Benchmark and Study of PSR Longitudinal Beam Dynamics

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 Goal: Benchmark ORBIT’s longitudinal impedance algorithm with experimental data. (Partial response to ASAC request for benchmark of ORBIT impedance capabilities). 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).

4. Wall current monitor signal Peak instability End injection Experimental Evidence of Microwave Instability • 3 inductive inserts @ 25º C. • Beam intensity = 650nC (4×1012 protons) • Instability peak @ 200 s (150 s after injection) Figure courtesy C. Beltran, doctoral thesis.

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

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

7. 650 nC (~4×1012 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. 8×106 macroparticles. ORBIT Simulations of Microwave Instability ORBIT Simulation Parameters:

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

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 Experimental Simulated 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. 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. Analysis of Instability Threshold Experimental Data 460 nC (strong instability) 80 nC (threshold) 70 nC (noise level) Simulated Data 500 nC (strong instability) 70 nC (threshold) 50 nC (noise level)

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

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

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

15. 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. Simulations of the 201 MHz Structure With Space Charge No Space Charge End of Injection 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 (72nd sub-harmonic). See clear formation of separatrix and “anti-buckets” at frequency of 201 MHz. slow slow fast fast fast Formation of 201 MHz “anti-buckets” Self-consistent, stationary analytical solution can be found for simple 2-state system.

17. 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. Observations of 201 MHz Structure Dynamics 200 nC …650 turns after injection End of injection… …250 turns after injection 100 nC End of injection… …250 turns after injection …650 turns after injection

18. 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: A Vlasov solver for one bucket

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