1. Coherent wave excitation in a high current storage ring.
Alexander Novokhatski
58th ICFA Advanced Beam Dynamics Workshop
on High Luminosity Circular e+e-Colliders
October 16, 2016
Cockcroft Institute at Daresbury Laboratory, UK

2. Introduction We will consider a bunched beam in a storage ring.
Inside a bunch all particles moving under almost identical conditions and the radiation field from each particle has same frequency spectrum, but differs in a phases depending on particles’ positions.
The sum of these fields gives a coherent effect under some conditions on the bunch current distribution.
We may assume that the bunch length must be smaller than the wavelength of the radiation wave in order to have approximately fixed relations between phases during the time of excitation.

3. Introduction
The range of frequencies, were coherence becomes important may vary.
We can find may be three different type of ways to have a coherent wave radiation in a storage ring.

4. Coherent Synchrotron Radiation
The first one corresponds to synchrotron radiation, mainly to the low frequency part of the spectrum, which is independent of the particle energy.
Particles of a short bunch will excite waves with a spatial coherence.
The power of these waves may be many orders higher than the power radiated by a single particle.
An example of coherent part of the spectrum of a 0.2 mm bunch, which “touches” 1 THz. A bunch has 4·108 particles. A critical frequency is usually much higher than a bunch frequency

5. Coherent Synchrotron Radiation
An estimate of the total coherent power per unit length
Of cause the low frequency spectrum of the synchrotron radiation is limited by the size of the beam pipe. This shielding will decrease the effect of the coherent synchrotron radiation (CSR) on the beam dynamics.
The spectrum of CSR as of propagating waves can be changed and the frequency range may go to higher frequency in comparison with a bunch length due to the back reaction of these fields and distortion of the bunch shape.

6. Propagating waves
The second type corresponds to excitation of the propagating waves due to obstacles in the beam pipe, like for example, a collimator.
An energy loss of a point charge due to diffraction of its own field on an obstacle, is proportional to the particle energy.
We may consider this is a high frequency part of the radiation, but we know that the low frequency part does not depend on the energy in the ultra relativistic case. Then with many particles the power will increase many times. (Same as for CSR)
To fulfil the coherence condition in this case we need a smaller bunch length in comparison with a wavelength of an excited wave.
1/g

7. Propagating waves
Simple estimate of the loss factor for an obstacle Dr in a pipe with r=a
Usually these waves propagate away before the next bunch comes to this place.
However these waves can be dangerous too as they can propagate long distances and be absorbed in low resistance elements like NEGs or vacuum pumps.
Wake fields due to roughness surface or dielectric layer can be also include in this category, as a representative of the Cherenkov radiation.
We don’t have to forget the resistive-wall wake fields and check if the beam pipe to be water cooled.

8. Condition for a coherent excitation:
Trapped modes
The third type of the coherent excitation corresponds to existence of trapped modes.
This modes could exist in a cavity-like element in the beam pipe.
A trapped mode has a smaller frequency than a cut-off frequency of the beam pipe for a corresponding electromagnetic field distribution (monopole, dipole, etc.).
Sometimes (Murphy's Law) a frequency of a trapped mode is exactly equal to some of the beam spectrum line. The amplitude of the field in this cavity increases linear with a number of bunches passing nearby until self-saturation due to resistivity of the metal walls.
Condition for a coherent excitation:
resonance and

9. Trapped modes
If electromagnetic power dissipates in the place without any outside cooling (water or air), the temperature can rise to very high level up to the melting point.
We will discuss later that some cavities can be hidden outside the beam pipe but have a coupling to the beam through small holes in the metal wall or ceramic windows
We can suggest that “incoherent” excitation happened when a time decay of a trapped mode is smaller than the bunch spacing
And power is
If the bunch spacing is equal to a mode decay time, the “coherent” power is only twice more that “incoherent”

10. CSR fields in the beam pipe.
We found that CSR fields in a beam pipe with a rectangular (or we suggest an elliptical) cross-section, have common features with wake fields excited due to beam pipe inhomogeneity or resistivity.
We believe that all these fields are solutions of Maxwell equations and must have many things in common.
Some theoreticians don’t want to solve Maxwell’s equations (or may be they cannot) and they suggest some “artificial” wake functions or methods. The result is usually confusing.

11. CSR numerical code
The CSR code, which was developed some time ago, now got new features:
Calculating transverse forces coming from CSR, Space charge and wake fields.
3D particle motion
Now it contains quads additionally to the bends.
It includes the edge bend defocusing (usually vertical) and a gradient in the bend.
The improved algorithm in the code makes it possible to simulate 20 micron or even10 micron bunches.
Here we present CSR transverse forces from for SPEAR and LCLS dump magnet

12. SPEAR and LCLS dump magnet
Beam energy 3 GeV
Vacuum chamber 83 mm x 34 mm
Bunch length 0.4 mm
Horizontal beam size 50 micron
Beam energy 13.6 GeV
Vacuum chamber
25 mm x 40 mm
Bunch 20 micron

13. Bunch and CSR fields in the beam pipe
SPEAR
LCLS

14. Bunch shape and CSR fields
Integrated longitudinal force
Bunch shape
tail
head

15. Energy loss averaged over a slice of a bunch
SPEAR
LCLS

16. Beam distribution on the phase plane Energy-Z
tail
head
energy
slice energy
spread
Longitudinal coordinate

17. Slice energy spread
LCLS measured
SPEAR

18. This effect is stronger
We may compare with incoherent energy spread due to quantum fluctuations of synchrotron radiation
Matthew Sands
“Physics of Electron Storage Rings”
SLAC Report No 121, 1970
382 kV for 13.6 GeV
38 kV for 3.4 GeV
This effect is stronger

19. Horizontal kick averaged over a slice of a bunch
The kick is opposite to the bend direction
SPEAR
LCLS

20. We may compare with a formula (R.Talman force without logarithm)
More than two times larger than CSR force

21. A gradient of a vertical (SPEAR) a horizontal (LCLS) kick
Tail is defocused
Head is focused
Similar to the wake field quad in a rectangular chamber
LCLS
SPEAR

22. Comparison with magnet edge defocusing
Karl L. Brown
“A first and second-order matrix theory”
SLAC Report No 75, 1972
Comparable with CSR
Blue line shows relative field distribution
Red lines show integrated edge defocusing

23. A kick due to a vertical (SPEAR) or horizontal (LCLS) displacement of the beam
Similar to the wake field kick
LCLS
SPEAR

24. What is better to keep in mind when designing a beam pipe for a high current machine. PEP-II experience, 3A beam current achieved.
I will argue with some “researchers”, who allow to have a small gap in the beam pipe.
A small gap in a vacuum chamber can be a source of high intensity wake fields, which may cause electric breakdowns.
Usually a small gap in a beam pipe couples the bunch field with an outside “cavity”.

25. A small gap and a trapped mode between connecting flanges
We suggested that the gasket (RF gap ring), which is placed in the joint between the vacuum valve and the vacuum chamber, could have dimensions that are incorrect thereby producing a very small gap. We suspected that the gap size could be of order 100 microns.
The positron beam through this size of a small gap excites a cavity formed by the flange sides and the gaskets. Maximum electric field is in the gap and reach breakdown limits.
When we opened the vacuum chamber we found traces of breakdowns.
We cured this effect with a better design of the gap ring
Wake fields due to a small 0.2 mm gap in a flange connection
Electric field in the gap
Breakdowns

26. RF seal design is very important!
Almost same happened with RF Seals in the High Energy Ring

27. RF seal temperature jumps and an outer cavity excitation
HER flanges and flexible omega seal. Red arrows show thermal radiation
Measured single bunch and multi bunch spectrum

28. Bunch-spacing resonance in the HER bellows with broken shielded fingers
HER current
Bellows temperature
Vacuum chamber temperature

29. A trapped mode when an RF seal is not installed well
(a gap ring with a cut)
Breakdowns traces

30. A quarter-wave resonance of a spoiler
Somebody decided to start with a bunch pattern by 4. This pattern contains a bunch spacing resonance, which has a frequency very close to a frequency of a trapped mode: 476/4*5=595 MHz. The LER current was relatively small. What happened? Next page.

31. How strong wake field effect in resonance could be: melting the TI foil at the current of only 500 mA.
Damage occurred during x4 running. No unusual power loss in by 2. Gouge is not caused by direct beam.

32. Resonance of the BPM button

33. HOM power from high currents
Bunch Spacing
Loss Factor
Current
Loss factor of only 0.1 V/pC “works” like a microwave
Every small irregularities of the vacuum chamber become very important

34. Propagating waves from a collimator
The shape was optimized in order to have a minimum longitudinal impedance, however the higher nonsymmetrical waves (dipole and quadrupole) have large impedance

35. HOM leaking from TSP heater connector
The power in the wake fields
was high enough to char beyond use the feed-through for the titanium sublimation pump (TSP).
antenna
HOM spectrum from
Spectrum analyzer

36. HOMs go through RF screen into the NEG chamber, heat it to high temperature and destroy the pumping.
RF spectrum
antenna
RF screens

37. Fight the propagating waves using shielded absorbers
Tiles braised to water cooled copper support columns
Ready for install

38. Recommendations (nothing new)
Electron and positron bunches generate electromagnetic fields at any discontinuity of the vacuum chamber
These fields can travel long distance and penetrate inside bellows, pumps and vacuum valves.
Vacuum chamber must be very smooth.
HOM absorbers must be installed in every region that has unavoidable discontinuity of the vacuum chamber
Maximum attention to the RF seal designs
Design of a BPM button with a cooling possibility
No open (to the beam) ceramic or ferrite tiles
Increase the bunch length as possible