Experimental measurements of a betatron difference resonance

Application of independent component analysis to Fermilab Booster

“…We place M BPMs uniformly around the ring, i.e., the phase advances at the ith (i 0; 1; ; M ÿ 1) BPM are ;i 2 i=M and ÿ;i 2 ÿ i=M. The readings this BPM will record are x i n A 1 cos n ;i A 2 cos ÿ n ÿ;i ; (13) z i n B 1 cos n ;i B 2 cos ÿ n ÿ;i ; (14) where n is the revolution index. We can introduce bad-BPM modes and add white Gaussian random noises to each individual BPM to simulate the effect of noises.…”

“…Figure 6 shows an example of a third order resonance mode corresponding to ÿ x 2 z ÿ 1 0. Although the tracking data can be used to construct the Poincaré surface of section [14], the physical meaning of the spatial vector has not been fully understood. Since the signals of higher order resonance in the nominal operational condition of the Fermilab Booster are usually buried under the noise floor, we will not study its importance in this study.…”

Evolution of Beam Distribution in Crossing a Walkinshaw Resonance

“…There had been theoretical analysis on the ν x −2ν z = ℓ resonance [2,3], and subsequent experimental measurements in storage rings [4,5]. These papers, however, deal essentially with single-particle motion near the resonance at fixed betatron tunes.…”

“…Figure 2 in Ref. [5] shows also experimental data of one of the separatrix for this resonance, where the CS and coupling circles are expressed in phase space coordinates: (X, P ) = (…”

“…In term of the horizontal and vertical action-angle phase-space coordinates (J x , φ x ) and (J z , φ z ), the Hamiltonian near the ν x − 2ν z = ℓ resonance can be approximated as [5,6] …”