Fig. 3: Finding fixed lines and their properties.
From: Observation of fixed lines induced by a nonlinear resonance in the CERN Super Proton Synchrotron
a–c, A case with the stable features of Fig. 2. d–f, A case where the machine conditions (in particular the machine tunes) are subject to unwanted drift. a, Evolution of the distance from the resonance calculated using the measured beam tunes, Qx and Qy. The harmonic number of the resonance is 79. b, Evolution of \({\bar{a}}_{x}\) and \({\bar{a}}_{y}\), the scaled Courant–Snyder forms (dark blue), and their corresponding moving averages (light blue; Methods). c, \({\bar{a}}_{y}\) versus \(\varOmega\) diagram, where \({\varOmega}={{{\phi}_x}}+2{{{\phi}_y}}\). This beam revolves over 3,000 turns around a ‘fixed point’. The red vertical line is the fitted value of \(\varOmega\). The black vertical line is the value of \(\varOmega\) found from the driving term calculations. d, Evolution of the distance from the resonance calculated using the measured beam tunes. There are larger oscillations over time. e, Evolution of \({\bar{a}}_{x}\) and \({\bar{a}}_{y}\). The variation is larger but the variables are still correlated. f, \({\bar{a}}_{y}\) versus \(\varOmega\) diagram. The meaning of the red and black vertical lines is the same as in c. The quantity \({\bar{a}}_{y}\) spans over a large range, whereas \(\varOmega\) keeps oscillating around one specific \({{{\varOmega }}}_{{{{\rm{fl}}}}}^{\exp }\) (the location of the fixed line). This finding suggests that the beam is trapped30 on the resonance. The jitter in the light blue curves in b and e is due to the random error of the BPMs (σBPM,x ≈ 0.66 mm and σBPM,y ≈ 0.5 mm). The same random error causes the variations in c, which make the overlapping trajectory of (\(\varOmega,\,{\bar{a}}_y\)) appear like a thick ring rather than a circle. The thickness of this ring is consistent with \(\pm 3\) times the error bar: \({\sigma }_{{\bar{a}}_{y}}\lesssim 2.81\) mm2 and \({\sigma }_{\varOmega }\) ≈ 0.0032, (in units of 2π). In f, the dynamic span of \({\bar{a}}_{y}\) is much larger than the error bar. Hence, the spiral pattern show resonance trapping.
