1. Lecture HT14 Beam-Beam II
ACCELERATOR PHYSICS HT
E. J. N. Wilson

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

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

4. Need for higher luminosity

5. The Beam-beam effect - E. Wilson
Luminosity
f = the rev frequency
k = number of bunches
s = rms beam width or height
Beam beam tune shift (linear effect on Q)
Dependence of one upon the other

6. Estimating field tolerances for growth at a resonance
Deflection in a dipole
Total ring
Reduced by
42 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.
There are two main coupled dipole modes

8. More about Coherent Beam Beam.
CAS - CERN Accelerator School : 5th Advanced Accelerator Physics Course   20 Sep - 1 Oct 1993  - Rhodes, Greece  / Turner, Stuart (ed.) Geneva : CERN, v. CERN-95-06

9. Fourth integer resonance
Substituting
Incidentally
Excites resonances
making islands
Amplitude Q dependence
brings chains close

10. Phase space for fourth order resonance
Island of stability

11. Satellite stop bands Fourier analysis of octupole pattern
Ripple or rf modulation of Q
Makes sidebands at
Resonance when:
EACH SIDEBAND MAKES NEW CHAIN
Resonance when

12. Effect of satellites in phase space

13. Experimental results of satellites

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, Nov 1990, pp
TED0.PCT

15. Crossing the stochastic limit
Increasing Q shift causes islands to merge in a stochastic sea of chaos

16. Importance of tune modulation
Above is with – below is without modulation

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 b which is the length of the waist (hourglass effect)

18. History of colliders

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