1. Theoretical Results for the CSR-driven Instability with
Theoretical Results for the CSR-driven Instability with Negative Momentum
Compaction Factor
Peter Kuske,
Humboldt-Innovation GmbH, Helmholtz-Zentrum Berlin, Germany
TWIICE 2, 8-10, February, 2016, The Cosener‘s House, Abingdon, GB

2. Plausibility of Results – Tune and Tune Spread, Mode Analysis
Content of the Talk
General Remarks on CSR Driven Single Bunch Instability – with Positive Momentum Compaction Factor, α >0
Theoretical Predictions for α < 0 from Multi Particle Tracking and VFP-Solver
Plausibility of Results – Tune and Tune Spread, Mode Analysis
Difficulties Encountered in the Calculations
Comparison with Experimental Results
Summary
Work in Progress – Preliminary Results Only 2

3. I.1 CSR-Driven Single Bunch Instability
Modeling the interaction of electrons via CSR which is shielded by conducting infinite parallel plates 2 h apart (J. B. Murphy, et al., Part. Acc. 1997, Vol. 57, pp 9-64)
Impact on particle distribution analyzed by
Numerical solution of the Vlasov-Fokker-Planck-equation (see my contribution to TWIICE 2014 Workshop, January, 2014, Soleil, France)
Multi particle tracking
General scaling laws exist for the following dimensionless parameters (K. Bane’s and Y. Cai’s contribution to TWIICE 2014 Workshop):
either the “shielding factor”:
or the “normalized resonance frequency” (K. Oide and K. Yokoya, KEK Preprint 90-10, April 1990) with the frequency given by (R.L. Warnock, PAC'91, PAC1991_1824,
with
and the strength parameter: 3 3

4. I.2 Theoretical Result for >0

5. I.3 CSR-Driven Longitudinal Instability for >0
Threshold current depends on
Π > 1: Scsr ~ ·Π
2Fres·o > 5: Finst/Fsyno= 2Fres·o

6. I.4 Summary of Results for >0
Threshold current for the CSR-driven longitudinal single bunch instability depends on shielding factor or normalized resonance frequency and the ratio of Tsyn to Tlong, the damping per synchrotron period. For 2 Fres 0 <2 the instability is weak.
For larger values of the normalized frequency the instability is strong and the threshold current is independent of =Tsyn/Tlong/2 and is given for by:
For longer bunches the frequency of the first unstable mode, Finst, is given by: ,
where inst is the bunch length at the onset of the instability.
Many experimental observations agree very well with these theoretical predictions.
As predicted stable islands at higher beam currents and above unstable regions have been observed. 6

7. II.1 Parameters for the Calculations
Calculations have been performed for the Metrology Light Source (PTB) because experimental results are available (M. Ries, PhD-thesis)
Parameter
MLS
Energy, E0/MeV
629
Bending radius, /m
1.528
Momentum compaction, α
variable
Cavity voltage, Vrf/kV
20, 100, 500
Accelerating frequency, rf/MHz
2 500
Revolution time, T0/ns
160
Natural energy spread, E
Zero current bunch length, 0/ps
Longitudinal damping time, l/ms
11.1
Synchrotron frequency, s/kHz
Height of the dipole chamber, 2h/cm
4.2 7 7

8. II.2 Results for <0
Vrf is fixed and bunch length varied via α. There is no simple linear scaling law. For all parameters studied the threshold depends on the damping per synchrotron period. Red solid line: threshold for >0. 8 8

9. III.1 Very Short Bunches – no Shielding
>0  positive Scsr, Sthr=0.5, <0  negative Scsr, Sthr~-4.
strong bunch lengthening with neg. alpha

10. III.2 Average Tune and Tune Spread at Threshold
Dip in rel. tune spread with >0 is one reason for the lower threshold at PI ~ 0.5
Tune spread larger for >0, with <0 and PI approaching 0 more then compensated for by bunch lengthening. Increased spread at PI~2.8 responsible for the increased threshold around PI ~ 1.8. Long bunches, large PI values, approach Gauss distribution

11. III.3 Mode Coupling for Shortest Bunches
Solution of the Sacherer integral equation with the Gaussian model [Y. Cai, Phys. Rev. ST Accel. Beams, 14, (2011)] = Gaussian distribution in harmonic potential, zero tune spread
Mode analysis using the Gaussian model. α>0 Scsr>0 and α>0 Scsr>0. Black lines represent coherent tune shifts, Re(Ω/ωsyn), and green lines the growth rates, Im(Ω/ωsyn).

12. III.4 Mode Coupling Thresholds for <0
Azimuthal mode coupling only for longer bunches. Thresholds of long bunches independent of sign of α.
Thresholds based on the Gaussian model seem to approach the line found for α>0. Black squares represent the threshold strength, open circles represent the unstable mode, the Re(Ω/ωsyn). Caution – very preliminary results – no good agreement with the other calculations! Fully self consistent calculations required!

13. IV.1 Some Observations and Difficulties
Vlasov-Fokker-Planck solver does not show instability for very short bunches.
Multi particle tracking used instead. Below 3ps bunch length energy widening depends on the number of particles and no clear threshold is seen.
Results for normalized bunch length and energy spread from multi particle tracking studies. With Vrf=50kV synchrotron periods approach the longitudinal damping time. Below 3 ps instability related coherent signals at quite high Scsr.

14. IV.2 Some Observations and Difficulties
4ps with 20 kV, Fsyn=813Hz, ß=
Usually quite clear coherent signals at the onset of the instability. With multi particle tracking appearance of coherent signals well below the threshold where the amplitude increases exponentially. 14 14

15. IV.3 Some Observations and Difficulties
4ps with 20 kV, Fsyn=813Hz, ß=
Normalized bunch length and energy spread from tracking calculations. Start at zero current with a bi-Gaussian distribution (both normalized σ=1) and track for up to 3 damping times. For increased current start with the previously found distribution.
I use binning with up to 40 bins per sigma and up to 106 particles. 15 15

16. IV.5 Some Observations and Difficulties
4ps with 20 kV, Fsyn=813Hz, ß=
Comparison of different calculations: numerical solution of the VFP-equation (red), particle tracking with different number of tracked particles, the length of tracking, and the resolution for the determination of the induced voltage (binning).
No instability observed with VFP-solver beyond island of instability (20 to 50 µA). 16 16

17. IV.6 Some Observations and Difficulties
4ps with 20 kV, Fsyn=813Hz, ß=
Usually tracking calculations start at I=0 from a bi-Gaussian particle distribution. For higher currents (I+ΔI) the calculation continues with the previously found distribution (I). If instead the Haissinski solution at 60 µA is used as initial distribution the black results are obtained – there appears a quite stable island between 50 and 70 µA. 17 17

18. IV.7 Some Observations and Difficulties
40 µA µA µA µA
Between 50 and 70µA there exist two “more or less stable” (top and bottom) particle distributions in the longitudinal phase space. In the low current unstable region (20 – 50 µA) the particle distribution rotates very regularly in the longitudinal phase space. Above 50µA the bunches are “boiling” or can be quite “silent” up to 70µA (bottom). 18 18

19. V.1 Experimental Results from M. Ries for <0
MLS-experiments: alpha fixed and Vrf varied 19 19

20. V.2 Experimental Results from M. Ries for <0
MLS-experiments: alpha fixed and Vrf varied 20 20

21. V.3 Experimental and Theoretical Results for <0
MLS-experiments: alpha fixed and Vrf varied 21 21

22. IV.4 Experimental and Theoretical Results for <0
For long bunches the double log-plot shows acceptable agreement with α > 0 results. For short bunches theory predicts higher thresholds than observed. Suppression of the bunch lengthening by the inductive impedance of the vacuum chamber? 22 22

23. VI.1 Summary of Results for <0
Experimental and theoretical data in very good agreement for long bunches, and in poor agreement for short bunches.
The CSR-driven longitudinal single bunch instability thresholds seem to depend always on the ratio between Tsyn and Tlong.
There is no simple scaling law relating the strength at which the instability sets in and the shielding parameter (or the normalized resonance frequency).
For short bunches my numerical solution of the VFP-equation does not yield meaningful instability thresholds – importance of shot noise?
Had to use particle tracking for shorter bunches 20 kV)
Thresholds depend on the number of tracked particles – the more particles the higher the threshold.
Only for long bunches particular spectral features in the spectra of the low frequency CSR-power indicate the onset of the instability directly. Even in this region no linear relationship between the dominant frequency and 2 ·Fres·o was found (Vrf=20kV).
Only the increase in energy spread can be used for the shortest bunches as an indication for the instability.
Work in Progress – Preliminary Results Only 23 23