4.9.5. Cyclotron acceleration of ions by the combination frequency
Considered below will be the effects of ion acceleration in magnetic field in the events of induced scattering. The energy conservation law in the case of scattering in magnetic field is of the form a
At v = 0 it is expedient to treat the scattering in the magnetic field; at v = ± 1 the cyclotron acceleration by combination frequency a-a will be treated. The cyclotron acceleration by the combination frequency (v = ± 1) in an intense magnetic field is much in excess of the acceleration owed to induced scattering (v = 0). In this case the ion velocity components perpendicular to the external magnetic field are predominantly increased. The estimate of the effect of acceleration by the plasma waves is of the form (Tsytovich, 1966c):
dtX 32 mik2 nlvr where vgr is the wave group velocity, and W is the energy density of the waves.
The work (Smith, 1976) considers the acceleration of particles in their interactions with various plasma waves in a magnetic field H when the ratio of electron gyro-frequency to the electron plasma frequency aLe/moe - 1. The kinetic equations are presented describing the evolution of the energy distribution of particles and plasmons including their forced scattering. Simplifying assumptions are made (the possibility of averaging over the non-relativistic and Maxwell distributions of ions and over the azimuth in a plane perpendicular to H and the possibility of limitation to the main terms of the expansion in («¿e/ aooe), and others) and the approximate expressions for the scattering probability and the ratio of the electric energy of waves to their total energy have been obtained. The general mode of the behavior of the plasmons filling numbers has been studied and the relaxation time of the plasmons distribution has been estimated. It has been shown that the nonlinear process of the forced scattering by polarized ion clouds results in wave collapse and in an almost one-dimensional number spectrum extended along H. The consecutive acceleration of relativistic and non-relativistic particles has been studied. It has been shown that such an acceleration is more effective for non-relativistic particles (protons); in this case, if the wave distribution is negatively sloped, the acceleration decreases for small velocities and increases for high velocities compared with the acceleration and isotropic distribution of the plasma waves in magnetic field. This should result in further changes of the wave spectrum and the value of the acceleration.
Gintzburg (1967, 1968) has obtained the solution for the equation of two-fluid magnetic hydrodynamics in the form of periodic ion waves of large amplitude with simultaneous rotation of the magnetic field vector and change of the magnetic field modulus. The analytic expressions for the wave profile have also been found. The study of nonlinear ion waves with relativistic velocities shows that they may accelerate particles up to high energies. It is not excluded that this mechanism may be of definite importance in the case of CR generation in chromospheric flares on the Sun and other stars as well as in Supernova explosions. Particle acceleration by nonlinear waves was considered in details in Sagdeev et al. (M1988), and He (1998, 1999, 2000, 2002, 2003). He (2003) considered dimensionless Newtonian equations for charged particle in an electric field:
where x and v are particle position and velocity respectively, charge number q = ±1, and potential ((x, t) is chosen as a solution of the driven/damped nonlinear drift-wave equation:
with a periodic system of length 2n and fixed constants of a = - 0.287, c = 1.0, f= - 6.0, y = 0.1 (there was considered only the effect of nonlinear waves of Eq. 4.9.13 on the particle, while neglecting other effects related to the physical system from which the drift-wave equation is derived). A lot of simulations for different cases lead He (2003) to following conclusions. In all the tested non-steady wave fields a slow particle can be accelerated in the orientation of the steady waves. However, in the spatially regular field the particle is finally trapped by a wave trough and eventually acquires the group velocity of the steady waves, whilst in the chaotic both in time and in space field the particle experiences trapped and free phases randomly, depending on the charge sign the averaged velocity can be larger or smaller than the group velocity of the steady waves. It is shown that the virtual pattern of saddle steady waves plays the role of an asymmetric potential, which together with nonzero perturbation waves are necessary for the acceleration.
Bloomberg and Gary (1973) have considered the particle acceleration by electrostatic waves with a phase velocity increasing as such wave propagates in the space. The one-dimensional motion of a charged particle in an electrostatic wave propagating in inhomogeneous medium has been analyzed. It is assumed that the wave is of a fixed frequency and has the wave vector decreasing in space. The particle is accelerated in two stages. At first it is trapped between the wave crests. Then the particle is accelerated without oscillations in the well. The particle velocity at the ends of the first and second stages has been estimated. The estimate for the first stage has been made using the adiabatic invariant and gives a velocity ^ E14, dx/dt = v, dv/dt = -qV((x, t ),
where E is the electrostatic field in the wave. For the second stage the velocity rc E12 . Numerical calculations have been carried out for a wave with wave vector depending on distance x according to the law k(x) = ko (1 + ax)-1. The calculations were carried out for 20 particles with the same initial velocities distributed uniformly along a segment equaling the period. The calculation results show that the particle may be considerably accelerated during the second stage, already after their oscillatory motion stops.
4.9.9. Stochastic Fermi acceleration by the turbulence with circularly polarized Alfven waves
Ostrowski and Siemieniec-Ozi^blo (1997) demonstrated that the forward-backward asymmetry of particle scattering (as measured in the scattering center rest frame) at randomly moving scattering centers can lead to a first order regular acceleration term, in addition to the one resulting from the momentum diffusion. A physical example of such asymmetric scattering provides a finite amplitude circularly polarized Alfven wave (Siemieniec-Ozi^blo et al., 1999). This research was continued in the paper of Michalek et al. (1999), in which were presented preliminary results of Monte Carlo modeling of the particle acceleration/diffusion process for protons interacting with finite amplitude circularly polarized Alfven waves. It was shown that the scattering's forward-backward asymmetry occurring for such waves allows for the first order acceleration effects to occur in the stochastic acceleration process, enabling in favorable conditions for more effective acceleration in comparison to the linearly polarized Alfven waves of the same amplitude.
4.10. Statistical acceleration of particles by electromagnetic radiation
4.10.1. Effectiveness of charged particle acceleration by electromagnetic radiation; comparison with the Fermi mechanism
Tsytovich (1963b,d), Nikolaev and Tsytovich (1976) studied the charged particle acceleration by electromagnetic radiation and compared the effectiveness of this mechanism with the Fermi mechanism. It has been shown that whilst the force affecting the charge in vacuum is determined by the light pressure proportional to the Thomson cross-section, the same force in the medium with the same radiation flux will increase by a factor of A/ro , where A is the wavelength and ro is the radius of a particle with charge Ze. The following form of relativistic expression has been obtained for the change of energy E of a particle with charge Ze in a medium with refractive index n in the field of isotropic radiation of density W with frequency ro, wavelength A = 2m/ro and propagation velocity u = c/n :
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