33 ACCELERATOR PHYSICS HT E. J. N.

Size: px

Start display at page:

Download "33 ACCELERATOR PHYSICS HT E. J. N."

Ira Dennis
5 years ago
Views:

1 Lecture 33 ACCELERATOR PHYSICS HT E. J. N. Wilson Lecture 33 - E. Wilson - 3/9/ Slide 1

2 Summary of last lectures Beam Beam Effect I The Beam-beam effect Examples of the limit Field around a moving cylinder of charge Elliptical beam section Force on test particle Beam Beam Force Remember Q-shift from quadrupole Beam beam Q shift Choosing lattice parameters Lecture 33 - E. Wilson - 3/9/ Slide 2

3 Beam-Beam Tune Shift II Coherent Beam-Beam limit Fourth integer resonance Satellite stop bands Phase space for fourth order resonance Example of tracking Importance of tune modulation Lecture 33 - E. Wilson - 3/9/ Slide 3

4 Need for higher luminosity Lecture 33 - E. Wilson - 3/9/ Slide 4

5 The Beam-beam effect - E. Wilson Luminosity L = N 2 fk 4πσ x σ y f = the rev frequency k = number of bunches σ = rms beam width or height Beam beam tune shift (linear effect on Q) ξ y = Nr 0 β y ( ) 2πγσ y σ x + σ y Dependence of one upon the other L = Nfkγ 2r 0 β ξ * y y σ x + σ y σ x Lecture 33 - E. Wilson - 3/9/ Slide 5

6 Estimating field tolerances for growth at a resonance Deflection in a dipole θ = ( B ) (Bρ) Total ring = 2πρ Reduced by x' = θ 2π N 1 N B B rms β = λ / 2π of betatron oscillations = ρ / Q 2π Q = a a = β θ a = ρ Qa 2π N B B rms Q = = ρ(= 1000) Q(= 25) a(= 0.01) Lecture 33 - E. Wilson - 3/9/ Slide turns = 1ms exit time

7 Coherent Beam-Beam limit So far we have considered the force on an individual test particle. It sees a different force on each encounter. The linear force it sees will cause a constant shift in tune The non linear force will excite resonances. We could also take the average of the Q shift over a distribution of test particles an it turns out to be 1/2 of the incoherent effect Now we should look at how the two bunches interact, deflecting each other depending on their relative position when they clash One can imagine that they relate by a four by four matrix. cos µ + β + sin µ z + β 1 + sin µ + cos µ z cos µ β sin µ z 0 0 β 1 sin µ cos µ z There are two main coupled dipole modes Lecture 33 - E. Wilson - 3/9/ Slide 7

8 More about Coherent Beam Beam. CAS - CERN Accelerator School : 5th Advanced Accelerator Physics Course 20 Sep - 1 Oct Rhodes, Greece / Turner, Stuart (ed.) Geneva : CERN, v. CERN Lecture 33 - E. Wilson - 3/9/ Slide 8

9 Fourth integer resonance Substituting x = acos Qθ x' = ( B )/ Bρ = ( B )x2 / 3!Bρ p = ˆ β x' 2π Q = p a cosqθ Incidentally ( a = p sin Qθ) 2π Q = ˆ β B 6 Bρ Q = ˆ β ( B ) 96π Bρ ( ) ( ) a2 cos 4 Qθ ( ) a2 ( 1 4cos2Qθ + cos4qθ) Amplitude Q dependence brings chains close Lecture 33 - E. Wilson - 3/9/ Slide 9 Excites resonances making islands

10 Phase space for fourth order resonance Island of stability Lecture 33 - E. Wilson - 3/9/ Slide 10

11 Satellite stop bands 2π Q = ˆ β B 48 Bρ ( ) ( ) a2 ( 1 4cos2Qθ + cos4qθ) = a a Fourier analysis of octupole pattern fp cos(pθ) Resonance when p = 4Q p Ripple or rf modulation of Q Q = Qo + ˆ Q sin Q s θ Makes sidebands at Q o ± nq s Resonance when: p = 4(Q ± nq s ) Lecture 33 - E. Wilson - 3/9/ Slide 11

12 Effect of satellites in phase space Lecture 33 - E. Wilson - 3/9/ Slide 12

13 Experimental results of satellites Lecture 33 - E. Wilson - 3/9/ Slide 13

Example of tracking Below the beam-beam limit and without tune modulation we see the many archipelagos of islands due to different multipole components in the beam-beam potential.

14 Example of tracking Below the beam-beam limit and without tune modulation we see the many archipelagos of islands due to different multipole components in the beam-beam potential. Increasing the beam-beam interaction will enlarge the islands so that they overlap in stochastic regions where particle may diffuse out in 4 dimensional phase space. Increasing amplitude tune slope will merge them too Hadron collider luminosity limitations Author(s) Evans, Lyndon R In: 4th US- CERN School on Particle Accelerators, Hilton Head Island, South CA, USA, 7-14 Nov 1990, pp Lecture 33 - E. Wilson - 3/9/ Slide 14 TED0.PCT

15 Crossing the stochastic limit Increasing Q shift causes islands to merge in a stochastic sea of chaos Lecture 33 - E. Wilson - 3/9/ Slide 15

16 Importance of tune modulation Above is with below is without modulation Lecture 33 - E. Wilson - 3/9/ Slide 16

17 Other details for linear colliders Luminosity enhancement (pinch) where D is disruption Beamstrahlung Make bunch length as large as possible but must be less than Courant s β which is the length of the waist (hourglass effect) Lecture 33 - E. Wilson - 3/9/ Slide 17

18 History of colliders Lecture 33 - E. Wilson - 3/9/ Slide 18

19 Summary Beam-Beam Tune Shift The Beam-beam effect Examples of the limit Field around a moving cylinder of charge Elliptical beam section Force on test particle Beam Beam Force Remember Q-shift from quadrupole Beam beam Q shift Choosing lattice parameters Coherent Beam-Beam limit Fourth integer resonance Satellite stop bands Phase space for fourth order resonance Example of tracking Importance of tune modulation Lecture 33 - E. Wilson - 3/9/ Slide 19

Beam-beam effects. (an introduction) Werner Herr CERN, AB Department. (/afs/ictp/home/w/wfherr/public/cas/doc/beambeam.pdf)

Beam-beam effects. (an introduction) Werner Herr CERN, AB Department. (/afs/ictp/home/w/wfherr/public/cas/doc/beambeam.pdf) Beam-beam effects (an introduction) Werner Herr CERN, AB Department (/afs/ictp/home/w/wfherr/public/cas/doc/beambeam.pdf) (http://cern.ch/lhc-beam-beam/talks/trieste beambeam.pdf) Werner Herr, beam-beam