Tune Shift Induced by Flat-Chamber Resistive Wall Impedance LHC Collimator Experiment in the SPS Frank Zimmermann Frank Zimmermann, GSI Meeting 31.03.2006

Frank Zimmermann, GSI Meeting 31.03.2006 introduction collimators are largest impedance in LHC ~1-m long graphite blocks (for survival), half gap ~1.5 mm 2004 experiment aimed at validating our impedance model best measured quantity: coherent tune shift Frank Zimmermann, GSI Meeting 31.03.2006

Frank Zimmermann, GSI Meeting 31.03.2006 prototype LHC collimator installed in the SPS (R. Assmann) Frank Zimmermann, GSI Meeting 31.03.2006

Frank Zimmermann, GSI Meeting 31.03.2006 parameters Frank Zimmermann, GSI Meeting 31.03.2006

Frank Zimmermann, GSI Meeting 31.03.2006 measurement data from Marek Gasior (BBQ monitor) For smallest gaps, intensity was reduced & transverse emittance increased Frank Zimmermann, GSI Meeting 31.03.2006

Frank Zimmermann, GSI Meeting 31.03.2006 comparing measurement with classical theory applies if with Chao, Physics of Collective Instabilities in High Energy Accelerators, J. Wiley, New York 1993 c~2x10-5 for s~105 W-1m-1 and half gap b~1.5 mm →OK from 1 MHz to 1 THz classical factor 2.5 difference at small gaps data Frank Zimmermann, GSI Meeting 31.03.2006

Frank Zimmermann, GSI Meeting 31.03.2006 A.Burov, V. Lebedev, EPAC’02 comparing with Burov-Lebedev theory includes effect of finite chamber thickness and so-called ‘inductive bypass effect’ (correct dependence at low frequency) applies if with and Burov-Lebedev factor 2 difference at small gaps Frank Zimmermann, GSI Meeting 31.03.2006

Frank Zimmermann, GSI Meeting 31.03.2006 consideration rms beam size ~1/4 half gap → nonlinear component of the wake field could be important Frank Zimmermann, GSI Meeting 31.03.2006

Frank Zimmermann, GSI Meeting 31.03.2006 wake potential & nonlinear deflection potential for nonlinear resistive-wall impedance between two parallel plates was derived by Piwinski (DESY 94-068, Eq. (52)) and re-written by Bane, Irwin, and Raubenheimer (NLC ZDR p. 594). 2b: full gap nonlinear kick to test particle: Piwinski formula applies if Frank Zimmermann, GSI Meeting 31.03.2006

Frank Zimmermann, GSI Meeting 31.03.2006 time dependence of kick along the bunch is described by fR tail head Frank Zimmermann, GSI Meeting 31.03.2006

introduce new coordinates and perform 2 integrations coherent tune shift: introduce new coordinates and perform 2 integrations Frank Zimmermann, GSI Meeting 31.03.2006

Frank Zimmermann, GSI Meeting 31.03.2006 the function G(X,Y) Frank Zimmermann, GSI Meeting 31.03.2006

Frank Zimmermann, GSI Meeting 31.03.2006 the function G(X,Y) ) note: change of sign for large X Frank Zimmermann, GSI Meeting 31.03.2006

Frank Zimmermann, GSI Meeting 31.03.2006 the function G(X,Y) note: divergence for Y→2b Frank Zimmermann, GSI Meeting 31.03.2006

Frank Zimmermann, GSI Meeting 31.03.2006 variation of coherent tune shift with emittance g example parameters Frank Zimmermann, GSI Meeting 31.03.2006

Frank Zimmermann, GSI Meeting 31.03.2006 comparing SPS measurement with tune shift expected from nonlinear wake field nonlinear wake field data 20% difference for small gaps Frank Zimmermann, GSI Meeting 31.03.2006

Frank Zimmermann, GSI Meeting 31.03.2006 comparing SPS measurement with tune shift expected from nonlinear wake field for a 50-mm closed orbit offset at the collimator c2 of the agreement increases from 0.80 to 0.83 nonlinear wake field with 50 mm c.o. offset Frank Zimmermann, GSI Meeting 31.03.2006

Frank Zimmermann, GSI Meeting 31.03.2006 generalized formula: combine correct frequency dependence of Burov-Lebedev with complete nonlinear on transverse coordinates from Piwinski, assuming that the two dependencies remain factorized generalized formula nearly perfect agreement measurement Frank Zimmermann, GSI Meeting 31.03.2006

Frank Zimmermann, GSI Meeting 31.03.2006 incoherent tune shift single-particle tune nonlinearly depends on transverse coordinates & on position along the bunch t Frank Zimmermann, GSI Meeting 31.03.2006

Frank Zimmermann, GSI Meeting 31.03.2006 incoherent tune spread b=1.0 mm b=1.5 mm Monte-Carlo evaluation of analytical formula for example parameters Frank Zimmermann, GSI Meeting 31.03.2006

Frank Zimmermann, GSI Meeting 31.03.2006 centroid motion from multi-particle tracking with nonlinear wake field (2000 particles over 11000 turns) no collimator b=1.5 mm, no synchr.osc. b=1.0 mm, no synchr.osc. b=1.5 mm, with synchr.osc. b=1.0 mm, no synchr.osc., 1000 particles b=1.0 mm, with synchr.osc. w/o synchrotron motion Frank Zimmermann, GSI Meeting 31.03.2006

Frank Zimmermann, GSI Meeting 31.03.2006 FFT of centroid motion for the same 6 cases no collimator b=1.5 mm, no synchr.osc. b=1.0 mm, no synchr.osc. b=1.5 mm, with synchr.osc. b=1.0 mm, no synchr.osc., 1000 particles b=1.0 mm, with synchr.osc. increased tune spread for small gaps Frank Zimmermann, GSI Meeting 31.03.2006

Frank Zimmermann, GSI Meeting 31.03.2006 conclusions nonlinear terms of resistive-wall wake field become important if aperture comparable to rms beam size generalized formula combining Burov-Lebedev (dependence on w) & Piwinski (dependence on x and y) in perfect agreement with SPS measurement for small gaps, incoherent tune spread from nonlinear wake field increases beam stability via enhanced Landau damping More details in CERN-AB-Note-2006-007 Frank Zimmermann, GSI Meeting 31.03.2006