Experimental determination of a nonlinear Hamiltonian in a synchrotron

Lee, S. Y.; Ball, M.; Brabson, B.; Caussyn, D.D.; Collins, John C.; Curtis, Scott; Derenchuck, Vladimir; DuPlantis, D.; East, G.; Ellison, M.; Ellison, T.; Friesel, D.L.; Hamilton, B.; Jones, W. P.; Lamble, W.; Li, D.; Minty, M.; Schwandt, P.; Sloan, T.; Xu, G.; Chao, Alexander W.; Tepikian, S.; Ng, K.Y.

doi:10.1103/physrevlett.67.3768

Cited by 47 publications

(12 citation statements)

References 3 publications

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“…Lately several theoretical ( [l], [2], [3]) and experimental studies ( [4], [5], [6]) have been performed to analyze the behavior of single particles in nonlinear fields typically present in modern accelerators.…”

Section: Introductionmentioning

confidence: 99%

Collective behavior of an ensemble of forced duffing oscillators near the 1{vert_ellipsis}1 resonance. Revision 1

Sen¹,

Ellison²,

Kauffmann³

1994

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Section: Introductionmentioning

confidence: 99%

Collective behavior of an ensemble of forced duffing oscillators near the 1{vert_ellipsis}1 resonance. Revision 1

Sen¹,

Ellison²,

Kauffmann³

1994

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“…The beam position at a high dispersion location was used to measure the momentum deviation, which is related to the off momentum closed orbit by Xco = Dxwith Dx ;:::;; 3.9 m. A total of 8 channels were used to obtain 6D phase space maps with 16 384 points recorded at 10 turn intervals. Details of our detection system had been reported earlier [1,7]. With a horizontal dipole (vertical field) modulation at location so, the horizontal closed orbit becomes [4], fm [Hz] 400 600 for a single trace, the bunch was indeed observed to split into two beamlets located at amplitudes corresponding to the steady state solutions of this dissipative parametric system.…”

mentioning

confidence: 99%

Experimental study of synchro-betatron coupling induced by dipole modulation

Syphers¹,

Ball²,

Brabson³

et al. 1993

Phys. Rev. Lett.

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Synchro-betatron coupling in a proton storage ring with electron cooling was studied experimentally by modulating a transverse dipole field close to the synchrotron frequency. The combination of the electron cooling and transverse field modulation on the synchrotron oscillation is equivalent to a dissipative parametric resonant system. The proton bunch was observed to split longitudinally into two pieces, or beamlets, converging toward attractors of the dissipative system. These phenomena might be important in understanding the effect of ground vibration on the Superconducting Super Collider beam, and the effect of power supply ripple on the Relativistic Heavy-Ion Collider beam.PACS numbers: 03.20.+i, 05.45.+b, 29.20.Dh In a recent experiment [1], we found that the longitudinal response of a beam to the phase modulation of the accelerating field exhibited characteristics of a parametric resonant system [2]. The equation of motion for phase oscillations [3] of a particle in a synchrotron, in the absence of forced oscillations, is given by ¢ + w;(sin 4>-sin 4>o) = 0.Here w8 = w0 ;; J] is the small amplitude synchrotron angular frequency at 4>o = 0, wo is the angular revolution frequency, and h,V, and ¢0 are, respectively, the harmonic number, the peak rf voltage, and the synchronous phase angle, (3c and E are, respectively, the speed and the energy of the particle, and rJ is the phase slip factor. Coupled motion between the transverse and longitudinal degrees of freedom is called synchro-betatron (SB) coupling.The SB coupling is important to electron storage rings [5], where the fractional parts of the synchrotron and betatron tunes are of the same order of magnitude. On the other hand, the fractional part of the betatron and synchrotron tunes differ substantially in proton storage rings, and the coupling between longitudinal and transverse motions is less important. To the knowledge of the authors, SB coupling has not been observed previously in proton storage rings. For the SSC, where the synchrotron frequency varies from 7 Hz at injection energy to 4Hz at 20 TeV, SB coupling may arise due to ground vibration. At Relativistic Heavy Ion Collider, the synchrotron frequency ramps through 60 Hz around 17 GeV/c for heavy ion beams, SB coupling may result from power supply ripple.The dominant effect of ground vibration or power supply ripple is a modulation of the dipole field. A slow adiabatic dipole modulation gives rise to a closed orbit modulation. Provided that the resulting closed or-

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“…The AC dipole strength is δ = (BL/(Bρ)) β D , where BL is the integrated field amplitude, (Bρ) is the rigidity, and β D is the Twiss function at the dipole. If z = z 0 just before the first dipole kick, then the net displacement phasor on turn T is The exact general solution for linear motion is [10] (5) where z = z 0 − δ − + δ + is a constant given by the initial conditions, and the complex AC dipole strengths are…”

Section: Linear Motionmentioning

confidence: 99%

“…Second, strong and long lasting signals must be generated. Third, the measurement technique should be non-destructive.Conventionally, a single turn kick moves the beam to large amplitudes, and turn-by-turn data are recorded from multiple beam position monitors (BPMs) [1][2][3][4][5][6]. Unfortunately, tune spread across the beam causes the center of charge beam signal to "decohere" on a time scale often less than 100 turns.…”

mentioning

confidence: 99%

Nonlinear diagnostics using AC dipoles

Peggs¹

Proceedings of the 1999 Particle Accelerator Conference (Cat. No.99CH36366)

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There are three goals in the accurate nonlinear diagnosis of a storage ring. First, the beam must be moved to amplitudes many times the natural beam size. Second, strong and long lasting signals must be generated. Third, the measurement technique should be non-destructive.Conventionally, a single turn kick moves the beam to large amplitudes, and turn-by-turn data are recorded from multiple beam position monitors (BPMs) [1][2][3][4][5][6]. Unfortunately, tune spread across the beam causes the center of charge beam signal to "decohere" on a time scale often less than 100 turns. Filamentation also permanently destroys the beam emittance (in a hadron ring). Thus, the "strong single turn kick" technique successfully achieves only one out of the three goals. AC dipole techniques can achieve all three. Adiabatically excited AC dipoles slowly move the beam out to large amplitudes. The coherent signals then recorded last arbitrarily long. The beam maintains its original emittance if the AC dipoles are also turned off adiabatically, ready for further use.The AGS already uses an RF dipole to accelerate polarized proton beams through depolarizing resonances with minimal polarization loss [7]. Similar AC dipoles will be installed in the horizontal and vertical planes of both rings in RHIC [8]. The RHIC AC dipoles will also be used as spin flippers, and to measure linear optical functions [9]. LINEAR MOTIONHorizontal motion is described using complex phasorsso that the unperturbed one turn motion is justwhere R = exp(i 2πQ X ). Here Q X is the betatron tune, and the normalized coordinates x and x both have the dimensions of length. An AC dipole just after the reference point gives a real normalized angular kick on turn t ofwhere Q D is the drive tune and ψ 0 is the initial phase. The AC dipole strength is δ = (BL/(Bρ)) β D , where BL is the integrated field amplitude, (Bρ) is the rigidity, and β D is the Twiss function at the dipole. If z = z 0 just before the first dipole kick, then the net displacement phasor on turn T is The exact general solution for linear motion is [10] (5) where z = z 0 − δ − + δ + is a constant given by the initial conditions, and the complex AC dipole strengths arewhereThe oscillating closed orbit is defined as that orbit which exactly repeats itself after one modulation period. The solution on turn T is obtained by putting z = 0, so thatgenerally following a tilted ellipse, not a circle, in normalized phase space. The semi-minor and semi-major axes are ||δ − | − |δ + || and |δ − | + |δ + | long. In practice the aspect ratio of the ellipse is close to 1 when the AC dipole is driven at a tune close to the fractional betatron tune (Q − ≈ 0).Motion in the rotating frame, which rotates with the AC dipole drive at 2πQ D per turn, is denoted by over-bars. Assuming the previous approximation to be accurate,That is, a test particle slowly circulates the vector δ − at a radius of constant length | z |, as illustrated in Fig. 1. When a bunch is considered, a distribution of z values must be used. A smoothly di...

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Experimental determination of a nonlinear Hamiltonian in a synchrotron

Cited by 47 publications

References 3 publications

Collective behavior of an ensemble of forced duffing oscillators near the 1{vert_ellipsis}1 resonance. Revision 1

Collective behavior of an ensemble of forced duffing oscillators near the 1{vert_ellipsis}1 resonance. Revision 1

Experimental study of synchro-betatron coupling induced by dipole modulation

Nonlinear diagnostics using AC dipoles

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