1. Multi-particle Tracking as a Tool for Studying the Microwave Instability Karl Bane Stanford Linear Accelerator Center Cornell Damping Ring Workshop 26 September 2006

2. the longitudinal broad-band impedance of a ring can cause current dependent bunch lengthening, energy spread increase, time dependent (e.g. bursting) behavior, and heating of elements numerical tools: - wake calculations– MAFIA, ABCI, GDFIDL, … - longitudinal phase space simulations— 1 multi-particle tracking (Renieri, 1976) 2 linearized Vlasov equation for threshold (Oide, Yokoya, 1990) 3 Vlasov—Fokker-Planck equation (Ellison, Venturini, Warnock, 1999) Introduction compared to VFP equation, multi-particle tracking is easy to implement and quick to run; ~0.4  s/particle/turn=> for N p = 40k, 50 kicks/T s, 40T s, takes ~30 sec (MATLAB 7, 3.4 GHz Pentium Xeon) however, random noise can lead to inaccuracies in sensitive cases

3. Outline of talk using multi-particle tracking, consider benchmark examples: old SLC damping ring wake new (current) SLC damping ring wake Q= 1 resonator wake series R+L wake coherent synchrotron radiation wake (CSR; no shielding) for SLC wakes, also discuss wake calculation and comparison with measurement not meant to be a thorough study see also S. Heifets’ talk added to Stupakov’s list presumably similar calculations will be performed for ILC rings from literature (Bane & Oide, 1995)

4. Updating equations (for particle i): (for particle i): radiation damping quantum excitation wake kick Leap-frog equations; parameters normalized:   s  t,   /   0,   /   0, v ind  V ind /(V rf   0 ), t d  t d /T s r i – number from Normal distribution, with = 0, = 1 v ind : bin particles to obtain (  b ); then convolve bin to bin:. normalized charge k= Nr e /(2  s   0 ), distribution    0

5. SLC Damping Rings 3 versions: (i) original, (ii) old (shielded bellows), (iii) new (current; new, smoother vacuum chamber) Nominal  z ~ 5 mm, half aperture a ~1 cm Old ring inductive (small objects dominated impedance); new ring resistive Layout of north damping ring. Circumference is 35 m. Cross-section of a bend chamber. Dashed circle shows the size of a quad chamber.

6. Old ring was inductive; generated a table of strength of inductive elements Pseudo-Green function: for a short Gaussian bunch (  z = 1 mm) find an accurate wake; to be used in potential well/instability calculations; used MAFIA; included QD, QF segments, RF cavities, BPM’s, etc (not e.g. septum) Vertical profile of QF segment (top) and QD segment (bottom). There are 20 of each in the ring. Dashes represent non-cylindrically symmetric objects. The inductive vacuum chamber objects. The total yields |Z/n|= 2.6 . Calculations: old ring

7. Pseudo-Green functionFourier transform of Green function. Dots give result when bellows are shielded. Green function convolved with  z = 6 mm Gaussian bunch. Wake is inductive. front

8. Comparison with measurement Haissinski solution for bunch shapes (head is to the left). Plotting symbols are measurement data. (a) Bunch length and (b) centroid shift. Plotting symbols are measurement data.

9. Tracking (a) Turn-by-turn skew when N=3.5e10. (b) Rms when N= 5e10. N p = 300k. Fourier transforms of plots at left. Position of peaks in skew signal FT vs. N. “Sextupole” mode seen in measurements with same d /dN. strong potential well distortion strong instability; “azimuthal mode coupling”; Boussard criterion applies

10. Bunch shape at two phases 180 deg. apart. N= 3.5e10 Shape of the mode: density of phase space when subtracted from the average.

11. New (current) ring New bend-to-quad transition New Green function Potential well calculation New, smoother vacuum chamber was installed

12. Vlasov equation calculation. Unstable mode begins at N= 1e10 with = 1.95 s0. Shape of unstable mode new type of mode—”weak instability”, radial mode coupling (Oide); not governed by Boussard criterion   just above threshold (a) and at 2e10 (b) when N p = 30,000 N th vs  d as obtained by tracking. Sign of weak instability.

13. Experimental oscilloscope traces (B. Podobedov). Simulated oscilloscope trace using the new SLC DR wake and the VFP program. N= 3e10. (Warnock and Ellison)

14. SLC damping ring summary original1.1e101.5e10 old2.0e103.0e10 new2.0e10*1.5e10 threshold version calculated measured “sextupole” mode quadrupole mode *if add 2nH (0.1  ) inductance How to understand: from old to new ring reduced the impedance and threshold dropped? old, inductive ring—strong mode—tune spread—weak modes Landau damped new, resistive ring—weak mode—little tune spread—no Landau damping Note: old ring, SLC operation limited to 3e10, new ring—5e10

15. Recent Benchmark Simulations A: Q=1 resonator with MATLAB program: 100 bins,  b = 0.1   0 ;  t= 0.02T s, t end = 40T s ; t d = 200T s ; with N p = 40k takes 30 s CPU set Q= 1; normalized frequency x=  0   0, strength S= k  0 R/Q [remember k= Nr e /(2  s   0 ), ] (Oide & Yokoya, 1990) my result for x= 0.5 Vlasov tracking threshold for Q=1 resonator

16. S= 7.5 S= 12.5 Q=1 resonator seems relatively easy to simulate by this method; no serious noise problems Haissinski solution

17. B: R+L i) pure inductor, L i) pure inductor, L  , averaged over last 10T s, vs kl: kl= 1.5: Haissinski solution pure inductor [v ind = -kl ’; kl= eNL/(V rf   0 3 )] is lossless, should be stable but numerical noise is enhanced in ’ [=( i+1 - i )/  b ] for almost inductive wake, Boussard criterion kl<~ (2  ) 1/2 for test ran pure inductor, N p = 400k, averaging over 5 bins.

18. incoherent tune vs amplitude for resistive impedance (K. Oide, KEK- Preprint 94-138) ii) R+ (weak) L K. Oide studied R+L impedance (1994): [kr= eNR/(V rf   0 2 )] he found R alone is always unstable, with a weak instability he found that small L stabilizes the beam: kl> 0.088k 2 r 2 (for small kr); for kr= 0.3 => kl> 0.026 his hypothesis: instability occurs when, in potential well, 2 amplitudes have same tune v ind = -k(r + l ’)..

19. damping turned off kr= 0.3, kl= 0.06 (twice Oide’s stability criterion) N p = 40k no smoothing N p = 40k 5 bin smoothing N p = 400k 5 bin smoothing N p = 4M 5 bin smoothing

20. C: CSR in free space wake: integrate by parts: strength parameter S= 2kC/(3 4/3  2/3 (c   0 ) 4/3 ) some difficulty in dealing with (x’) -1/3 properly 1/3 Stupakov’s S

21. N p = 800k S th ~ 1.5 (compare with Stupakov S th = 5.5/3= 1.8) S= 1.5 S= 2.5

22. Discussion and Conclusion simulations using the SLC damping ring wakes reproduced many features that were found in measurement (bunch lengthening, instabilities, etc) --however, important vacuum objects were basically 2D; if 3D objects are significant => more difficulty in obtaining an accurate wake as a complement to the more accurate VFP and Vlasov solvers, multi- particle tracking is a fast, easy-to-implement method for studying the microwave instability for benchmarking examples, it seems that: --method works well for SLC DR wakes, Q=1 resonator --for sensitive impedances, e.g. L, R+(weak)L works less well (filtering?) --for CSR (in free space) wake, algorithm needs refinement