New Approach to Resonance Crossing

Abstract

Time-varying nonlinear oscillatory systems produce phenomena of resonance crossing and trapping of particles in resonance islands. Traditionally, such processes have been analyzed in terms of adiabatic conditions. Considering, as an example, a simplified one-dimensional model describing the ''electroncloud pinch'' during a bunch passage in a particle accelerator, here we present an approach to resonance trapping which does not require any adiabatic condition. Instead we introduce the concept of the attraction point and investigate invariance and scaling properties of motion close to the attraction point, considering a single resonance crossing. DOI: 10.1103/PhysRevLett.109.234102 PACS numbers: 41.75.Ài, 05.45.Àa, 29.27.Bd Nonlinear oscillatory systems subject to time-varying parameters give rise to the phenomenon of resonance crossing. Trapping or detrapping of a moving object into a resonance occurs during the separatrix crossing process [1,2]. This mechanism is found in almost all areas of physics, for example, in tokamaks [3], in planetary science [4][5][6], in the orbital evolution of asteroid fragments [7], in fluid dynamics [8], in spacecraft research [9], in ion traps [10], and also in accelerator beam physics [11][12][13][14][15][16].We here consider resonance-crossing phenomena occurring in an accelerator for a proton beam afflicted by an electron cloud [13,17], which are of relevance, e.g., for the facility for antiproton and ion research in Europe project (FAIR) [18] and for the Large Hadron Collider [19]. During the passage of a proton bunch, the cloud electrons perform nonlinear oscillations in the nonlinear beam potential, which induces a particular ''pinch'' structure of the negatively charged electron cloud. The electric field from this charge distribution feeds back onto the beam, creating structure resonances as well as phenomena of periodic resonance crossing [13,17,20,21].The one-turn map describing a specific proton dynamics due to a localized one-dimensional electron layer in a circular accelerator [17,20,21] can be written asThis equation refers to protons at a certain longitudinal position along the bunch such that they experience the force from a thin electron-cloud layer located in the vertical midplane (at y ¼ 0). The phase-space variables y and p in (1) are the so-called normalized Courant-Snyder coordinates [22], ! ¼ 2 Q y is the betatron phase advance per turn (Q y denotes the betatron tune), andF represents the effect of the electric field generated by the sheet of electrons located at y ¼ 0. The ''sgn'' function incorporates the discontinuity of the Coulomb electric field across the electron sheet layer, approximated as infinitely thin and delta-function-like.The map (1) creates a series of structure resonances N Q y ¼ M, with N and M integers. We denote with ¼ ! À 2 M=N the distance, in betatron frequency, of Q y from the resonance located at Q r ¼ M=N . By representing the particle coordinates in the complex variable Y 0 ¼ y þ ip, the map (1) takes the convenient form Y 0;1 ...

Fundamental beam-beam limit from head-on interaction in the Large Hadron Collider

“…Therefore, during a synchrotron period (or half period), these particles can be trapped, moved, and released by the resonances. This process, which repeats every synchrotron period, causes an enhancement of the emittance growth [18,23]. The same diffusion mechanism appears if the dispersion at the collision point or the chromaticity are nonzero, as we will illustrate later.…”

Electron cloud in accelerators

“…Depending on the cloud density, 3 the instability could appear as a beam breakup with a rise time much shorter than the synchrotron period (τ T s ), as a transverse mode coupling instability with a rise time comparable to the synchrotron period (τ ≈ T s ), or as a conventional head-tail instability, which typically has a slower growth rate (τ T s ). In addition, the electron nonlinear fields induce a tune spread which may lead to resonance crossing resonance excitation and cause an incoherent emittance growth, [78][79][80] this mechanism can explain some beam observations at KEKB, SPS and LHC.…”