Multi-particle Tracking as a Tool for Multi-particle Tracking as a Tool for Studying the Microwave InstabilityStudying the Microwave Instability

Multi-particle Tracking as a Tool for Multi-particle Tracking as a Tool for Studying the Microwave InstabilityStudying the Microwave Instability

Karl BaneStanford Linear Accelerator CenterStanford Linear Accelerator Center

Cornell Damping Ring Workshop26 September 2006

• 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)

IntroductionIntroduction

• compared to VFP equation, multi-particle tracking is easy to implement and quick to run; ~0.4 s/particle/turn=> for Np= 40k, 50 kicks/Ts, 40Ts, takes ~30 sec (MATLAB 7, 3.4 GHz Pentium Xeon)

• however, random noise can lead to inaccuracies in sensitive cases

Outline of talkOutline 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)

Updating equationsUpdating equations (for particle (for particle i)i)::

radiation damping

quantumexcitation

wakekick

• Leap-frog equations; parameters normalized: st, /0, /0, vindVind/(Vrf 0), td td/Ts

• ri– number from Normal distribution, with <r>= 0, <r2>= 1

• vind: bin particles to obtain (b); then convolve bin to bin:

.

• normalized charge k= Nre/(2s0), distribution 0

SLC Damping Rings 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. Dashedcircle shows the size of a quad chamber.

• 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 ringCalculations: old ring

Pseudo-Green function Fourier transform of Green function. Dotsgive result when bellows are shielded.

Green function convolved with z= 6 mm Gaussian bunch. Wake is inductive.

front

Comparison with measurementComparison 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.

TrackingTracking

(a) Turn-by-turn skew when N=3.5e10. (b) Rms when N= 5e10. Np= 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

Bunch shape at two phases 180 deg. apart.

N= 3.5e10

Shape of the mode: density of phase space when subtracted from the average.

New (current) ringNew (current) ring

New bend-to-quad transition

New Green function

Potential well calculation

• New, smoother vacuum chamberwas installed

Vlasov equation calculation. Unstable mode begins at N= 1e10 with = 1.95s0.

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 Np= 30,000

Nth vs d as obtained by tracking. Sign of weak instability.

Experimental oscilloscope traces(B. Podobedov).

Simulated oscilloscope trace using thenew SLC DR wake and the VFP program. N= 3e10. (Warnock and Ellison)

SLC damping ring summarySLC damping ring summary

original 1.1e10 1.5e10

old 2.0e10 3.0e10

new 2.0e10* 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

Recent Benchmark SimulationsRecent Benchmark Simulations

A: QA: Q=1 resonator=1 resonator

• with MATLAB program: 100 bins, b= 0.10; t= 0.02Ts, tend= 40Ts; td= 200Ts; with Np= 40k takes 30 s CPU

set Q= 1; normalized frequency x= 00, strength S= k0R/Q [remember k= Nre/(2s0), ]

(Oide & Yokoya, 1990)

my result for x= 0.5

Vlasov

tracking

threshold for Q=1 resonator

S= 7.5

S= 12.5

• Q=1 resonator seems relatively easy to simulate by this method; no serious noise problems

Haissinski solution

B: R+LB: R+L i) pure inductor, i) pure inductor, LL

, averaged over last 10Ts, vs kl:

kl= 1.5:

Haissinski solution

• pure inductor [vind= -kl ’; kl= eNL/(Vrf03)] 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, Np= 400k, averaging over 5 bins

.

incoherent tune vs amplitude for resistive impedance

(K. Oide, KEK-Preprint 94-138)

ii) R+ (weak) Lii) R+ (weak) L

• K. Oide studied R+L impedance (1994): [kr= eNR/(Vrf0

2)]

• he found R alone is always unstable, with a weak instability

• he found that small L stabilizes the beam: kl> 0.088k2r2 (for small kr); for kr= 0.3 => kl> 0.026

his hypothesis: instability occurs when, in potential well, 2 amplitudes have same tune

vind= -k(r + l ’)..

damping turned off

kr= 0.3, kl= 0.06 (twice Oide’s stability criterion)

Np= 40kno smoothing

Np= 40k5 bin smoothing

Np= 400k5 bin smoothing

Np= 4M5 bin smoothing

C: CSR in free spaceC: CSR in free space

• wake:

• integrate by parts:

• strength parameter S= 2kC/(34/32/3(c0)4/3)

•some difficulty in dealing with (x’)-1/3 properly

1/3 Stupakov’s S

Np= 800kSth~ 1.5 (compare with StupakovSth= 5.5/3= 1.8)

S= 1.5

S= 2.5

Discussion and ConclusionDiscussion 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