Frequency and Diffusion Maps for the Sns Ring

Using a single particle dynamics approach, the major magnetic non-linearities of the SNS accumulator ring are studied. Frequency maps are employed in order to display the global dynamics of the beam, for several working points. By means of diffusion maps the major resonances are explored and their bandwidth is estimated. The global tune diffusion coefficient is finally used to compare and choose the best working point. 1 THE SNS ACCUMULATOR RING The SNS ring is designed to accumulate a high-intensity beam with a beam power of up to 2MW. The ring lattice has a 4-fold symmetry and consists of FODO arcs and quadrupole doublet straight sections [1]. The beam dynamics is mostly dominated by multi-particle effects such as space-charge [2] and other collective instabilities (e.g. electron-proton [3]), which are mostly intensity dependent. On the other hand, the strict fractional beam loss, dictated by the permissible radiation level for hands-on maintenance [4], imposes a very good control and correction of all non-linear effects. This includes the ones coming from magnet imperfections, the impact of which is amplified due to the large beam emittance. Indeed, in the early commissioning stages where the beam intensity will be low, the dominant effects will be of a single-particle nature. One of them is the chromaticity: in order to reduce the effect of collective instabilities the longitudinal phase space is painted to a large momentum spread of p=p = 0:7%, with the bucket size set at p=p = 1% and the momentum aperture at p=p = 2%. Taken the bucket size as the reference, the chromatic tune-shift is around 0.08, which is close to the space charge tune-shift of 0:15 0:20. Another key issue for the performance of the SNS ring is the proper working point (w.p.) choice. After an extensive tuning campaign [5], we have concentrated on 4 potential w.p. candidates, due to optical matching and aperture considerations. Two of them have a tune-split of more than half a unit, in order to avoid strong coupling from spacecharge and/or magnet errors: (Qx; Qy) = (6:3; 5:8), the old “nominal” w.p. for which most of the past studies have been performed, and (Qx; Qy) = (6:23; 5:24). The other two are in a quadrant of the tune space where structural resonances are not present: (Qx; Qy) = (6:23; 6:20), close to the coupling resonance which avoids all the main 3rd order resonances and (Qx; Qy) = (6:4; 6:3). In order to choose the best w.p., with respect to the non-linear particle Work performed under the auspices of the US Department of Energy y [email protected] behavior, we use Laskar’s frequency map analysis [6]. In brief, the frequency map is a detailed tune footprint. Due to the very high precision of the tune determination through a refined Fourier analysis using the NAFF algorithm, all the excited resonances appear as line distortions of the footprint. The tune variation along the particle trajectory can give a good estimation of the diffusion rate, which can then be mapped to the real space, for each particle initial condition. Finally, the average tune diffusion coefficient can be used as a global quality factor for comparison between the dynamics of the different designs. 2 WORKING POINT COMPARISON We choose a standard SNS ring lattice matched for the four w.p. in study. The fringe fields in the ends of the quadrupoles are modeled as thin kicks using 5th order maps issued by MaryLie 5 [7]. Systematic and random magnet errors at the required level of 10 4 of the main magnet field are also included [8]. The chromaticity sextupoles are switched off in order to study the effect of the chromatic tune-shift and then tuned to give 0 chromaticity. In all the simulations, the RF voltage is set to 0, for simplicity. Finally, space-charge forces are not included, as we would like at a first stage to study the single particle resonance effects to the SNS beam. A total of 1500 particles are uniformly distributed on the phase-space with zero initial momentum and different initial transverse coordinates, up to 480 mm mrad (the transverse physical aperture) and tracked with the FTPOT module of the UAL environment [9]. They are given 9 different p=p values, from +2% to -2% corresponding to the beam momentum aperture. The tracking is conducted for 500 turns to provide a good precision in the tune determination. Frequency maps for the w.p. (6.4,6.3) and natural chromaticity are displayed in Fig. 1, where different colors correspond to different p=p. The triangular shape of the maps reflects the octupole-like nature of the leading order quadrupole fringe fields [10], which apparently dominate the single-particle non-linear dynamics. The first evident observation is that the beam is crossing a multitude of excited resonant lines, which appear as distortions of the frequency map, due to the huge chromatic tune-shift of more than 0.3, for 2% . This can be clearly viewed in Fig. 2, where we zoom on one of the frequency maps of Fig. 1. For dp=p=-1%, the chromatic tune-shift moves the beam very close to the half-integer resonance which is excited by quadrupole errors and quadrupole fringe fields, as well. Its width is quite big and can be measured on the frequency map to be around 0.004. In addition, the structural 5th or0-7803-7191-7/01/$10.00 ©2001 IEEE. 462 Proceedings of the 2001 Particle Accelerator Conference, Chicago