Collision Effects in Ultra High Illuminated Plasmas
A.A. Balakin¤ and G.M. Fraiman¤
¤Institute of Applied Physics RAS, Nizhny Novgorod, Russia
Abstract. Electron-ion collisions in strong laser fields in the ultra-relativistic limit are considered. The accurate canonical
transformations to the relativistic drift coordinates are found. Expressions connecting electron momentums in drift and
laboratory coordinate system are obtained. The analysis of the expressions allows us to predict appearance of ultra fast
particles with the energies up to the third power of a pump wave vector potential. Electron distribution on energies and
its density are derived for collisional matter of high-energy (“hot”) electrons.
Keywords: relativistic laser-plasma interaction, electron-ion collision, hot electrons
PACS: 45.20.Jj, 52.20.Dq, 52.27.Ny, 52.35.Mw
It is known that electron-ion collisions in strong electromagnetic (EM) fields even with non-relativistic levels of
intensity yield quite a number of interesting effects: growth of joule heating , harmonics  and fast particles
generation . Passing to relativistic intensities of EM fields leads to unexpected modifications of the above-mentioned
effects. In particular as it is shown in this paper, fast electrons with the energies up to p3
osc/m2c appear at the electronion
collisions in ultra-relativistic fields (posc = −eA/c is the oscillator momentum of the electron, A is the vector
potential of an electromagnetic wave). Note, that the maximal energy of the fast particles grows even faster than in
the non-relativistic case, when the particles energy is restricted by the magnitude of 2p2
osc/m. The traces of such ultrafast
particles, apparently, can be found in the experiment , where, appearance of electrons with the energies up
to hundreds MeV is observed in the laser plasma interaction with laser intensity of the order of 1020 Wt/cm2. The
presence of fast electrons in these experiments is traditionally explained in the framework of collision-free models in
which the acceleration is realized by means of the excited plasma waves [5, 4, 6].
In the paper we’ll estimate the efficiency of fast electron generation in the transparent plasma at electron-ion
collision in the field of a relativistically strong laser pulse. All analysis will be performed without regard to the radiation
losses in the framework of classical motion equations. The paper consists of two parts. In the first one the problem of
instant (caused by the collisions) electron “injection” inside of a smooth in both longitudinal and transversal directions
wave packet is discussed. It is assumed, that the laser pulse propagates in vacuum with the light velocity and the
task is to estimate the maximal energy of an electron after its passing out of the wave packet area. This problem was
repeatedly solved by many authors (see  and the cited bibliography). The novelty of the part is in the fact that
passing to drift coordinates, corresponded to averaged pondermotive description in non-relativistic approximation,
is realized within the framework of the exact canonical transformation. That makes it possible to estimate the nonadiabatic
effects of the energy exchange between the electrons and the laser field. In the second part of the paper
the problem of electron injection, caused by the coulomb collisions is analyzed. An electron-ion impact befalls in a
moment during the time much less than the laser period. So the electron scattering one can consider as its injection
with a new cinematic momentum after scattering. By other words, in this part the influence of a laser field on the last
impact may be neglected. Taking into account this fact, we’ll make estimations for distribution of collisional electrons
and a density of the electrons current. In conclusion estimations and comparison with the known experiments will be
One of the reasons of writing this paper was the question about the magnitude of maximal energy, which can get a
particle in relativistically strong fields. For example, it is known  that during laser ionization the particles can have
the energy of the order of p2
osc/m = mc2a2. Here a = eA/mc2 is the frequently used notation for the normalized vector
potential. The main question is, whether it is possible to reach higher energy at collisions?
One can use well-known formulas for particle motion in the field of a plane monochromatic wave :
virtual electron-positron pairs and so on. Though these effects prove to be not so important. The birth of virtual
electron-positron pairs during the scattering leads to effective change of the ion charge on small value of the order of
10−3  and, more over, it enters into the dependence of distribution on energies (15) only as a factor. The expressions
for scattering matrix in quantum and classical problems coincide  (also like in non relativistic case), i.e. quantum
effects are inessential in broad diapason of parameters. The radiation of particle moving in the wave field tells on only
at pump intensities about 1028 Wt/cm2 , i.e. much higher of that being achieved for the present time. The question
about the influence of particles radiation during the electron-ion collision itself is still open. Though, this will rather
lead to variation of the scattering diagram only and to insignificant decrease of the scattered particle momentum. As a
result, dependence (16) may prove to be modified slightly, but the effect of super intensive particles appearance in itself
will stay. In particular, the presence of super fast particles with power law of distribution on energies in experiment 
In any case, expressions for drift coordinates in relativistically strong laser field is interesting in itself both for
collisions integral derivation and numerical simulation of elecron-ion collisions in relativistic fields.
This work has been supported by RFBR (grants 05-02-17367, 04-02-16684).
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