Effect of nonlinearity on Head-Tail instability 3/18/04.
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Transcript of Effect of nonlinearity on Head-Tail instability 3/18/04.
![Page 1: Effect of nonlinearity on Head-Tail instability 3/18/04.](https://reader035.fdocuments.us/reader035/viewer/2022062301/5697bf851a28abf838c878e6/html5/thumbnails/1.jpg)
Effect of nonlinearity on Head-Tail instability
3/18/04
![Page 2: Effect of nonlinearity on Head-Tail instability 3/18/04.](https://reader035.fdocuments.us/reader035/viewer/2022062301/5697bf851a28abf838c878e6/html5/thumbnails/2.jpg)
contents
• 2-particle model and head-tail instability
• Nonlinear force– One particle under nonlinear force– Decomposition of the results– Nonlinear resonance under sinusoidal driving– Is this a way to simplify the dynamics of 2-
particle model under nonlinear force
• Conclusion
![Page 3: Effect of nonlinearity on Head-Tail instability 3/18/04.](https://reader035.fdocuments.us/reader035/viewer/2022062301/5697bf851a28abf838c878e6/html5/thumbnails/3.jpg)
2-particle model
• A bunch made of two macro-particles, each of them executes synchrotron oscillation.
• The tail-particle sees the wake field produced by head-particle.
• After each half synchrotron period, the two particles switch longitudinal positions
![Page 4: Effect of nonlinearity on Head-Tail instability 3/18/04.](https://reader035.fdocuments.us/reader035/viewer/2022062301/5697bf851a28abf838c878e6/html5/thumbnails/4.jpg)
Equations of motion:
122
2
12
1 0
xFxx
xx
w
• Where, is the wake force, and is related to by: *
• is a dimensionless parameter • Solutions stable condition: , when
wF
s
wF2
2
* See A. Chao “physics of collective beam instabilities in high energy accelerator”
s
![Page 5: Effect of nonlinearity on Head-Tail instability 3/18/04.](https://reader035.fdocuments.us/reader035/viewer/2022062301/5697bf851a28abf838c878e6/html5/thumbnails/5.jpg)
Modes of the model
22sin
odd ,2
even ,2
ll
ll
ss
ss
tune
![Page 6: Effect of nonlinearity on Head-Tail instability 3/18/04.](https://reader035.fdocuments.us/reader035/viewer/2022062301/5697bf851a28abf838c878e6/html5/thumbnails/6.jpg)
Sextupole and damping
122222
22
21211
21
2
0 2
xFxgxxx
xgxxx
w
Can be unstable when is smaller than 2
s
wF2
![Page 7: Effect of nonlinearity on Head-Tail instability 3/18/04.](https://reader035.fdocuments.us/reader035/viewer/2022062301/5697bf851a28abf838c878e6/html5/thumbnails/7.jpg)
Head particle motion in phase space
![Page 8: Effect of nonlinearity on Head-Tail instability 3/18/04.](https://reader035.fdocuments.us/reader035/viewer/2022062301/5697bf851a28abf838c878e6/html5/thumbnails/8.jpg)
FFT of equation-1 solution
![Page 9: Effect of nonlinearity on Head-Tail instability 3/18/04.](https://reader035.fdocuments.us/reader035/viewer/2022062301/5697bf851a28abf838c878e6/html5/thumbnails/9.jpg)
Tune shift vs. Xmax
![Page 10: Effect of nonlinearity on Head-Tail instability 3/18/04.](https://reader035.fdocuments.us/reader035/viewer/2022062301/5697bf851a28abf838c878e6/html5/thumbnails/10.jpg)
Ratio of amplitudes vs. Xmax
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.0 0.5 1.0 1.5 2.0 2.5
Xmax (mm)
Rat
io o
f am
pli
tud
esg2=10000 g2=20000
![Page 11: Effect of nonlinearity on Head-Tail instability 3/18/04.](https://reader035.fdocuments.us/reader035/viewer/2022062301/5697bf851a28abf838c878e6/html5/thumbnails/11.jpg)
Ratio of amplitudes vs. tune shift
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
tune shift
Rat
io o
f am
plitu
de
g2=10000 g2=20000
![Page 12: Effect of nonlinearity on Head-Tail instability 3/18/04.](https://reader035.fdocuments.us/reader035/viewer/2022062301/5697bf851a28abf838c878e6/html5/thumbnails/12.jpg)
Sinusoidal driving nonlinear oscillation
)cos('2'' 22
2 daxgxxx
![Page 13: Effect of nonlinearity on Head-Tail instability 3/18/04.](https://reader035.fdocuments.us/reader035/viewer/2022062301/5697bf851a28abf838c878e6/html5/thumbnails/13.jpg)
Fold over effect
Driving frequency
Xm
ax
![Page 14: Effect of nonlinearity on Head-Tail instability 3/18/04.](https://reader035.fdocuments.us/reader035/viewer/2022062301/5697bf851a28abf838c878e6/html5/thumbnails/14.jpg)
Transition point
Xm
ax
X_initial
![Page 15: Effect of nonlinearity on Head-Tail instability 3/18/04.](https://reader035.fdocuments.us/reader035/viewer/2022062301/5697bf851a28abf838c878e6/html5/thumbnails/15.jpg)
Bi-stability
![Page 16: Effect of nonlinearity on Head-Tail instability 3/18/04.](https://reader035.fdocuments.us/reader035/viewer/2022062301/5697bf851a28abf838c878e6/html5/thumbnails/16.jpg)
In phase space
![Page 17: Effect of nonlinearity on Head-Tail instability 3/18/04.](https://reader035.fdocuments.us/reader035/viewer/2022062301/5697bf851a28abf838c878e6/html5/thumbnails/17.jpg)
In phase space with larger driving
![Page 18: Effect of nonlinearity on Head-Tail instability 3/18/04.](https://reader035.fdocuments.us/reader035/viewer/2022062301/5697bf851a28abf838c878e6/html5/thumbnails/18.jpg)
In phase space with larger driving
![Page 19: Effect of nonlinearity on Head-Tail instability 3/18/04.](https://reader035.fdocuments.us/reader035/viewer/2022062301/5697bf851a28abf838c878e6/html5/thumbnails/19.jpg)
Compare the approximation with the total wake case
Total w
ake
Fundam
ental com
ponent only
![Page 20: Effect of nonlinearity on Head-Tail instability 3/18/04.](https://reader035.fdocuments.us/reader035/viewer/2022062301/5697bf851a28abf838c878e6/html5/thumbnails/20.jpg)
2-particle model
![Page 21: Effect of nonlinearity on Head-Tail instability 3/18/04.](https://reader035.fdocuments.us/reader035/viewer/2022062301/5697bf851a28abf838c878e6/html5/thumbnails/21.jpg)
Conclusion
• The wake force from head particle is approximated with a simple harmonic solution.
• The equation of motion of tail particle becomes a sinusoidal driving nonlinear oscillation.
• Based on understanding of motion of the single particles, further work instability for 2-particle model under nonlinear force can be better investigated.