1. INTENSITY LIMITATIONS IN THE LHC INJECTORS Discussion on Landau damping Ibon Santiago González Summer Student Session 2007

2. Why is the quality of the beam limited? Set by nature PARAMETERS: SIZE, INTENSITY

3. Luminosity Number of particles per bunch Number of bunches per beam Revolution frequency Relativistic velocity factor Normalized transverse beam emittance β-function at the collision point Geometric reduction factor INTENSITY SIZE

4. protons are like elephants They have good memory They won’t forgive you. No natural COOLING. Easily perturbed and mistakes add up. Cool down as they travel around Beam keeps itself focused and straight. p+ e-

5. …and they’re not alone! PS dept.

6. Main intensity limitations Beam pipe presents an IMPEDANCE Z (geometry dependant) Bunch induces EM field. Currents act back on the beam. INTENSITY dependant EXAMPLES SPACE CHARGE ELECTRON CLOUD EFFECTS TRANVERSE MODE COUPLING INSTABILITIES Etc. Courtesy of K.Schindl

7. Why do accelerators work? Large number of collective instability mechanisms. But the beam seems to be basically stable. Natural stabilizing mechanism:

8. Landau damping Sacherer formalism applied to the PS Booster

9. Basic idea: swing… Stiff frame. Frequency spread. COHERENT C.M. motion decays quickly compared to INCOHERENT motion of children. STABLE MOTION Flexible frame INSTABILITY GROWTH RATE higher than FREQUENCY SPREAD. INSTABILITY Courtesy of A. Hofmann

10. Basic idea (encore)

11. 1.Frequency shift (due to beam forces, proportional to impedance) 2.Frequency spread (due to non-linear RF forces and momentum spread) Two ingredients

12. The Sacherer Cracker (1964)

13. A longitudinal stability criterion. Task: See when Landau damping is lost for a given mode. Given by Sacherer formulae1. 2. Dispersion relation

14. Stability diagram m=1 m=2 In the PS Booster SPACE CHARGE is dominant

15. Conclusion Landau damping is very important. This model is not realistic but gives hints to loss of Landau damping in the PSB. The bridge collapsed due to loss of Landau damping... Minneapolis bridge. Aug. 07

16. Any questions?