Search Results (14 total)

A few years ago, a novel multiturn extraction scheme was proposed, based on particle trapping inside stable resonances. Numerical simulations and experimental tests have confirmed the feasibility of such a scheme for low order resonances. While the third-order resonance is generically unstable and those higher than fourth order are generically stable, the fourth-order resonance can be either stable or unstable depending on the specifics of the system under consideration. By means of the normal form, a general approach to control the stability of the fourth-order resonance has been derived. This approach is based on the control of the amplitude detuning and the general form for a lattice with an arbitrary number of sextupole and octupole families is derived in this paper. Numerical simulations have confirmed the analytical results and have shown that, when crossing the unstable fourth-order resonance, the region around the center of the phase space is depleted and particles are trapped in only the four stable islands. A four-turn extraction could be designed using this technique.

We consider the dynamical properties of a particle-core model for a uniformly filled triaxial ellipsoid in a periodic lattice of a high intensity linac. The mismatched oscillation modes are analytically computed in the smooth approximation and are compared with the numerical results of a tracking program. The study of the phase space in the mismatched case is performed by the frequency map analysis. In particular, we can analyze the effect of the nonlinear resonances between the envelope modes and the single particle sincrobetatron frequencies. A chaoticity criterion based on the frequency map analysis allows one to compute the stability region around the beam core. An estimate of the transport and its enhancement due to mismatch is provided by tracking orbits at the border of the stability region.

A simple 1D model is proposed to explore the resonant extraction of intense beams from a synchrotron as performed in the SIS synchrotron in GSI (Darmstadt). The model Hamiltonian consists of a constant focusing, a thin sextupole, and a smooth space charge field. Hyperbolic normal forms are used to estimate the extraction times and the emittance of the extracted beam; the quality of the reconstruction is tested in absence of space charge. The effect of space charge on the dynamical behavior of the beam near the 1/3 betatron resonance is numerically investigated using the frequency map analysis and qualitatively explained with perturbation theory. A polynomial approximation to the one turn map is obtained by replacing the exact space charge force with a sequence of polynomial kicks, and the resonant normal forms reproduce quite accurately the nonlinear tunes and the fixed points position. At low order an analytical estimate of the area of the stable region is proposed to recover the self-consistency of the model.

The network of resonances in the action plane for a four-dimensional map is obtained by computing the actions from the Fourier coefficients of the orbits, and it is compared with the results of Birkhoff normal forms. This method, which combines the positive features of standard frequency analysis and normal forms, is suitable to study the one turn map of a particle accelerator.

The presence of a natural boundary in the angle complex plane of the function conjugating an area preserving map with a rotation is a signature of nonintegrability. Numerical results suggest that the boundary arises from the condensation of singularities when a real nonresonant frequency is approached by a sequence of complex resonant frequencies. For the quadratic map of the complex plane F: z’=λz+z² the function conjugating the linear part of F with F has q cuts with end points on a spiral if λ=e^2πip/q-α with α>0. When p/q tends to a (quadratic) irrational number and α→0 the points coalesce on the boundary of a circle, whose image is the Siegel disk.

We investigate the diffusion in the action variable when the frequency of an integrable isochronous map is modulated. Purely stochastic, hyperbolic, or periodic deterministic modulations are considered. The diffusion coefficient in the invariant for the unperturbed map is exactly determined and shown to be nonzero, except in the last case, when the modulation is smooth and nonzero, due to the presence of topological barriers.

The Mellin transform of the correlation integral is introduced and proved to be equal to the energy integral whose divergence abscissa is a lower bound to the Hausdorff dimension. For some Julia sets exact results are obtained. For the linear Cantor sets on the real axis it is shown that the energy integral is meromorphic, and the real pole, determining the divergence abscissa, has a sequence of satellite poles equally spaced on a line parallel to the imaginary axis, which explain the oscillations observed in numerical calculations of the correlation integral. The order-d generalized energy integrals are introduced as Mellin transforms of the order-d correlation integrals and for the Cantor sets they are proved to have the same singularities as the ordinary energy integrals. Letting r_d be the residue of the real pole corresponding to the divergence abscissa it is proved that lim_d→∞(-d^-1lnr_d) is the second Renyi entropy. Some numerical results obtained for the energy integrals are discussed.

Let F(q²)=(4π / Zq)∫0^∞rsinqr ρ(r) dr be a charge density form factor for an atomic nucleus. Then it is shown how rigorous upper and lower bounds can be imposed on a variety of indefinite integrals involving ρ(r) starting from information about F(q²) which may itself derive from experimental measurements. Examples are provided in the context of the ³He nucleus to demonstrate the feasibility of the method.

NUCLEAR STRUCTURE Moment theory, nuclear form factors.

We fully develop the content of the Schwinger variational principle for the phase shifts in potential scattering. We introduce matrix Padé approximations built up from the perturbation expansion of the Green's function. They appear to lead to a new type of (Padé) approximation when optimized through the variational principle. These new approximations, which are no longer rational fractions in the expansion parameter, appears to have the full analytical richness of the exact solution. For the case of a nonchanging-sign potential these new types of approximations provide the best bounds to the phase shifts and bound states. The extension to arbitrarily singular potentials is also discussed. A numerical example confirms the extreme efficiency of the method Typically, for values of the coupling giving rise to one or two bound states the phase shifts are obtained within 10^-3 of their exact values, and this on the full range of energy, by including only the first and second Born terms of the perturbation series.

A modification in the definition of one invariant in our earlier paper is presented which allows an easier computation of the vector-meson contribution in the one-loop approximation.

We discuss the problem of which diagrams should be taken for the matrix Padé analysis and for the Bethe-Salpeter equation. We relate the "potential" of the Bethe-Salpeter equation with bare intermediate nucleons to the "potential" of the Bethe-Salpeter equation with dressed intermediate nucleons. We discuss the origin of the differences in recent calculations of the nucleon-nucleon phase shifts based on matrix Padé approximants.

From a unitary approximation to the nucleon-nucleon Green's function we derive the phase shifts for the Yukawa model and the nonlinear σ model. The results agree with the experimental data except for the lowest waves: ¹S₀, ³S₁, ³P₀ where, however, we get good scattering lengths.

A formalism for a complete partial-wave expansion of the four-nucleon Green's function is given by generalizing a standard method. All of the constraints imposed by the symmetry properties (parity, time reversal, and exchange symmetry) are worked out, and an extended unitarity condition is shown to be satisfied. The use of Padé approximations to the Green's function is shown to provide a physical amplitude which is unitary, has correct threshold behavior in all waves, and has good analyticity properties. Therefore, such a scheme is well suited to explore beyond the Born term the dynamical content of various Lagrangian models proposed for the nucleon-nucleon interaction.

The unitary Padé approximants, successfully introduced in strong-interaction physics for the pion and kaon systems, are now applied to the nucleon-nucleon problem. It is assumed that the interaction between two nucleons is described by the renormalizable Lagrangian L_I=igψ̅ γ₅τψΦ+λ(Φ·Φ)². We present the result of the complete calculation of the [1,1] unitary Padé approximant, which does not involve the second term in the Lagrangian: This implies that no free parameters appear in our model. A complete description of low-energy nucleon-nucleon physics is then obtained which qualitatively and often quantitatively agrees with experiment. Bound states appear only in S waves, and a real pole is found in the deuteron amplitude at 4.8 MeV when the pion-nucleon coupling constant is taken at its physical value g² / 4π=14.7. The Regge trajectories rise with energy: The deuteron recurrence does not become physical, while the recurrences of the virtual ¹S₀ state give rise to narrow resonances in the ¹D₂ and ¹G₄ waves. For all waves (with the exception of the ¹S₀ which in the [1,1] Padé approximation has a wrong threshold behavior), the calculated phase shifts are in good qualitative agreement with the experimental phase-shift analysis.