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Type of presentation requested : Poster Classification : [D02] Nonlinear Dynamics  Resonances, Tracking, Higher Order 449  Fundamental Aspects of the Moment Problem for a Particle Density evolving under Nonlinear Forces J.M. De Conto, C. Peaucelle, ISN, Grenoble A statistical description (moments) of a beam can be foreseen to study its evolution when the force is not linear. In this case, higher order moments must be estimated from the known moments. We present the fundamental aspects of this problem, showing how high order moments can be extrapolated by using orthogonal polynomials associated to the beam density, in a twodimension phase space, and showing what kind of maximum information can be obtained from a finite number of moments. In particular, this gives a good estimate of the particle localisation and density. Its shows also the limits of such methods. Type of presentation requested : Poster Classification : [D02] Nonlinear Dynamics  Resonances, Tracking, Higher Order 600  Analytical Formalism for the Longitudinal Acceleration including Particle Velocity Changing Effect* M. Doleans, S.H. Kim, E. Tanke, ORNL, Oak Ridge To calculate the energy gain of a particle passing through an accelerating element, the Panofsky equation which introduces the concept of transit time factor and average phase is widely used. For the transit time factor the assumption of a constant beta through the element is usually made. In the case of large acceleration or long and complex accelerating element this approach can lead to some inaccuracies. To address this problem a method taking into account the variation of the beta within the accelerating element has been developed. This method is applicable to any element by using decomposition of the electrical field into fourier components. The average phase concept is adapted to the new formulation and the passage from the physical entrance phase to the average phase is clearly stated. The energy gain and time of flight functions are analytical and allow the treatment of a bunch of particles by expansion around a reference particle. The accuracy of the method is presented by comparison with a slow and precise numerical approach. *SNS is managed by UTBattelle, LLC, under contract DEAC0500OR22725 for the U.S. Department of Energy. SNS is a partnership of six national laboratories: Argonne, Brookhaven, Jefferson, Lawrence Berkeley, Los Alamos, and Oak Ridge. Type of presentation requested : Poster Classification : [D02] Nonlinear Dynamics  Resonances, Tracking, Higher Order 1598  Quasiperiodic Spinorbit Motion in Storage Rings and a Rigorous Definition of Spin Tune* J.A. Ellison, UNM, Albuquerque; D.P. Barber, K. Heinemann, DESY, Hamburg We present a rigorous discussion of the concept of spin tune for integrable orbital motion in a storage ring. Spin motion on the periodic closed orbit of a storage ring can be analyzed in terms of the Floquet theorem for equations of motion with periodic parameters and the spin tune emerges in a Floquet exponent as an extra frequency of the system. To define spin tune on synchrobetatron orbits we exploit the important concepts of quasiperiodicity, the uniform precession frame and the uniform precession rate. These allow a generalization of the Floquet theorem whereby the uniform precession rate appears in a Floquet exponent as an extra frequency in the system in analogy with the case of motion on the closed orbit. The spin tune is a uniform precession rate obtained when certain conditions are fulfilled. Having defined spin tune we define spinorbit resonance on synchrobetatron orbits and can then examine its consequences. We give conditions (e.g. where small divisors are controlled by applying a Diophantine condition) for the existence of a spin tune and illustrate with several examples. The formalism suggests the use of Fourier analysis to "measure" spin tune during computer simulations of spin motion on synchrobetatron orbits. *Work of J.E. supported by DOE contract DEFG0399ER41104. Type of presentation requested : Poster Classification : [D02] Nonlinear Dynamics  Resonances, Tracking, Higher Order 1638  Applying the Birkhoff Ergodic Theorem to Stroboscopic Averaging of Spin Fields and Liouville Densities* J.A. Ellison, UNM, Albuquerque; K. Heinemann, DESY, Hamburg A class of orbital systems with azimuthperiodic vector fields and volume preserving flows is defined. Spin degrees of freedom are added and for the resulting class of spinorbit motions spin fields are considered. By performing 'stroboscopic" averaging, i.e. a ${\rm Ces\grave{a}ro}$ summation of the spin field at an increasing sequence of azimuth values, one gets, via the Birkhoff Ergodic Theorem, polarization fields which are periodic in the azimuthal variable. This fulfills the main task of this paper in that it demonstrates that the tracking algorithm, encoded in the program SPRINT and used in the simulation of spin polarized storage rings, is mathematically wellfounded. Examples are considered which illustrate certain points. The machinery developed also works for "Liouville" densities associated with the orbital dynamics. A tracking algorithm is derived which provides periodic Liouville densities. *Work of J.E. supported by DOE contract DEFG0399ER41104. Type of presentation requested : Poster Classification : [D02] Nonlinear Dynamics  Resonances, Tracking, Higher Order 985  Analytic Studies of the Long Range BeamBeam Tune Shifts and Chromaticities B. Erdelyi, T. Sen, FNAL, Batavia A formula was derived, which allows efficient analytical evaluation of the long range beambeam tune shifts with amplitude and the chromaticities. It is assumed that the beams are infinitely short, oppositely charged, and with Gaussian transversal profile. The formula employs a quite rapidly converging infinite sum, with convergence rate depending only on the amplitudes, making it well suited especially for the long range case. For a deeper understanding of the beambeam effects, the structure of the formula is analyzed. It is shown that the tune shifts change sign at certain values of the separation and/or amplitude, and folds in the tune footprint are predicted. Applications to the Tevatron are presented. Type of presentation requested : Poster Speaker : Dr. T. Sen Classification : [D02] Nonlinear Dynamics  Resonances, Tracking, Higher Order 1862  Orbital Motion in Storage Rings: Algebraical Criteria of Stability and Localization A. Fedorova, M. Zeitlin, IPME RAS, St. Petersburg We consider orbital beam motion in storage rings with and without control as variational problem with constraints in arbitrary but finite rational npole approximation. Set of reduced algebraical problems and machinery of multiresolution representation in maximally localized bases [1] allow to formulate algebraical and computable criteria of stability and localization for nonlinear beam motion. As example we consider modeling and control for shortterm dynamical aperture. 1.A.Fedorova, M.Zeitlin, Seven papers in PAC01 Proc., arXiv preprints: physics/0106022, 0106010, 0106009, 0106008, 0106007, 0106006, 0106005. Type of presentation requested : Poster Classification : [D02] Nonlinear Dynamics  Resonances, Tracking, Higher Order 43  Observations of Beambeam Effects in RHIC* W. Fischer, J.M. Brennan, P. Cameron, R. Connolly, C. Montag, S. Peggs, V. Ptitsyn, S. Tepikian, J. van Zeijts, BNL, Upton, Long Island, New York Beambeam effects in RHIC were observed in operation and in dedicated experiments with both gold ions and protons. Observations include measurements of beambeam induced tune shifts, Schottky spectra with and without beambeam interaction, and lifetime and emittance growth measurements with and without beambeam interaction. Transverse emittance growth due to beambeam interaction in conjunction with orbit modulations is simulated. During ramps unequal radio frequencies in the two rings cause the crossing points to move longitudinally. Thus bunches experience beambeam interactions only in intervals and transverse tunes are modulated. In proton operation, RHIC is the only existing hadron collider in which strongstrong beambeam effects can be expected. Measurements of coherent modes were attempted with proton beams. *Work supported by U.S. DOE under contract No DEAC0298CH10886 Type of presentation requested : Poster Classification : [D02] Nonlinear Dynamics  Resonances, Tracking, Higher Order 46  Measurements of Resonance Driving Terms in RHIC* W. Fischer, F. Pilat, BNL, Upton, Long Island, New York; F. Schmidt, CERN, Geneva Measured resonance driving terms may be used to correct for nonlinear field errors in accelerators and storage rings. This is especially interesting for local nonlinear corrections in the interaction regions of machines like RHIC and LHC. Measurements are presented that are based on tunebyturn data of the beam centroid with different betatron amplitudes. Resonance strengths were varied with a single sextupole corrector in one interaction region. *Work supported by U.S. DOE under contract No DEAC0298CH10886 Type of presentation requested : Poster Classification : [D02] Nonlinear Dynamics  Resonances, Tracking, Higher Order 62  Adiabatic Capture of Charged Particles in Islands of Phase Space: A New Method for Multiturn Extraction M. Giovannozzi, R. Cappi, CERN, Geneva A new method for multiturn extraction from a circular particle accelerator is presented. It is based on adiabatic capture of particles into islands of transverse phase space generated by nonlinear resonances. By appropriate use of nonlinear elements, such as sextupoles and octupoles, stable islands can be created at small amplitude. By inducing an appropriate slow variation of the linear tune, particles can be captured inside these islands. This allows the beam to be split into different smaller beamlets in the transverse phase space. These beamlets can be finally transported towards higheramplitudes to prepare them for extraction, performed by means of kicker and septum magnets. Results of numerical simulations are presented and discussed. Type of presentation requested : Poster Classification : [D02] Nonlinear Dynamics  Resonances, Tracking, Higher Order 245  Axisymmetric Vortices in Axisymmetric Inhomogeneous Beams* Y. Golub, MRI, Moscow We analyzed localized vortices in nonneutral inhomogeneous by density and velocity electron beams propagating in vacuum along the external magnetic field. These vortices distinguish from wellknown vortices of LarichevReznik or Reznik types, which used in [1]. New types of vortex are obtained by new method of nonlinear equations solution. That method distinguish from standard LarichevReznik or Reznik method, which used in [1]. It has been found new expression for electric field potential of vortex in a wave frame. The expression is axisymmetric in a wave frame. New vortices are the result of external disturbances or the appearance and development of instabilities like for example a diocotron instability in hollow beams and a slippinginstability in solid beams. [1] Golub Yu.Ya., Nikulin M.G., Rozanov N.E. In: Nonlinear world: IV Intern. Workshop on Nonlin. and Turbul. Proc. in Phys., (ed. by V.G. Bar'yakhtar et all) World Scientific Publishing Co. Pte. Ltd., Singapore, 1990, vol. 2, p.857 Type of presentation requested : Poster Classification : [D02] Nonlinear Dynamics  Resonances, Tracking, Higher Order 306  Tolerances of the Spool Piece Correction System for the LHC M. Hayes, CERN, Geneva The LHC main supeconducting dipoles are provided with a spool piece correction system which is to be used to correct locally the nonlinear multipole errors b3, b4 and b5. The tolerances of this correction system to failures are discussed. Also discussed are simple methods of ensuring they are set correctly during commissioning. Type of presentation requested : Poster Classification : [D02] Nonlinear Dynamics  Resonances, Tracking, Higher Order 997  Beam Tails in the PLS Storage Ring E.S. Kim, K.M. Ha, D.T. Kim, S.J. Park, PAL, Pohang NonGaussian transverse beam tails are investigated from measurements of the beam lifetime in the PLS storage ring. Gasscattering lifetime and Touschek lifetime are estimated from the measured beam lifetimes. The beam tails are also compared with those that are obtained by a simulation method. Type of presentation requested : Poster Speaker : EunSan Kim Classification : [D02] Nonlinear Dynamics  Resonances, Tracking, Higher Order 1037  Study of Beam Lifetime in PLS Storage RIng E.S. Kim, PAL, Pohang We investigate measurements and interpretation of the beam lifetime at the PLS storage ring. The beam lifetimes are observed under variations of several machine and beam parameters such as gap height of insertion devices, bunch filling pattern and coupling constant. Type of presentation requested : Poster Speaker : EunSan Kim Classification : [D02] Nonlinear Dynamics  Resonances, Tracking, Higher Order 549  Beam Dynamics Formation in Magnetic Field E. Kotina, St. Petersburg State University, St. Petersburg In this paper methods of construction of magnetic fields forming given beam dynamics are developed. We consider this problem as the inverse problem of determination of magnetic field by given motion of charged particles. Different inverse problems of electrodynamics have always been the subject of attention of many researchers. Attempts to find approaches to the solving of problems of such kind had been undertaken for a long time. In particular, similar problems had been considered in the works of G.A.Grinberg, A.R.Lucas, B.Meltzer, V.T.Ovcharov, V.I.Zubov. In this paper numericalanalytical method of finding of magnetic field is developed. The investigated problem is reduced to the solving of the Goursat problem for the quasilinear system of differential equations with partial derivation. To solve this problem the method of characteristics and the Masso method are developed. Type of presentation requested : Poster Classification : [D02] Nonlinear Dynamics  Resonances, Tracking, Higher Order 809  On the Possibility of Fast Charged Particles Channeling in Dusting Crystals N. Maksyuta, V. Vysotskii, National Taras Shevchenko University of Kyiv, Kyiv To obtain an effective shortwave radiation based on channeling we could realize a positrontype regime for electrons. This regime would cause an anomalous dissipation of electron beam between hypothesis crystallographic planes. So for the case of natural crystals such regime doesn't work. However some artificial structures would make true the possibility of positrontype regime channeling. Coulomb crystals named dusting ones well studied in physics nowadays could suit best of all. Byelectrode layers tumed out to realize conditions for periodic system origin and restrain. Its points of lattice contain dusting particles of a fractal structure with great negative charges. These charge values are determined by a concentration and dusting particles dimensions. Besides the possibility of being used for shortwave radiation in dusting crystals they could be used as macroscopic analogs of natural crystals suitable for orientation effects study. We could propose a periodic structure which points of lattice could accumulate electron neutral particles (e.g. negative ions, surrounded by charged ones). It would reveal new possibilities for electron orientational motion realization. 