1. Benchmark and Study of PSR Longitudinal Beam Dynamics Sarah M. Cousineau, Jeffrey A. Holmes, Slava Danilov MAP Workshop March 18, 2004

2. Longitudinal Instability Benchmark Part I – Benchmark of PSR longitudinal microwave instability.

3. ORBIT Benchmark of Microwave Instability in PSR Background: 1999 – 3 ferrite inductive inserts placed in PSR to provide longitudinal space charge compensation.  Inserts lead to unacceptably large microwave instability, were removed. 1999, 2000 – Heating of the inserts was shown to cure instability while still providing space charge compensations. Two heated inserts now used in PSR. 2003 – Chris Beltran models impedances of both sets of inserts using MAFIA (PhD thesis). Results allow for detailed simulations of instability (ESME, ORBIT, etc). Goal: Benchmark ORBIT’s longitudinal impedance algorithm with experimental data. (Partial response to ASAC request for benchmark of ORBIT impedance capabilities).

4. Experimental Evidence of Microwave Instability Wall current monitor signal End injection Peak instability 3 inductive inserts @ 25º C. Beam intensity = 650nC (4×10 12 protons) Instability peak @ 200  s (150  s after injection) Figure courtesy C. Beltran, doctoral thesis.

5. 2 Turn Wall Current Monitor Signal (Experimental) Signal for 2 turns at end of injection Signal for 2 turns at peak of instability Instability Frequency = 72 MHz (harmonic = 26) Figures courtesy C. Beltran, doctoral thesis.

6. Impedance of Inductive Inserts Impedance of room temperature inductive inserts (C. Beltran, thesis 2003)

7. ORBIT Simulations of Microwave Instability ORBIT Simulation Parameters: 650 nC (~4×10 12 protons). 150 us accumulation time (~400 turns), + 200 us storage (~600 turns). Z/n as computed by C. Beltran.  p/p as bi-Gaussian, 66% with  =6.9×10 -4, and 34% with  = 2.8×10 -4. Longitudinal tracking only. From a numerical convergence study performed, used: –256 longitudinal bins. –8×10 6 macroparticles.

8. ORBIT Benchmark Results End injection Peak of instability Instability peak 150  s after injection (Same as experiment ).

9. Simulated One-Turn Wall Current Monitor Signal Signal for 2 turns at end of injection Signal for 2 turns a peak of instability Experimental Simulated

10. Evolution of Dominant Harmonics Exponential growth of harmonics observed. Dominant harmonic is h=26, same as experiment. Growth time of instability,   42  s; Experiment result is   33  s Slope=1/   1/42  s

11. Analysis of Instability Threshold Data set taken in 2002 to understand threshold; 2 inductors at room temp. Define threshold by beam intensity at which relevant harmonics rise coherently above noise level. Experimental threshold=80 nC; Simulated threshold=60-70 nC. Experimental Data Simulated Data 70 nC (noise level) 80 nC (threshold) 460 nC (strong instability) 500 nC (strong instability) 70 nC (threshold) 50 nC (noise level)

12. Linac Microbunch Dynamics Part II – Linac microbunch dynamics in the PSR.

13. The PSR 201 MHz Phenomenon 201 MHz structure in PSR should disappear in  30 turns Microwave instability data shows this structure sticking around for ~1000 turns. End of Injection Chopped beam Coasting beam End of Injection

14. Experimental Analysis of the 201 MHz Structure Analysis of 201 MHz harmonic shows structure increasing after injection. Longitudinal profile 300 turns after end of injection. 70 nC 210 nC Analysis also shows 201 MHz structure is stronger at higher intensity End of injection Chopped beam Coasting beam

15. Simulations of the 201 MHz Structure 1D tracking simulations with ORBIT show same long-lived 201 MHz microstructure; structure present with or without impedance. Structure quickly decoheres in simulations without space charge. With Space Charge No Space Charge End of Injection

16. Long-lived “bubble” structures noticed in CERN PS Booster ring; suspected due to linac micro-structure. Much theoretical work published to explain long-lived structure (resistive wake, etc). In 2000 CERN PSB experiment used RF to insert larger, lower frequency “holes” and observe structure during acceleration. Paper by Koscielniak et al argues that longevity of holes due to space charge. The PSR machine a special case for which the ring frequency is an exact sub- harmonic of the linac frequency (72 nd sub-harmonic). See clear formation of separatrix and “anti-buckets” at frequency of 201 MHz. Formation of 201 MHz “anti-buckets” fast slow fast slow fast Self-consistent, stationary analytical solution can be found for simple 2-state system.

17. Observations of 201 MHz Structure Dynamics Near steady-state condition for certain balance of  p/p and intensity. Rate of injection also an important condition for establishing steady state. We are in the process of investigating these dynamics for the PSR case. End of injection… …250 turns after injection …650 turns after injection End of injection… …250 turns after injection …650 turns after injection 200 nC 100 nC

18. A Vlasov solver for one bucket Set up fast Vlasov solver to look for steady-state solutions. Solve: with, (self-consistency) and periodic boundary conditions in . For steady state, should have:

19. Hamiltonian contours Can anti-buckets live forever?  At least 10000 turns, under right conditions Long-lived solutions 0 turn longitudinal phase space 10,000 turn longitudinal phase space Density profiles for 0 and 10000 turn distributions Distribution function dE Phi