The fastslow mode coupling instability for coasting beams
The fast-slow mode coupling instability for coasting beams N. Biancacci, E. Métral, M. Migliorati ABP Group Information Meeting – 26 th March 2020
Outline Introduction • Coasting beam instability simulations: • • • Instability theory for: • • • Fast and slow waves Effect of driving impedance Effect of driving and detuning impedances Uncoupled fast and slow waves Conclusions and next steps 2
Introduction Collective headtail horizontal instabilities have been measured and characterized versus chromaticity in the past at injection in the PS. Head. Tail simulations were performed in [1] and could reproduce the expected behavior versus chromaticity. 3
Introduction Collective headtail horizontal instabilities have been measured and characterized versus chromaticity in the past at injection in the PS. Head. Tail simulations were performed in [1] and could reproduce the expected behavior versus chromaticity. Questions remained: • What is the effect of space charge? and the more fundamental: • Which impedance is driving the instability? • Why horizontal plane more critical than vertical? 4
Introduction Dedicated study performed in 2019 with M. Migliorati from the University of Rome “La Sapienza”. Performed Py. Head. Tail simulations accounting for the impedance of the PS elliptical beam pipe (to start with). driving impedance 5
Introduction Dedicated study performed in 2019 with M. Migliorati from the University of Rome “La Sapienza”. Performed Py. Head. Tail simulations accounting for the impedance of the PS elliptical beam pipe (to start with). driving impedance X plane “Inductive bypass” regime Y plane Classical “thick-wall“ regime 6
Introduction Dedicated study performed in 2019 with M. Migliorati from the University of Rome “La Sapienza”. Performed Py. Head. Tail simulations accounting for the impedance of the PS elliptical beam pipe (to start with). driving impedance X plane No instability Y plane Instability 7
Introduction Dedicated study performed in 2019 with M. Migliorati from the University of Rome “La Sapienza”. Performed Py. Head. Tail simulations accounting for the impedance of the PS elliptical beam pipe (to start with). driving and detuning impedance + 8
Introduction Dedicated study performed in 2019 with M. Migliorati from the University of Rome “La Sapienza”. Performed Py. Head. Tail simulations accounting for the impedance of the PS elliptical beam pipe (to start with). driving and detuning impedance X plane Y plane 9
Introduction Dedicated study performed in 2019 with M. Migliorati from the University of Rome “La Sapienza”. Performed Py. Head. Tail simulations accounting for the impedance of the PS elliptical beam pipe (to start with). driving and detuning impedance X plane Instability Y plane No instability 10
Effect of detuning impedance 11
Coasting beam simulations: slow/fast waves 12
Coasting beam simulations: slow/fast waves 13
Coasting beam simulations: slow/fast waves 14
Coasting beam simulations: slow/fast waves Lowest slow wave unstable (due to resistive wall) 15
Effect of driving impedance Scenario: driving impedance, intensity scan, PS-like parameters V plane more critical! Good agreement with existing theory (Laclare’s [6]) 16
Effect of driving and detuning impedance Scenario: driving and detuning impedance, intensity scan, PS-like parameters H plane more critical! • Horizontal plane is destabilised, the vertical a bit stabilised. • Standard theory is insufficient to describe this effect. 17
Theory: uncoupled waves X plane Y plane (slow wave) (fast wave) (slow wave) with (fast wave) 18
Theory correction: coupled waves X plane Y plane Coupling(s)! (slow wave) (fast wave) with 19
Theory correction: coupled waves X plane Y plane Coupling(s)! (slow wave) (fast wave) with Detuning impedance sampled at DC! See importance of inductive bypass ; ) 20
Effect of driving and detuning impedance We had this… 21
Effect of driving and detuning impedance … now this Good agreement accounting for fast-slow waves coupling! 22
Conclusions and next steps A significantly destabilising effect of the detuning impedance has been observed in the frame of the study of the PS instability at injection. Problem attacked with a “simpler” case: coasting beams. Of course it was everything but “simpler”… but ok! What we found: o Destabilising effect observed also in coasting beam simulations. o Inductive bypass essential to extrapolate the effect of detuning impedance (as it is for the LHC collimators until now). o A new theory was developed accounting for coupling of slow and fast waves. o Slow and fast waves couple due to the detuning impedance and destabilise. What’s next: o Finalise benchmark for other working points. o Adding real life: longitudinal coasting beam distributions are not perfect, effect of chromaticity and momentum spread? o Where is the coupling threshold for LEIR? 23
Thank you for your bandwidth! Any questions? 24
Backup 25
Theory - driving •
Theory - driving Let’s consider now a perturbation mode along the ring as: We can check if this mode is stable/unstable computing the force acting on a particle in the beam driven by this oscillation: The integrated force term is: Which leads to:
Theory - driving In the end the equation of motion can be written as
Theory - detuning For the detuning impedance, we need to integrate along the unperturbed longitudinal distribution: For small perturbations:
Theory – driving + detuning Driving part Detuning part From which we get: As the detuning impedance in 0 is purely imaginary, the total impedance is effectively adding only to the imaginary part of the dipolar one. This leads to the complex tune shift:
Theory – two modes Keeping only the closest fast/slow waves we have:
Mauro’s simulations A destabilising effect of the detuning impedance has been observed in simulations by M. Migliorati (see e. g. https: //indico. cern. ch/event/824835/) The instability is observed for the PS parameters at injection energy. We set-up the same simulations in coasting beams to try to understand the mechanism in a simpler physical context.
- Slides: 32