D. Teychenne et al., OSCILLATORY RELATIVISTIC MOTION OF A PARTICLE IN A POWER-LAW OR SINUSOIDAL-SHAPED POTENTIAL WELL, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 49(4), 1994, pp. 3253-3263
We report on an analytical work about the one-dimensional undamped rel
ativistic motion of a particle located in a potential well whose space
dependence is given by a power law or a sinusoidal law. This analysis
is of interest in the present-day research field on interaction of ul
tra-high-intensity laser pulses with plasmas. First, we consider the o
scillating motion of a particle trapped inside a potential-energy prof
ile of the form K\x\n where K is a constant, x is the space coordinate
, and n is a positive real number. The cases n = 1 and n =2 are emphas
ized since the former is related to the motion of electrons that exit
an homogeneous electrically neutral plasma towards vacuum and the latt
er is related to the usual electron plasma oscillation. Second, we stu
dy the potential-energy profile cos(x) in connection with the accelera
tion of an electron inside an electron plasma wave. Both trapped and u
ntrapped particle motions are considered. For all the potential-energy
shapes, analytic expressions of both the period of the oscillating mo
tion and the particle trajectory are provided. The particle motion in
the weakly relativistic case is discussed. The acceleration length of
a particle trapped inside a sinusoidal drifting wave is finally calcul
ated.