ABSTRACT
High intensity linacs and storage rings are being considered for fundamental research on matter at very high energy density and a variety of applications ranging from neutron spallation sources to nuclear wastes transmutation and inertial fusion. ln high intensity accelerators the major problems are the field quality and the control of small losses. ln the storage rings, where the mtmber of visited FODO cells can be as high as 106 , the resonances between the collective motion of the core aud nonlinear betatron motion may cause slow diffusion and affect the dynamic aperture. Moreover the collisional effects need to be considered because the changes induced in the distribution may be relevant on a long time scale. The semi-analytical tools available are mainly based on the particle-core models developed to study the behavior of a test particle in a given self consistent field of the core. The application of the frequency map analysis to these models allowed to detect the key role of the mismatch oscillations in the diffusion process which may explain the growth of the halo [I, 2, 31. The analytical self consistent solutions of the 2 D and 3D Poisson-Vlasov ( P. V.) equation for the particles distribution and the corresponding electric field are confined to the constant focusing case [4]. in which a a linear and nonlinear stability analysis of small perturbations have also been developed [5, 13[. The excitation of the lowest unstable mode was proposed as a possible escape mechanism from the core, generating the halo [6, 7j. For the periodic focusing case
numerical PIC solutions are extensively used since the only known analytic solution is KV distribution in the 2D case. We have developed a 2D PIC code for a lartice of identical FODO cells in order investigate its convergence properties, the onset of instabilities and the dynamics of test particles. This program has been undertaken using the KV as a reference solution in the periodic focusing case. Our 20 PIC solver is based on a symplectic integrator and a FFT Poisson solver which allows to impose Dirichelet conditions on a arbitrary closed boundary. For a matched KV solution the error on the electric field and on the moments of the distribution are analyzed as a function of the number K of Fourier components in each dimension and the number of pseudo-particles N. ln this case the rms radii, the emittance and the field error exhibit a linear growth. For a fixecl K we observe a decrease with N with a possible asymptotic limit; for K fixed the error decreases reaches a minimum and increases again. The linear growth of the moments is related to the accumulation numerical errors ( due to the grid size and to the density fluctuations), which behave as a noise in the equations of motion[! OJ; For a circular machine to keep the error growth sufficiently small is a computational challenge.!, In this case the eflect of Coulomb collisions cannot be neglected. To this end we have developed a molecular dynamics algorithm to integrate symplectically a system of N macro-particles, reducing the computational complexity from N 2 to N 312 (with a multipolar expansion of the far field). We have introduced a hard core T"H so that an elastic collision occurs whenever there is a contact. ln the absence of hard core the close encounters are resolved by increasing the integration accuracy. It is found that an initial KV distribution evolves towards the self consistent Maxwell-Boltzmann distribution. This relaxation process cannot be obtained within the framework of a mean field equation like P.V. The relaxation time increases as r 8 decreases and has a finite limit for r 8 ___. 0. An estimate of the relaxation time for the lattice of identical FODO cells previously considered is given. We have also developed a fully 3D parallel PIC code in order in order to study the evolution of a bunch in the linac lSCL, a project of the [NFN laboratories of Legnaro (Italy) .. Ln that case a Neuffer-KV initial distribution evolves numerically, with our PIC solver. towards a Fermi-Dirac distribution wbose tails are .'vlaxwellian. [II] We present a 3D particle core model defined by a Langevin equation with a drift and noise given by the Landau integrals in view of an inclusion of the collisions in the PIC code as recently proposed [12].
