Intensity limitations from combined effects and/or (un)conventional impedances Giovanni Rumolo thanks to Oliver Boine-Frankenheim and Frank Zimmermann . CARE-HHH-APD Workshop (CERN,10.11.2004)

1
Intensity limitations from combined effects
and/or (un)conventional impedancesGiovanni
Rumolothanks to Oliver Boine-Frankenheim and
Frank Zimmermann .CARE-HHH-APD Workshop
(CERN,10.11.2004)

• Unconventional impedances
• Summary of the features of the electron cloud
  wake fields
• Detrimental combined effects e-cloudspace
  charge or e-cloudbeam-beam
• Effects of a conventional broad band impedance on
  an unconventionally shaped bunch.
• Head-tail phenomena in barrier buckets
• Centroid and envelope motion, tune shifts and
  envelope spectrum line shifts.
• Regular head-tail instability driven by negative
  Q and threshold for the strong head-tail
  instability
• Comparison with a parabolic bunch

2
Impedance sources in an accelerator ring

• Conventional impedances (longitudinal and
  transverse)
• Space charge
• Resistive wall (including the effect of small
  holes, surface roughness and finite wall
  thickness)
• Narrow- or broad-band resonators (modeling
  specific objects, like cavities, kickers,
  pick-ups, etc., or the whole accelerator
  environment)
• Unconventional impedances
• Synchrotron radiation (high energy machines,
  mainly longitudinal)
• Electron cooler (high intensity ion machines,
  mainly transverse)
• Electron cloud (high intensity hadron/positron
  machines, mainly transverse, highly
  unconventional)

3

• Conventional impedance and beam stability
• V.K. Neil and A.M. Sessler, Rev. Sci. Instrum.
  36, 429 (1965)
• V.G. Vaccaro, CERN-ISR-RF 65-35 (1966)
• ..... A. Ruggiero, R.L. Gluckstern, A.W. Chao, L.
  Palumbo, S.S. Kurennoy, A.V. Fedotov, J.L.
  Laclare, et al.
• B.W. Zotter and S.A. Kheifets Impedances and
  Wake Fields in High Energy Particle Accelerators
  World Scientific Singapore 1998
• ..... A. Al-khateeb, O. Boine-Frankenheim, F.
  Zimmermann, K. Oide, S. Petracca, et al.

4

• Unconventional impedances (I)
• On the Impedance Due to Synchrotron Radiation,
  by S. Heifets and A. Michailichenko, SLAC/AP-83
  (1990), refers to previous detailed works by R.Y.
  Ng, R. Warnock, P. Morton.
• A qualitative description of the impedance
  caused by SR gives the value of the threshold
  frequency and the maximum value of the impedance.
• Electron cooler impedances by A.V. Burov and
  V.G. Vaccaro in Proceedings of the Workshops on
  Beam Cooling and Related Topics (1993) and on the
  Crystalline Beams (1995)
• The e-cooler can induce a blow-wind transverse
  instability when the e-cooler is detuned.

5

• Unconventional impedances (II)
• Electron cloud
• ? Coasting beams electrons from residual gas
  ionization accumulate around the beam to some
  neutralization degree and can drive a two-stream
  instability.
• P. Zenkevich, D.G Koshkarev, E. Keil, B. Zotter,
  L.J. Laslett, A.M. Sessler, D. Möhl (1970-80)
• Transverse Electron-Ion Instability in Ion
  Storage Rings with High-Current, P. Zenkevich,
  Proc. of Workshop on Space Charge Dominated Beam
  Physics for HIF (1999) proposes first the concept
  of two-stream transverse impedance
• ? Bunched beams an electron cloud due to
  multiplication of primary electrons through
  secondary emission causes head-tail coupling
  within one bunch.
• K. Ohmi, F. Zimmermann, G. Rumolo, E.
  Perevedentsev, M. Blaskiewicz (2000-04)
• Wake fields and broad band model, generalized
  two-frequency impedance model for TMCI threshold
  calculation.

6
(No Transcript)
7
(No Transcript)
8
(No Transcript)
9

• Two remarkable features render the e-cloud wake
  field different from a conventional wake field
• Averaged wake functions and wake functions on
  axis are differently shaped and differ in
  amplitude by about 2 orders of magnitude
• The shape of the wake depends on the location of
  the displaced source slice
• This needs to be taken into account in the TMCI
  theory ? see Head-tail Instability Caused by
  Electron Cloud, E. Perevedentsev, in Proc. of
  ECLOUD02, CERN, Geneva Switzerland (2002)

10
Dependence on the electron distribution
Distributions with vertical stripes (one or two)
can exist in dipoles. Different initial
distributions lead to different resulting wake
fields.
11

1. The frequency of the wake decreases as the
   separation between the two stripes increases.
2. The vertical wake is weakened by the two stripes.
3. The horizontal wake, which is anyway much weaker
   due to the dipole field, is not much affected.

12
Dependence on the transverse proton distribution
Average wake field
Wake field on axis

• For a uniform transverse distribution the
  oscillation of the wake field is not damped,
  which corresponds to a broad-band oscillator with
  an infinite quality factor Q.
• The frequency of the wake from a Gaussian proton
  distribution is higher.

13
Dependence on the boundary conditions
Average wake field
Wake field on axis

• For a pipe chamber 10 times larger than the beam
  rms-sizes, the boundary conditions do not seem to
  affect significantly shape or frequency of the
  wake fields
• The electron space charge has been found only to
  slightly lower the frequency of the wakes.

14
(No Transcript)
15
(No Transcript)
16
Example of calculation of a double frequency
impedance
17
(No Transcript)
18
Summary of part I (features of the electron cloud
wake field)

• The dipole wake field of an electron cloud
  depends on the transverse coordinates (x,y)
• Differently located displacements along a bunch
  create differently shaped wake fields
• The wake field depends
• Strongly
• On the initial electron distribution
• On the bunch particle transverse distribution
• Weakly
• On the boundary conditions for a wide pipe
• On the electron space charge for low degrees of
  neutralization

19
Summary of part I (continues) and possible future
work

• A description in terms of double frequency
  impedance Z(w,w) is necessary for a correct TMC
  analysis
• Numerical tool to handle the calculation of
  Z(w,w) has been developed.
• To be yet investigated
• Dependence of the wake on the longitudinal shape
  of the bunch
• The electron cloud wake field for long bunches
  might strongly depend on the trailing edge
  electron production and multiplication

20

• Detrimental combined effects
• electron cloud space charge
• electron cloud beam-beam
• G. Rumolo and F. Zimmermann Electron cloud
  instability with space charge or beam-beam in
  Proc. of the Two-stream Instabilities Workshop,
  KEK, Tsukuba, Japan (2001)
• K. Ohmi and A. Chao Combined Phenomena of
  Beam-Beam and Beam-Electron Cloud Effects in
  Circular ee- Colliders, in Proc. of ECLOUD02,
  CERN, Geneva, Switzerland (2002)

21
(No Transcript)
22
3-4 particle models to explain the combined
effects of electron cloud and space charge or
beam-beam
23
Head-tail phenomena in barrier buckets Background

• Motivation and previous work
• Use of flattened bunches (e.g., in double or
  multi-harmonic rf buckets) against space charge
  or coupled bunch instability, or for luminosity
  increase head-tail properties closer to a
  barrier bucket
• T. Takayama, ICFA-HB2004 (more results presented
  here) first successful experiments of
  acceleration of rf bunches with induction
  cavities (confinement ?).
• H. Damerau has suggested the use of long and flat
  bunches (longer than nominal bunches (x 10) but
  not as long as super-bunches) for LHC luminosity
  upgrade.
• Head-tail instability for a super-bunch by Y.
  Shimosaki from KEK (ICFA-HB2004 )

24
Head-tail phenomena in barrier buckets Model

• Model
• We consider a square wave form as bunch shape and
  the bunch particles get instantly elastically
  reflected at the walls of the bucket (ideal case
  of infinite electric field at the bucket ends ?
  barrier bucket)
• The bunch feels the action of a broad-band
  impedance
• Simulation work carried out with HEADTAIL
• The bunch is subdivided into slices, and each
  slice feels the sum of the wakes of all preceding
  slices.
• Dipole and quadrupole components of the wake
  fields weighed with the Yokoya coefficients are
  used to study the effect of a flat chamber.
• Centroid and envelope oscillations are analyzed
  varying the bunch intensity and chromaticity.

25
Simulation parameters (SPS)
Protons per bunch 0.1 to 2 x 1011
Proton energy 26 GeV
Bunch length 2 m
Momentum spread 2 x 10-3
Momentum compaction factor 1.92 x 10-3
Transverse beam sizes (sx,y) 1.2 mm
Tunes (Qx,y) 26.185/26.13
Chromaticities (xx,y) 0 to -1
Shunt impedance 20 MW/m
Resonator frequency 1.3 GHz
Quality factor 1
Scan for tune shift with current
?
Long. emittance 0.8 eVs
Scan for growth rates
?
Broad-band resonator
26
Threshold for strong head-tail instability
Rectangular bunch in a barrier bucket
Gaussian bunch in a sinusoidal bucket

• Bunches with the same longitudinal emittance (0.8
  eVs)
• A regular Gaussian bunch in a sinusoidal bucket
  has a clear threshold above which TMC occurs
• A bunch in a barrier bucket exhibits a slow
  growth (threshold ?), but no violent instability
  sets in

27
Coherent tune shift as a function of the bunch
current (I)
We look at the tune shift through Fourier
analysis of the transverse motion of a
(transversely) kicked bunch. We analyze the bunch
centroid motion....
Horizontal tune
Vertical tune
Proton number is scanned from 0.1 to 2 x 1011,
chamber is round
28
Coherent tune shift as a function of the bunch
current (II)
We look at the tune shift through Fourier
analysis of the transverse motion of a
(transversely) kicked bunch. .... and the
spectrum of envelope oscillation
Vertical modes
Horizontal modes
Proton number is scanned from 0.1 to 2 x 1011,
chamber is round
29
Coherent tune shift as a function of the bunch
current (III)
We look at the tune shift through Fourier
analysis of the transverse motion of a
(transversely) kicked bunch. We analyze the bunch
centroid motion....
Horizontal tune
Vertical tune
Proton number is scanned from 0.1 to 2 x 1011,
chamber is flat
30
Coherent tune shift as a function of the bunch
current (IV)
We look at the tune shift through Fourier
analysis of the transverse motion of a
(transversely) kicked bunch. .... and the
spectrum of envelope oscillation
Horizontal modes
Vertical modes
Proton number is scanned from 0.1 to 2 x 1011,
chamber is flat
31
Coherent tune shift as a function of the bunch
current (V)

• Parametric dependence
• on the shunt impedance ? the slope doubles when
  doubling RS
• on the bunch length ? the slope halves when
  doubling the bunch length

32
Coherent tune shift as a function of the bunch
current (VI)

• Parametric dependence (continues)
• on the chamber shape ? for flat chamber the tune
  shift in x vanishes at all currents, the tune
  shift in y is the same as in the case of round
  chamber.

33
Coherent tune shift as a function of the bunch
current (VII)
Dependence on the shunt impedance ? the slope of
the secondary line doubles when doubling RS, but
the main line stays unchanged
34
Coherent tune shift as a function of the bunch
current (VIII)

• Dependence on the chamber shape
• The main line, which does not move for a round
  chamber, shifts toward higher tunes in x and
  toward lower tunes in y
• The slope of the secondary line does not change
  with the chamber shape.

35
Coherent tune shift as a function of the bunch
current (IX)
DQ
Comparison with a Gaussian matched bunch in a
sinusoidal bucket (theoretical prediction on the
right side, pink line) The slope is identical
for low currents, then the coherent main mode
shifts to higher order modes for a bunch in
sinusoidal bucket.
Current
36
Growth times for a head-tail instability in low
current (I)

• Rise times of head-tail instability for negative
  chromaticities
• The rise times are inversely proportional to the
  shunt impedance
• For flat chamber, vertical rise times are almost
  unchanged, whereas horizontal rise times are
  about a factor 2 larger.

37
Growth times for a head-tail instability in low
current (II)
t (s)
x Q/Q

• Comparison with a Gaussian matched bunch in a
  sinusoidal bucket (theoretical prediction for
  round chamber, plot on the right side)
• Gaussian bunch and barrier bucket have similar
  growth times
• For flat chamber horizontal rise times are about
  the half of the vertical rise times.

38
Growth times for a head-tail instability in low
current (III)
Dependence of the rise times on the bunch
length Doubling the bunch length, the rise times
of the instability become about double, too.
39
Summary of part II (Tune line shifts in a barrier
bucket with a BB-impedance)

• The coherent tune shift DQ of a bunch in a
  barrier bucket as a function of the bunch current
  depends on
• Shunt impedance (proportional)
• Bunch length (inversely proportional) and maybe
  momentum spread (proportional ?)
• Chamber shape (only in x)
• The DQ follows that of a usual bunch in a
  sinusoidal bucket and low current with the same
  longitudinal emittance (theoretical line)
• Coherent envelope modes depend on the chamber
  shape
• Round chamber has two modes both in x and y, one
  current dependent and one current independent.
• Flat chamber has one mode in x with a positive
  shift with increasing current, and two modes in
  y, both with a negative shift with current.

40
Summary of part II (Instabilities of barrier
buckets with a BB-impedance)

• The threshold for strong head-tail instability is
  not found for bunches in a barrier bucket, but
  there is rather a regime of slow growth at high
  currents.
• Regular head-tail instability driven by negative
  Q (above transition) exhibits similar features
  as for bunches in sinusoidal buckets.
• Growth rates are proportional to the shunt
  impedance
• The quickest instability occurs when wxwr
• In a flat chamber growth times in the x direction
  are about double of the growth times in the y
  direction
• Longer bunches slow down the instability (because
  of the decay of the wake along the bunch or
  because of the lower synchrotron frequencies ?)
• Analytical model (maybe few particles model or
  kinetic model based on Vlasov equation) needed.