1. Multibunch instabilities from TMBF data at Diamond
R. Bartolini
Diamond Light Source
and
John Adams Institute, University of Oxford
thanks: R. Fielder, S. Pande, G. Rehm, V. Smalyuk (BNL)
Beam dynamics meets diagnostics
Firenze, 6 November 2015

2. Beam dynamics meets diagnostics
Outline
(Short) review of main results of the theory of multi-bunch instability
Data from Diamond TMBF system
Impedance analysis and comparison with CST simulations (WIP)
Open issues
Beam dynamics meets diagnostics
Firenze, 6 November 2015

3. Transverse multibunch oscillations - no wakefields
Each bunch oscillates with the betatron frequency x,y
In a multi-bunch train without wakefields all bunches are independent
The motion can be expanded in Fourier modes, (M bunches = M modes, and in full fill, M = harmonic number)
However, without wakefields there is no reason to prefer this set to any other “basis”
(i.e. modes are degenerate)
Beam dynamics meets diagnostics
Firenze, 6 November 2015

4. Multibunch oscillations - with transverse wakefields
Bunches are now coupled by the wakefields
Each bunch oscillates with the betatron frequency x,y
In the limit of small wakefields (linear wakes) the Fourier modes are eigenvectors of the time evolution, i.e. modes are preserved during the time evolution
The betatron oscillation frequency becomes complex
Im  damping or anti-damping
Re  frequency shift
The degeneracy of the Fourier modes is broken, i.e.  = ()

5. Complex frequency shift and driving impedance
For zero length (point-like) bunches (see e.g. A. Chao)
e.g.
coherent tuneshift
(Re = ImZ)
growth rate
(Im  = ReZ)
mode  is driven by the impedance computed at (pM + )0 + 
Beam dynamics meets diagnostics
Firenze, 6 November 2015

6. Complex frequency shift and driving impedance
For Gaussian rigid bunches with rms bunch duration 
e.g.
coherent tuneshift
(Re = ImZ)
growth rate
(Im  = ReZ)
mode  is driven by the impedance computed at (pM + )0 + 
Beam dynamics meets diagnostics
Firenze, 6 November 2015

7. Beam dynamics meets diagnostics
Non rigid bunches
For bunches with finite length – and internal modes (see e.g. A. Chao)
Beam dynamics meets diagnostics
Firenze, 6 November 2015

8. Complex frequency shift and driving Impedance
For bunches with finite length – and internal modes (see e.g. A. Chao)
e.g.
coherent tuneshift
(Re = ImZ)
growth rate
(Im  = ReZ)
mode (, l) is driven by the impedance computed at (pM + )0 +  + ls
Beam dynamics meets diagnostics
Firenze, 6 November 2015

9. Complex frequency shift and driving Impedance
With non-zero chromaticity the bunch frequency spectrum is shifted by
 =  / and naming ,l = (pM + )0 +  + ls
e.g.
coherent tuneshift
(Re = ImZ)
growth rate
(Im  = ReZ)
The bunch samples the impedance at ,l but the bunch frequency spectrum is shifted at ,l – 
Beam dynamics meets diagnostics
Firenze, 6 November 2015

10. Sampling in time and aliasing in frequency
A mode  sampled turn-by-turn corresponds to the frequency
frac(x,y) aliased in [0, 0]  frac(x,y)
i.e. fractional part of the tune
A mode  sampled bunch-by-bunch corresponds to the frequency
0 + x,y aliased in [0, M0]  i.e. baseband of the RF
For real signal the spectrum is further folded symmetrically in [0 , M0/2] 1 2 3
Modes  = 0 , 1, , ..., M – 1
Modes  = 0 , 1, , ..., M – 1
folded, i.e.
– 1 = M – 1
– 2 = M – 2
... -1 1 -2 2 -3 3

11. Beam dynamics meets diagnostics
Grow damp experiment 1)
Artificially excite mode  by using a stripline driven at the frequency (pM + )0 +  2)
Stop the excitation and measure free oscillations (damped or anti damped) 3)
Run feedback to damp any unstable mode or any residual oscillation
Repeat for all modes  = 0 , 1, ..., M - 1
Beam dynamics meets diagnostics
Firenze, 6 November 2015

12. Grow damp experiment at Diamond
936 bunches - 2 ns (500 MHz) – full fill M = 936
530 kHz revolution frequency –
Current up to 300 mA; bunch length ps rms – with current 1)
Artificially excite mode  for 250 turns at the frequency (pM + )0 + 
p is optimised on the specific stripline design and operating frequency (at Diamond p = 0 corresponding to 0 – 250 MHz band) 2)
Stop the excitation and measure free oscillations for 250 turns
transfer data ***
impossible to transfer fast all bunch-by-bunch turn-by-turn data
only the Fourier component (amplitude and phase) at the frequency of interest for mode  is stored 3)
Run feedback to damp any unstable mode or any residual oscillation for 250 turns

13. Grow damp experiment at Diamond
Example of mode that is naturally damped: recording the complex amplitude on a turn-by-turn basis only of the mode previously excited
Data reduced from 1.3 GB to 5.6 MB (see G. Rehm et al IBIC14)
amplitude
phase
fit slope here
250 turns
250 turns
250 turns
Repeat for all modes  = 0 , 1, ..., M – 1
Offline post processing gives the frequency shift and damping or growth rate (take away radiation damping and chromatic damping (if any))

14. Growth rates of vertical coupled bunch modes
Vertical TMF data
full fill – zero chromaticity – ID gap open
Radiation damping subtracted
blu measured – red fit
data suggest resistive wall and few high Q resonators 3 4
Resistive Wall 2
β = m, b = 13.5 mm,
ρ = 7.3·10-7 Ω·m 5 1
Resonator

15. Search for resonator-like structures

16. BPM buttons and enclosure1 2 3
Resonance 1
fr = (19Nb – 22)f0 = GHz
Rs = 2.8 MΩ/m Q = 2000
Resonance 1 – mode -22
fr = GHz; Q = 2000; Δf = 61.7 MHz
Resonance 2 – mode -64
fr = GHz; Q = 20000; Δf = 51.4 MHz
Resonance 2
fr = (18Nb – 64)f0 = GHz
Rs = 1.4 MΩ/m Q = 20000
Resonance 3
fr = (13Nb + 119)f0 = GHz
Rs = 0.8 MΩ/m Q = 1000
Resonance 3 – mode 119
fr = GHz; Q = 1000; Δf = MHz

17. Effect of closing the IDs
Closing the gap of all IDs changes the geometric and RW impedance
Vertical coupled bunch modes 3 4
geometry
ID res. walls
Forest of spikes at modes has been associated to IDs

18. First tentative interpretation of IDs
ID gap – I07: CPMU
different vessel enclosure
ID gap – I04
Phase I - in vacuum ID
Changes with gap only in specific modes
ID gap – I13:
In vacuum ID

19. More recent measurements
Large resonant peak moved in the last months from n = 22 now at n = 81, e.g.
Same conditions 150 mA – full fill 936 bunches – 0 chromaticity – IDs open
Other peaks (e.g. n = 61) are reproducible through the measurements
Investigating causes:
unlikely from BPM buttons and enclosures
shutdown interventions? septum move; RF cavity swap

20. Transverse wakefields in the RF cavity
Vertical impedance in RF cavities
Largest secondary peak at GHz.
does not match any of the excited modes
Three stub tuners?
Different geometry of coupling to waveguides?
Issue still unresolved

21. Further analysis on IDs
Occasional jumps in the mode spectrum with ID gap
These are however “better understood” as they are clearly correlated to the ID gap changes – jumps of 10 or more modes !
Beam dynamics meets diagnostics
Firenze, 6 November 2015

22. Beam dynamics meets diagnostics
ASP CB mode analysis
R. Dowd et al., IPAC15
Largest mode number
shifts with ID gap
Beam dynamics meets diagnostics
Firenze, 6 November 2015

23. Other measurements (WIP)
Using the functionality of the TMBF, grow damp measurement can be made
on many machine configuration repeatedly with time
Different RF
i.e. different bunch lengths
Different chromaticities
i.e. different chromatic damping 2 1

24. Open issues and conclusions
The Diamond TMBF allowed extensive and repetitive grow damp measurements allowing parameter scan and a regular monitoring of the impedance of the machine
These studies have produced rather surprising results.
Although all coupled bunch modes are eventually stabilised by the TMBF system, it is clear that the machine impedance is changing with time in way that are difficult to interpret and predict.
The specific resonances at Diamond will be further investigated to identify possible causes beyond IDs, BPMs and RF cavities (e.g. Collimators, ...)
More quantitative evaluation of the impedance – beyond the qualitative identification of the frequency excited in the spectrum – is forthcoming
Beam dynamics meets diagnostics
Firenze, 6 November 2015