Transcript Pergine-Presentation

Collective Effects and Beam
Measurements in Particle Accelerators
beam diagnostics and fundamental limitations to
the performance of high-intensity accelerators
http://ab-abp-rlc.web.cern.ch/ab-abp-rlc/
See also slides on Measurements, ideas, curiosities
F. Ruggiero
CERN
Univ. “La Sapienza”, Rome, 20–24 March 2006
Outline
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Wakefields and beam coupling impedances
Beam Instabilities in circular accelerators
Landau damping, and stability diagrams
Instability cures
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low-impedance machine design and impedance
localization
Landau octupoles and tune spread
feedback and chromaticity control
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synchrotron radiation
Intra-Beam Scattering
Schottky signals and beam echoes
Incoherent effects and beam diagnostics
F. Ruggiero
CERN
Collective Effects and Beam Measurements
TRISIM3D simulation of a LEP bunch
E = 22 GeV, Qs = 0.092, I = 0.3 mA
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3-dimensional computer simulation of single bunch
collective effects in particle accelerators
top and side view of the particle distribution turn by
turn, over 60 turns
Courtesy Andreas Wagner
http://awagner.home.cern.ch/awagner/
F. Ruggiero
CERN
Collective Effects and Beam Measurements
Particle beams are sets of oscillators
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Under the focusing action of the magnetic
quadrupoles, charged particles in an
accelerator perform transverse (betatron)
oscillations around the closed orbit.
For bunched beams, they also perform
longitudinal (synchrotron) oscillations
around the synchronous positions defined
by the RF-system.
Therefore a particle beam can be considered
as a collection of oscillators, characterized
by their betatron and synchrotron
frequencies or by the corresponding tunes,
defined as the ratio of these frequencies to
the ring revolution frequency.
In the LHC there will be 2835 proton
bunches per beam, each bunch with a
nominal population of about 1011 particles.
F. Ruggiero
CERN
Collective Effects and Beam Measurements
Impedances and tune shifts
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The single and multi-bunch coherent
oscillation frequencies are modified by the
electromagnetic field created by the beam
interacting with its enclosure: the latter can
be characterized by longitudinal and
transverse complex impedances, representing
the decelerating or deflecting voltage for a
monochromatic unit beam current or
transverse dipole current, respectively.
A rule of thumb is that the coherent tune shift
of a given beam oscillation mode is
proportional to the convolution integral of the
impedance times the mode spectrum, i.e. to
the so-called effective impedance.
For longitudinal oscillation modes, the
relevant convolution integral involves the
longitudinal impedance divided by the
frequency or Z/n.
F. Ruggiero
CERN
Collective Effects and Beam Measurements
Tune shifts, instabilities, Landau damping
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The real part of the longitudinal impedance
is responsible for parasitic power losses.
A tune shift with positive imaginary part
corresponds to a beam instability and, for
transverse oscillation modes, is associated
with a negative real effective impedance.
Another rule of thumb is that an instability
with a given growth rate is “Landau
damped” if there is a comparable (or larger)
incoherent tune spread in the beam.
Such a tune spread is associated with
lattice non-linearities and can be controlled
by octupolar magnets. Also fluctuations in
the bunch population can give rise to a
bunch-to-bunch tune spread and provide
some Landau damping.
F. Ruggiero
CERN
Collective Effects and Beam Measurements
Catalog of Coherent Effects
SINGLE-BUNCH
LOSSES
TUNE SHIFTS
(low-Q structures)
MULTI-BUNCH
(high-Q structures)
broad-band impedance
(resistive wall, random slots)
photo-electrons
broad-band impedance
narrow-band impedance
(HOM's of RF-cavities, …)
secondary electrons
depend on multi-bunch mode
loss of Landau damping
INSTABILITIES
head-tail modes
mode coupling
microwave
F. Ruggiero
CERN
dipole modes (resistive wall, …)
multi-bunch head-tail
multi-bunch mode coupling
electron-cloud instability
Collective Effects and Beam Measurements
Trapped modes for tertiary LHC
collimator chambers (A. Grudiev, 2006)
F. Ruggiero
CERN
Collective Effects and Beam Measurements
Resistive wall impedance
Z T w  
w a  w 
 w  
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R
2c
3
1  j 
2
transverse resistive wall
impedance
for a thick circular beam pipe of
radius a and resistivity .
skin depth.
ow
R: ring radius
At very low frequency the transverse resistive wall
impedance increases as w-1/2 and behaves as a
narrow-band impedance (high-Q or long memory
resonator)
It can drive multi-bunch instabilities: for a betatron
tune Qb above the integer, the spectrum of the most
unstable transverse oscillation mode contains the
line (1-[Qb]) frev. At the LHC this is about 8 kHz.
F. Ruggiero
CERN
Collective Effects and Beam Measurements
LHC graphite collimators
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One may think that the classical “thick-wall” formula
applies also for 2 cm thick graphite collimatos about
2 mm away from the beam
In fact it is not  The resistive impedance is ~ 2
orders of magnitude lower at ~ 8 kHz!
New regime: d  a ,   d
Usual regime: d ,   a
beam
d
beam
d
a
 aeff  a
a
 aeff  a when δ  d
F. Ruggiero
CERN
Induced
current
Collective Effects and Beam Measurements
Induced
current
Resistive wall collimator impedance
comparison between Vos and Burov-Lebedev
F. Ruggiero
CERN
Collective Effects and Beam Measurements
LHC stability diagram (maximum octupoles) and collective
tune shift for the most unstable coupled-bunch mode
(E. Metral, 2004)
Im Q 
0.00015
All the machine
with Cu coated (5 μm)
collimators
0.000125
0.0001
All the machine
0.000075
0.00005
Without collimators
(TCDQ+RW+BB)
0.000025
Re  Q 
0.0008
F. Ruggiero
0.0006
CERN
0.0004
0.0002
Collective Effects and Beam Measurements
Transverse Mode-Coupling Instability
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A broad-band impedance induces different tune shifts for the
various head-tail oscillation modes of a single bunch and may
cause a mode-coupling instability above a threshold intensity.
The same formula is obtained (within a factor ~2) from 5
seemingly diverse formalisms for a Broad-Band resonator
impedance (Q = 1) :
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Coasting-beam approach with peak values (e.g. Laclare 1985)
Fast blow-up (Ruth&Wang1981)
Beam break-up (Brandt&Gareyte1988, for 0 chromaticity)
Post head-tail (Kernel&al.2000)
Transverse Mode Coupling with 2 modes in the “long-bunch” regime
(Zotter 1982, for 0 chromaticity)
chromatic frequency shift
f

y
th
N b    l  1  BB
fr

slippage factor
F. Ruggiero
long emittance
CERN




Proposed by E. Metral
Cross-checks with MOSES
(Chin 1984) and HEADTAIL
(Rumolo&Zimmermann 2002)
impedance peak frequency
Collective Effects and Beam Measurements
TRANSVERSE – HIGH INTENSITY EFFECTS IN THE SPS
LOW CHROMATICITY MEASUREMENTS (Courtesy E. Metral)
SPS MEASUREMENTS
 Travelling-wave pattern along the bunch
Head
Tail
1st trace (in red) = turn 2
F. Ruggiero
 y  0.14
CERN
Last trace = turn 150
Every turn shown
Collective Effects and Beam Measurements
TRANSVERSE – HIGH INTENSITY EFFECTS IN THE SPS
HIGH CHROMATICITY (Courtesy E. Metral)
SPS MEASUREMENTS
1st trace (in red) = turn 2
F. Ruggiero
CERN
Last trace = turn 150
 y  2.04
Every turn shown
Collective Effects and Beam Measurements
TRANSVERSE – HIGH INTENSITY EFFECTS IN THE SPS
LOW CHROMATICITY SIMULATIONS (Courtesy E. Metral)
HEADTAIL SIMULATIONS
Nb  1.2  1011 p/b
f r  1 GHz
Ry  20 M/m
 t  0.7 ns
Q 1
Head
Tail
 Travelling-wave pattern
along the bunch
1st trace = turn 1
F. Ruggiero
CERN
Last trace = turn 50
 y  0.14
Every turn shown
Collective Effects and Beam Measurements
TRANSVERSE – HIGH INTENSITY EFFECTS IN THE PS
Fast vertical single-bunch instability with protons
at the PS near transition in 2000
E  6 GeV Nb  4 1012 p/b
, R, V signals
Tail unstable
Head stable
~ 700 MHz
Time (10 ns/div)
 Instability suppressed by increasing the longitudinal
emittance
F. Ruggiero
CERN
Collective Effects and Beam Measurements
TRANSVERSE – HIGH INTENSITY EFFECTS IN THE PS
SIMULATION OF LONG BUNCHES (Courtesy E. Metral)
HEADTAIL SIMULATIONS
f r  1 GHz
Ry  3 M/m
Nb  4  1012 p/b
Q 1
 t  7.5 ns
Head
Tail
1st trace = turn 1
F. Ruggiero
CERN
Last trace = turn 130
Every turn shown
Collective Effects and Beam Measurements
Measurements vs simulations
MEASUREMENT
Head
Tail
HEADTAIL SIMULATION
SPS Head
Tail
PS
, R, V signals
Head
Tail
Head
~ 700 MHz
Time (10 ns/div)
F. Ruggiero
CERN
Collective Effects and Beam Measurements
Tail
Synchrotron Radiation
Uo  C E 4 / 
energy loss per turn
2
4p
rA
Z
m
17
C 

0
.
778

10
3
3 mAc 2 
A4 GeV 3
 bending radius
E beam energy
rA classical radius
Z atomic number C ring circumference
A atomic weight J partition number
mA particle mass trad emittance damping time
4


3
m
c
1
C
8
322
h
C
A
t rad J  2 A3 2 2

2e c rA Z B E 2p E/TeV B/T 2 2p Z 4
2 3
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Radiation damping times decrease inversely with the beam
energy and the square of the magnetic field
For the LHC proton beam: E =7 TeV, B =8.3 T, C/2p~1.5,
A =Z =1, Jx~Jy~1, Jz~2  txrad~ tyrad ~26 h, tzrad ~13 h
F. Ruggiero
CERN
Collective Effects and Beam Measurements
Synchrotron Radiation
Photon Fluctuations
3 3 c
 ph  
2 
N ph 
critical photon energy
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E
 ph
photons emitted in
one damping time
   1 / N ph
equilibrium rms relative
momentum spread
The critical energy of synchrotron radiation photons in the LHC
at 7 TeV is 45 eV. Such photons can produce photo-electrons.
In electron storage rings, quantum fluctuations due to discrete
photon emissions excite betatron and synchrotron oscillations.
Radiation damping leads to a quick relaxation (say within ms)
towards equilibrium values of the beam emittances.
For hadron rings such as the LHC these equilibrium values are
much smaller than the initial beam emittances and the damping
times are very long (10-20 h). Therefore the beam emittances
vary during a physics store and depend on additional effects
such as Intra-Beam Scattering or Beam-Beam diffusion.
F. Ruggiero
CERN
Collective Effects and Beam Measurements
Negative mass instability
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In a circular accelerator operated above transition energy,
the longitudinal particle dynamics is characterized by a
“negative mass”
Indeed the modest relativistic speed increase of an offmomentum particle is insufficient to overcome the dispersive
orbit lengthening
dz  1 1  Δp
  2  2 
ds  γ γt  p
  γt 
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γt  Qx
a positive increase p of the particle momentum
corresponds to a longitudinal slow-down dz/ds < 0
In a coasting uniform beam, a local increase of charge density
causes a space charge acceleration for leading particles and a
deceleration for trailing particles
Above transition this leads to a dynamic self-bunching effect,
known as negative mass instability
F. Ruggiero
CERN
Collective Effects and Beam Measurements
Intra-Beam Scattering and “negative
temperature” thermodynamic instability
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Above transition energy, the longitudinal particle dynamics is
characterized by a “negative mass”
A beam with a near-Boltzmann distribution is therefore
characterized also by a “negative longitudinal temperature”
A negative temperature is very-very hot…
(similar to optically pumped lasers with population inversion)
Intra-Beam Scattering couples the hot longitudinal negativetemperature reservoir to the positive-temperature reservoir
corresponding to the transverse betatron motion
Above transition, this situation leads to a thermodynamic
instability: thermal energy is continuously transferred from
the longitudinal to the transverse degrees of freedom
The longitudinal temperature decreases and the
corresponding reservoir becomes hotter… dS = dQ/T
As a result both the longitudinal and the transverse beam
entropies, i.e. the beam emittances, grow: no state of
thermodynamic equilibrium is possible above transition!
F. Ruggiero
CERN
Collective Effects and Beam Measurements
Intra-Beam Scattering
(simplified formulae by K. Bane, EPAC 2002)
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t zIBS
Hx
8 3 3 / 2 z δ3
1
 2
b av

2
rp cN b (log )
δ

t
t
IBS
x
IBS
z

 δ2 H x
For the LHC, the average
dispersion invariant is
<Hx>~0.03 m, bav~100 m, and
the Coulomb logarithm (log)~24
Note that the transverse IBS
growth rate is about
proportional to the particle
density in the 6D phase space
Nb/(2z
F. Ruggiero
CERN
IBS growth times for long
super-bunches in the LHC
as a function of the rms
momentum spread 
Collective Effects and Beam Measurements
Collective Effects
Summary
• Particle beams in an accelerators can be considered as
sets of oscillators. Coherent oscillation frequencies are
modified by the electromagnetic fields created by the
beam interacting with its enclosure, characterized by
complex coupling impedances.
• A broad-band impedance induces different tune shifts
for the various head-tail oscillation modes of a single
bunch and may cause a mode-coupling instability or a
loss of Landau damping, when a mode is shifted away
from the incoherent tune spread.
• Narrow-band impedances are associated with high-Q
structures. They may cause multi-bunch instabilities
and large power losses, unless properly damped.
• As a consequence of Intra-Beam Scattering, above
transition the beam emittances grow.
F. Ruggiero
CERN
Collective Effects and Beam Measurements