Control of Chaotic Particle Motion Using Adiabatic Thermal Beams

Charged-particle motion is studied in the self-electric and self-magnetic fields of a well-matched, intense charged-particle beam and an applied periodic solenoidal magnetic focusing field. The beam is assumed to be in a state of adiabatic thermal equilibrium. The phase space is analyzed and compared with that of the well-known Kapchinskij-Vladimirskij (KV)-type beam equilibrium. It is found that the widths of nonlinear resonances in the adiabatic thermal beam equilibrium are narrower than those in the KV-type beam equilibrium. Numerical evidence is presented, indicating almost complete elimination of chaotic particle motion in the adiabatic thermal beam equilibrium. INTRODUCTION Several kinetic equilibria have been discovered for periodically focused intense charged-particle beams. Well-known equilibria for periodically focused intense beams include the Kapchinskij-Vladimirskij (KV) equilibrium in an alternating-gradient (AG) quadrupole magnetic focusing field [1,2] and the periodically focused rigid-rotor Vlasov equilibrium of the KV type in a periodic solenoidal magnetic focusing field [3,4]. Both of these beam equilibria [1-4] have a singular (   function) distribution in the four-dimensional phase space. Such a   function distribution gives a uniform density profile across the beam in the transverse directions, and a transverse temperature profile which peaks on axis and decreases quadratically to zero on the edge of the beam. Because of the singularity in the distribution functions, these beam equilibria are not likely to occur in real physical systems and cannot provide realistic models for theoretical and experimental studies and simulations except in the zero-temperature limit. For example, the KV equilibrium model cannot be used to explain the beam tails in the radial distributions observed in recent high-intensity beam experiments [5]. Recently, adiabatic thermal beam equilibria have been discovered in a periodic solenoidal magnetic focusing field [6-8] and an AG quadrupole magnetic focusing field [8,9]. The measured density distribution [5] matches that of the adiabatic thermal beam equilibrium in a spatially varying solenoidal magnetic focusing field [6,8]. There have been many studies of charged-particle dynamics in the KV-type equilibria [10-14]. These studies have shown that the phase space for the KV-type equilibria exhibits rich nonlinear resonances and chaotic seas for charged particles outside the beam envelope [10,11]. If charged particles cross the beam envelope due to perturbations, they may enter chaotic seas to form a beam halo or cause beam losses [12-14]. THEORY AND SIMULATION We study charged-particle dynamics in the adiabatic thermal equilibrium of an intense charged-particle beam propagating with constant axial velocity z bcê  in the periodic solenoidal magnetic focusing field [15]       y x z z z ext y x ds s dB s B e e e B       2 1 , (1) where z s  is the axial coordinate,     s B S s B z z   is the axial magnetic field, S is the fundamental periodicity length of the focusing field, and c is the speed of light in vacuum. The adiabatic thermal beam equilibrium has been derived under the paraxial approximation with the following assumptions: 1) S rbrms  , where brms r is the RMS beam radius and 2) 1 / 2 3  b b   , where 2 2 / mc N q b   is the Budker parameter of the beam, q and m are the particle charge and rest mass, respectively,   rdr s r n N b b  2 , 0    = const is the number of particles per unit axial length, and 2 / 1 2 ) 1 (    b b   is the relativistic mass factor. In the adiabatic thermal beam equilibrium [6-8], the beam density distribution is given by