1. 1 Instabilities and Phase Space Tomography in RR Alexey Burov RR Talk May 19 2010

2. Instability as a cooling limitation Transverse instability limits cooling possibilities for pbars in RR. Conventionally, these limitations are described as a threshold for the “density” parameter This single-parameter description does not reflect dependence of the threshold on RF structure. Dependence on RF was studied Apr 13. RF and RWM data allows to reconstruct the phase space density for every bunch, and compare density thresholds for various RF configurations. 2

3. Instability, general ideas Instability is caused by the machine impedance (mainly resistive wall). Stabilization factors: –Landau Damping (~step-like function, effective above some frequency) –Damper (step-like function, effective below 70 MHz) 3 Impedance Damper Landau => cooling

4. Landau damping Landau damping is provided by the tail particles: At 70 MHz in RR,. Exact threshold for the density D depends on the distribution tails at which is hard to detect. Since tails depend on the depth of potential well, threshold density should reflect this dependence. 4

5. 5 Steady State Distribution The problem to solve is to find phase space density with an action as the argument I, from measured linear density and the voltage shape. Having this problem solved, the questions about 90% emittance etc. are immediately answered. This problem leads to the Abel integral equation on the phase space density as a function of Hamiltonian. This equation is independent of the voltage, which is needed at the second step only, for function. Abel equation can be solved either numerically by the matrix inversion, or the known analytical solution can be used (Leo Michelotti, PRST-AB, 6, 024001 (2003) and Refs. therein). For Tevatron, this problem has been solved by V. Lebedev and A. Tollerstrup.

6. 6 Hamiltonian Particle energy offset ε and its time-position in the beam τ can be treated as canonical variables. Then the Hamiltonian can be written as Here and are the synchronous particle momentum and the slippage factor and is the potential. The steady state distributions can depend on its arguments only through the integral of motion:.

7. 7 Abel Equation The beam linear density relates to the distribution function as Inverting the dependence, and assuming, the equation on the distribution function follows: Substitution transforms this integral equation to the Abel equation, solved by this Norwegian mathematician in 1823.

8. 8 Action variable To find out how high is 90% or any N% emittance, a canonical transformation from the original variables ε, τ to the action-phase variables I, φ is needed. Relation between the action and Hamiltonian follows from the phase space conservation under canonical transformations: Using that, the phase space density can be expressed in terms of the action Note that having expressed the energy offset ε in MeV, and the time position τ in μs, gives the action in conventional eV∙s.

9. 9 N% Emittance Portion of particles inside the phase space 2πI is given by the integral of the normalized distribution: An inverse function gives the phase space occupied by the given portion of particles N.

10. Case 3 Every one of 4 small 2.5 MHz bunches was analyzed. 10

11. Synchronization Raw data for RF and RWM are not perfectly synchronized. The error can be corrected, taking into account that the current is a unique function of the potential. For any given micro-bunch, left and right sides of the bunch profile must give the same dependence. 11 raw data RF retards by 5 ns

12. Case 3, tomography results Integral phase space densities for 4 micro-bunches: same cores, different tails. 12

13. Threshold Densities 13 Relative values of D 99% agrees with the instability results and shed a light on the actual threshold values for this RF configuration: Note: is 6 times rms emittance fit for FW measurements, mm*mrad.

14. Case 2: threshold unmeasured It was expected to see better densities for Case 2 than in Case 3, but it was not happen. Emittance growth was observed without any signal outside the damper bandwidth. It may be either external perturbation or a damper’s failure. So threshold density for case 2 should be considered as unmeasured. 14

15. Case 1 Case 1 is similar to operation’s mined bunches – same depth of the potential well. 15

16. Case 1 results 16

17. Case 4 17 Potential well is ~ 4 times deeper then in case 1. Better threshold density was expected.

18. Case 4 results 18 Threshold densities are ~ 3 times higher than for the case 1 !

19. Operations, Apr 27 2010 19

20. Mined bunch #9 20 Depth of the potential well is identical to the case 1 (only 1.5% deeper), but fast particles spend less time outside the bucket.

21. Same beam, extraction bunch #2 21

22. Table of Densities 22 D95D99comment Case 13.8±0.32.6±0.2threshold Case 411.9.threshold Case 37.2±0.25.1±0.3threshold Op 94.5±0.33.1±0.3stable Op 8/24.12.4stable

23. Conclusions Threshold density shows significant dependence on RF configuration. Increasing potential well allows to cool deeper (case 1 vs case 4). Comparison of case 1 with operational case (mined bucket #9) shows marginally visible benefit of #9. Since #9 was at unknown distance from the threshold, more studies needed to make a conclusion. Perhaps, reduction of “zero-potential” may be helpful. Increasing chromaticity (for cold beam stage) should help. Increasing it twice should allow to have 50% higher density. 23