Transverse-longitudinal coupling by space charge in cyclotrons

Baumgarten, C.

doi:10.1103/physrevstab.14.114201

Cited by 25 publications

(65 citation statements)

References 20 publications

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“…In high current cyclotrons, the combination of space charge and the external focusing forces of the cyclotron's azimuthally varying main magnetic field can lead to the formation of a stable, round (in the radial-longitudinal plane) bunch. For the case of the PSI Injector 2 cyclotron, this has been extensively studied and discussed (see for example [6,7,61]). The effect was dubbed vortex effect or vortex motion because the beam exhibits a vortex-like rotation in a local coordinate system (origin shifted to the bunch centroid position and local y-axis aligned with the bunch mean momentum).…”

Section: Vortex Motionmentioning

confidence: 99%

High intensity cyclotrons for neutrino physics

Winklehner

Bahng

Calabretta

et al. 2018

Nuclear Instruments and Methods in Physics Research Section A:

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In recent years, the interest in high intensity proton beams in excess of several milli-amperes has risen. Potential applications are in neutrino physics, materials and energy research, and isotope production. Continuous wave proton beams of five to ten milli-amperes are now in reach due to advances in accelerator technology and through improved understanding of the beam dynamics. As an example application, we present the proposed IsoDAR experiment, a search for so-called sterile neutrinos and non-standard interaction using the KamLAND detector located in Japan. We present updated sensitivities for this experiment and describe in detail the design of the high intensity proton driver that uses several novel ideas. These are: accelerating H + 2 instead of protons, directly injecting beam into the cyclotron via a radio frequency quadrupole, and carefully matching the beam to achieve so-called vortex motion. The preliminary design holds up well in PIC simulation studies and the injector system is now being constructed, to be commissioned with a 1 MeV/amu test cyclotron.

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Section: Vortex Motionmentioning

confidence: 99%

High intensity cyclotrons for neutrino physics

Winklehner

Bahng

Calabretta

et al. 2018

Nuclear Instruments and Methods in Physics Research Section A:

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“…Furthermore, we like to have a mathematically positive angular velocity and hence for positive charge we need to have a negative field B z , so that we define B = −B z . A rotation in the horizontal plane is described by the following generator matrix 8,9 :…”

Section: Bending Magnetsmentioning

confidence: 99%

“…In some sense this work is a continuation of Ref. 8,9 , where we derived new methods to "solved problems" with the general Hamiltonian of a two-dimensional harmonic oscillator. Here we start with the general Lagrangian description of an harmonic oscillator and derive the Hamiltonian from it.…”

Section: Introductionmentioning

confidence: 98%

A new look at linear (non-?) symplectic ion beam optics in magnets

Baumgarten

2014

Nuclear Instruments and Methods in Physics Research Section A:

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We take a new look at the details of symplectic motion in solenoid and bending magnets and rederive known (but not always well-known) facts. We start with a comparison of the general Lagrangian and Hamiltonian formalism of the harmonic oscillator and analyze the relation between the canonical momenta and the velocities (i.e. the first derivatives of the canonical coordinates). We show that the seemingly non-symplectic transfer maps at entrance and exit of solenoid magnets can be re-interpreted as transformations between the canonical and the mechanical momentum, which differ by the vector potential.In a second step we rederive the transfer matrix for charged particle motion in bending magnets from the Lorentz force equation in cartesic coordinates. We rediscover the geometrical and physical meaning of the local curvilinear coordinate system. We show that analog to the case of solenoids -also the transfer matrix of bending magnets can be interpreted as a symplectic product of 3 non-symplectic matrices, where the entrance and exit matrices are transformations between local cartesic and curvilinear coordinate systems.We show that these matrices are required to compare the second moment matrices of distributions obtained by numerical tracking in cartesic coordinates with those that are derived by the transfer a matrix method.

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“…In Ref. 4 the author presented a toolbox for the treatment of two coupled harmonic oscillators that is based on the use of the real Dirac matrices (RDMs) as generators of the symplectic group Sp(4, R) and a systematic survey of symplectic transformations in two dimensions. This toolbox enabled the developement of a straightforward recipe for the decoupling of positive definite two-dimensional harmonic oscillators.…”

Section: Introductionmentioning

confidence: 99%

Geometrical method of decoupling

Baumgarten

2012

Phys. Rev. ST Accel. Beams

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The computation of tunes and matched beam distributions are essential steps in the analysis of circular accelerators. If certain symmetries -like midplane symmetrie -are present, then it is possible to treat the betatron motion in the horizontal, the vertical plane and (under certain circumstances) the longitudinal motion separately using the well-known Courant-Snyder theory, or to apply transformations that have been described previously as for instance the method of Teng and Edwards 1,2 . In a preceeding paper it has been shown that this method requires a modification for the treatment of isochronous cyclotrons with non-negligible space charge forces 3 . Unfortunately the modification was numerically not as stable as desired and it was still unclear, if the extension would work for all conceivable cases. Hence a systematic derivation of a more general treatment seemed advisable.In a second paper the author suggested the use of real Dirac matrices as basic tools for coupled linear optics and gave a straightforward recipe to decouple positive definite Hamiltonians with imaginary eigenvalues 4 . In this article this method is generalized and simplified in order to formulate a straightforward method to decouple Hamiltonian matrices with eigenvalues on the real and the imaginary axis. The decoupling of symplectic matrices which are exponentials of such Hamiltonian matrices can be deduced from this in a few steps. It is shown that this algebraic decoupling is closely related to a geometric "decoupling" by the orthogonalization of the vectors E, B and P , that were introduced with the so-called "electromechanical equivalence" 4 .A mathematical analysis of the problem can be traced down to the task of finding a structure-preserving block-diagonalization of symplectic or Hamiltonian matrices. Structure preservation means in this context that the (sequence of) transformations must be symplectic and hence canonical.When used iteratively, the decoupling algorithm can also be applied to n-dimensional systems and requires O(n 2 ) iterations to converge to a given precision.

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Transverse-longitudinal coupling by space charge in cyclotrons

Cited by 25 publications

References 20 publications

High intensity cyclotrons for neutrino physics

High intensity cyclotrons for neutrino physics

A new look at linear (non-?) symplectic ion beam optics in magnets

Geometrical method of decoupling

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