TRANSVERSE INSTABILITIES E Mtral CERN The purpose of

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TRANSVERSE INSTABILITIES E. Métral (CERN) è The purpose of this course is to explain

TRANSVERSE INSTABILITIES E. Métral (CERN) è The purpose of this course is to explain (theoretically) such pictures of “transverse (single-bunch) instability” Following Laclare (and longitudinal) Observation in the CERN PSB in ~1974 (J. Gareyte and F. Sacherer) Observation in the CERN PS in 1999 Time (20 ns/div) Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3 rd, 2009 1 /22

SINGLE PARTICLE TRANSVERSE MOTION (1/3) u A purely linear synchrotron oscillation around the synchronous

SINGLE PARTICLE TRANSVERSE MOTION (1/3) u A purely linear synchrotron oscillation around the synchronous particle is assumed (with no coherent oscillations) u For the transverse betatron oscillation, the equation of unperturbed motion, e. g. in the horizontal plane, is written as u The horizontal betatron frequency is given by Chromaticity with Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3 rd, 2009 2 /22

SINGLE PARTICLE TRANSVERSE MOTION (2/3) and Horiz. chromatic frequency Elias Métral, CERN Accelerator School,

SINGLE PARTICLE TRANSVERSE MOTION (2/3) and Horiz. chromatic frequency Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3 rd, 2009 3 /22

SINGLE PARTICLE TRANSVERSE MOTION (3/3) u In the absence of perturbation, the horizontal coordinate

SINGLE PARTICLE TRANSVERSE MOTION (3/3) u In the absence of perturbation, the horizontal coordinate satisfies u In the presence of electromagnetic fields induced by the beam, the equation of motion writes When following the particle along its trajectory Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3 rd, 2009 4 /22

SINGLE PARTICLE TRANSVERSE SIGNAL (1/2) u The horizontal signal induced at a perfect PU

SINGLE PARTICLE TRANSVERSE SIGNAL (1/2) u The horizontal signal induced at a perfect PU electrode (infinite bandwidth) at angular position in the ring by the off-centered test particle is given by u Developing into exponential functions and using relations given in the longitudinal course, yields Complex conjugate Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3 rd, 2009 5 /22

SINGLE PARTICLE TRANSVERSE SIGNAL (2/2) with u The spectrum is a line spectrum at

SINGLE PARTICLE TRANSVERSE SIGNAL (2/2) with u The spectrum is a line spectrum at frequencies u Around every betatron line synchrotron satellites m , there is an infinite number of u The spectral amplitude of the mth satellite is given by u The spectrum is centered at the chromatic frequency Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3 rd, 2009 6 /22

STATIONARY DISTRIBUTION (1/2) u In the absence of perturbation, and are constants of the

STATIONARY DISTRIBUTION (1/2) u In the absence of perturbation, and are constants of the motion u Therefore, the stationary distribution is a function of the peak amplitudes only u No correlation between horizontal and longitudinal planes is assumed and the stationary part is thus written as the product of 2 stationary distributions, one for the longitudinal phase space and one for the horizontal one Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3 rd, 2009 7 /22

STATIONARY DISTRIBUTION (2/2) u Since on average, the beam center of mass is on

STATIONARY DISTRIBUTION (2/2) u Since on average, the beam center of mass is on axis, the horizontal signal induced by the stationary distribution is null Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3 rd, 2009 8 /22

PERTURBATION DISTRIBUTION (1/3) u In order to get some dipolar fields, density perturbations that

PERTURBATION DISTRIBUTION (1/3) u In order to get some dipolar fields, density perturbations that describe beam center-of-mass displacements along the bunch are assumed u The mathematical form of the perturbations is suggested by the singleparticle signal § Low-intensity Coherent betatron frequency shift to be determined Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3 rd, 2009 9 /22

PERTURBATION DISTRIBUTION (2/3) u In the time domain, the horizontal signal takes the form

PERTURBATION DISTRIBUTION (2/3) u In the time domain, the horizontal signal takes the form (for a single value m) Fourier transform with Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3 rd, 2009 10 /22

PERTURBATION DISTRIBUTION (3/3) § High-intensity Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3

PERTURBATION DISTRIBUTION (3/3) § High-intensity Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3 rd, 2009 11 /22

TRANSVERSE IMPEDANCE All the properties of the electromagnetic response of a given machine to

TRANSVERSE IMPEDANCE All the properties of the electromagnetic response of a given machine to a passing particle is gathered into the transverse impedance (complex function => in Ω / m) Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3 rd, 2009 12 /22

EFFECT OF THE PERTURBATION (1/10) u Vlasov equation Linearized Vlasov equation Elias Métral, CERN

EFFECT OF THE PERTURBATION (1/10) u Vlasov equation Linearized Vlasov equation Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3 rd, 2009 13 /22

EFFECT OF THE PERTURBATION (2/10) u The expression of equation of motion can be

EFFECT OF THE PERTURBATION (2/10) u The expression of equation of motion can be drawn from the single-particle horizontal Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3 rd, 2009 14 /22

EFFECT OF THE PERTURBATION (3/10) u Using the definition of the transverse impedance, the

EFFECT OF THE PERTURBATION (3/10) u Using the definition of the transverse impedance, the force can be written u Developing the into exponential functions, keeping then only the slowly varying term, making the approximation and using the relations and one from the longitudinal course, yields Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3 rd, 2009 15 /22

EFFECT OF THE PERTURBATION (4/10) For each mode m, one has Spectrum amplitude at

EFFECT OF THE PERTURBATION (4/10) For each mode m, one has Spectrum amplitude at frequency with Multiplying both sides by and integrating over using the relation Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3 rd, 2009 16 /22

EFFECT OF THE PERTURBATION (5/10) u Note that the horizontal stationary distribution disappeared and

EFFECT OF THE PERTURBATION (5/10) u Note that the horizontal stationary distribution disappeared and only the longitudinal one remains => Only the beam center of mass is important (in our case). This should also be valid for the perturbation, which can be written Averaged peak betatron amplitude Final form of the equation of coherent motion of a single bunch: Contribution from all the modes m Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3 rd, 2009 17 /22

EFFECT OF THE PERTURBATION (6/10) with u Coherent modes of oscillation at low intensity

EFFECT OF THE PERTURBATION (6/10) with u Coherent modes of oscillation at low intensity (i. e. considering only a single mode m) Multiplying both sides by Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3 rd, 2009 and integrating over 18 /22

EFFECT OF THE PERTURBATION (7/10) u Following the same procedure as for the longitudinal

EFFECT OF THE PERTURBATION (7/10) u Following the same procedure as for the longitudinal plane, the horizontal coherent oscillations (over several turns) of a “water-bag” bunch interacting with a constant inductive impedance are shown in the next slides for the first head-tail modes (Note that the index x has been removed for clarity) Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3 rd, 2009 19 /22

EFFECT OF THE PERTURBATION (8/10) DIPOLE Elias Métral, CERN Accelerator School, Darmstadt, Germany, October

EFFECT OF THE PERTURBATION (8/10) DIPOLE Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3 rd, 2009 QUADRUPOLE 20 /22

EFFECT OF THE PERTURBATION (9/10) Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3

EFFECT OF THE PERTURBATION (9/10) Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3 rd, 2009 21 /22

EFFECT OF THE PERTURBATION (10/10) (Laclare’s) theory Elias Métral, CERN Accelerator School, Darmstadt, Germany,

EFFECT OF THE PERTURBATION (10/10) (Laclare’s) theory Elias Métral, CERN Accelerator School, Darmstadt, Germany, October 3 rd, 2009 22 /22