependent, linear transport equation, with a Fokker-Planck collision operator to describe Coulomb collisions between the 'test population' and a thermal background of charged particles, using a finite differencing scheme. The model provides information on how non-Maxwellian features develop in the distribution function in the transition region from collision dominated to collisionless flow. By taking moments of the distribution the evolution of higher order moments, such as the heat flow, can be studied.
Lie-Svendsen, O.↗
Ion acoustic waves in a collisional plasma
Damping of ion-acoustic waves by Coulomb collisions of ion and electron combinations
COULOMB COLLISION↗
High-frequency waves in a collisional plasma with magnetic field
Fokker-Planck equation used to investigate damping of high frequency waves in hot plasma in external uniform magnetic field in presence of weak Coulomb collisions
HIGH FREQUENCY↗
Effect of Fokker-Planck collisions on plasma wave echoes
Fokker-Planck Coulomb collisions effect on plasma wave echoes
Johnston, G. L.↗
Multiple nucleon knockout by Coulomb dissociation in relativistic heavy-ion collisions
The Coulomb dissociation contributions to fragmentation cross sections in relativistic heavy ion collisions, where more than one nucleon is removed, are estimated using the Weizsacker-Williams method of virtual quanta. Photonuclear cross sections taken from experimental results were used to fold into target photon number spectra calculated with the Weizsacker-Williams method. Calculations for several projectile target combinations over a wide range of charge numbers, and a wide range of incident projectile energies, are reported. These results suggest that multiple nucleon knockout by the Coulomb field may be of negligible importance in galactic heavy ion studies for projectiles lighter than Fe-56.
Cucinotta, Francis A.↗
Quantitative tests of a steady state theory of solar wind electrons
A comparison is made of IMP 6, 7 and 8 electron data with the predictions of a solar wind electron steady state theory in which the control of transport by the macroscopic interplanetary electric and magnetic fields, as well as elastic Coulomb collisions with solar wind protons and thermal electrons, is assumed. While a ratio of forward to backward phase density for field-aligned extrathermal electrons of 6:1 is predicted, electron distribution measurements within the high speed solar wind show this ratio to be typically about an order of magnitude larger. A set of solar wind bulk speed anticorrelations predicted by the theory on the basis of a larger set of assumptions cannot be found in the IMP electron data set, so that improved agreement may require such modifications of the theory's assumptions as the inclusion of inelastic Coulomb and/or wave electron collisions.
Feldman, W. C.↗
Electrostatic waves and anomalous transport in the solar wind
In situ measurements of fluctuation spectra and particle distribution functions have now been carried out throughout interplanetary space. The link between these observations is established by theories of wave particle interaction. Linear instability analysis for the actual nonMaxwellian particle distribution functions and an examination of the velocity dependence of microscopic diffusion coefficients form the basis of such an investigation. It is described in more detail for the short wavelength, ion acoustic like turbulence which is found by linear instability analysis to correspond to the observed electrostatic fluctuations. Of the transport processes associated with these fluctuations, electron heat conduction and electron ion energy transfer are of particular importance for macroscopic solar wind expansion. These effects are studied with the aid of an anomalous transport theory. This theory (Dum, 1978 a,b) is based on the dominance of elastic scattering of electrons by fluctuations, similar to (enhanced) electron ion collisions. It has a much wider range of applicability than classical transport theory, which assumes dominance of Coulomb collisions for elastic and inelastic scattering.
Dum, C. T.↗