“tune”
description
Transcript of “tune”
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“tune”Frank Zimmermann
DITANET School, Royal Holloway,2 April 2009
02/04/09 DITANET School
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outlineintroduction
tune, coherent & incoherent tune, detectors
integer betatron tunefractional betatron tune
precision measurement, tune tracking, multiple bunches
modifications of tune signaldamping, filamentation, chromaticity, linear coupling
some “complications”colliding beams, space charge, measuring incoherent tune
applicationstune shift with amplitude,high-order resonances,tune
scans, b function measurement, nonlinear field errors
synchrotron tunedisplay of complex tune signals
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introduction
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schematic of betatron oscillation around storage ringtune Qx,y= number of (x,y) oscillations per turn
C s
dsCQ
)(2
1
2
)(
focusing elements:quadrupole magnets
quadrupole magnet(many)
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schematic of “longitudinal oscillation” around storage ring
tune Qs = number of synchrotron oscillations per turn
synchrotron tune Qs << 1, typically Qs ~ 0.01-0.0001betatron tune Qx,y > 1, typically Qx,y ~ 2 - 70
focusing elements:energy-dependent path lengthrf cavities
RF cavity
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incoherent and coherent tune K.H. Schindl
incoherent betatron motion of a particleinside a static beam with its center of massat rest
amplitude and phase are distributed at random over all particles
coherent motion of the whole beam after a transverse kick
the source of the direct space charge is now moving, individual particles still continue incoherent motion around the common coherent trajectory
at low beam intensity these two tunes should be about the same
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button pick ups
button electrode for use between theundulators of the TTF II SASE FEL(courtesy D. Noelle and M. Wendt, 2003)
unterminated transmission line
transmission line terminated (rhs) to a matched impedance
strip line pick ups
the LEUTL at Argonne shorted S-band quarter-wave four-plate stripline BPM (courtesy R.M. Lill, 2003)
reference: “Cavity BPMs”, R. Lorentz (BIW, Stanford, 1998)
TM010, “common mode” ( I)TM110, dipole mode of interest
cavity BPMs
detectors to measure the coherent beam oscillations
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Direct Diode Detection Base-Band Q (3D-BBQ) Measurement in CERN Accelerators - Principle
Apart from detectors, the filter is most important element of the system. It attenuates revolution frequency and its harmonics, as well as low frequencies.
Marek Gasior
pick up
diodedetectors
FE electronics:amplifiers & filter
detector FE electronics
SPS installation
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integer betatron tune
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02/04/09 DITANET School
integer part of betatron tune
• first turn injection oscillation
• or difference orbit after exciting a single steering corrector
count number of oscillations(directly or via FFT) integer value of tune Q
)(cos sAx
C s
dsCQ
)(2
1
2
)(
oscillation
more intricate method: use multi-turn BPM data to measure f at each BPM; then find Df between BPMs
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integer tunes 64 and 59 equal to their design values!(vertical FFT has second peak!?) – basic check of optics
J. Wenninger
example - checking the integer tuneLHC beam commissioning 12 September 2008
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fractional betatron tune
• precision measurement• tune tracking• multiple bunches
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rms vertical beam size of the electron beam extracted from the SLC damping ring as a function of the vertical betatron tune, under unusually poor vacuum conditions.
all nonlinear and high-intensity effects are very sensitive to the fractional tune - best performance requires optimum tune!
fractional part of the tune – why is it important?one example
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two categories:
• precision tune measurements
• tune tracking to monitor & control fast changes e.g. during acceleration
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FFT (Fast Fourier Transform)(1) excite transverse beam motion + detect transverse position on a pick up over N turns(2) compute frequency spectrum of signal; identify
betatron tunes as highest peaks
xrev
x fN
fnQ
step 1/N between points
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N
j
inQj
jeQnx1
2)()(
FFT signal = expansion coefficient
Q error of FFT: N
Q2
1
due to discreteness of steps
50010 3 NQ
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checking the fractional tuneLHC beam commissioning 10 September 2008
signal decay(due to de-bunching)
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multi-turn orbit measurement for the motion of a single bunch in a 3-bunch train at LEP-1
BPM in a dispersive arc region (wheretransverse positionvaries with beam momentum)
BPM in a straight sectionwithout dispersion
signal decay (due to fast head-tail damping)
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FFT power spectra for the two previous measurements
BPM in a dispersive arc region
BPM in a straight sectionwithout dispersion
b tune
synchrotron tune
b tune
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about 1000 turns are required for adequate tune measurement with FFT, but
• filamentation, damping,… (see later)spurious results?!
further improvement in resolution, e.g. by
interpolating the shape of the Fourier spectrum around main peak
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)(sin
)(sin)(
int
int
jF
jFj QQN
QQNQ
NQQ
NQ
N
kQ
kk
kF
cos)()(
sin)(arctan
1
1
1int
2N
constQ
:1N )()(
)(arctan
1
1
1int
kk
kF QQ
Q
NN
kQ
assumption: shape = pure sinusoidal oscillation
error:
uses Q at peak and highest neighbor
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further improvement by interpolated FFT with data windowing
N
n
inQj
jennxN
Q1
2)()(1
)(
N
nAn lll
sin)(
2)()(
)()1(1
1
1int
l
Ql
NN
kQ
kk
kF
2 lN
constQ
filter
e.g., Hanning filter of order l:
resolution:
however, with noise 2N
constQ as for the simple interpolation
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example: refined FFTs (M. Giovannozzi et al.)
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j
n
j
nN nnQ
nnQ
nnQ
nnQQP
)(2sin
)(2sin
)(2cos
)(2cos
2
1)(
02
2
0
02
2
0
2
N
iinn x
Nxd
1
1
N
nndN 1
22
1
1
n
n
Qn
QnQn
4cos
4sin4tan 0
“Lomb normalized periodogram”
another approach to obtain higher accuracy than FFT
no constraint on # data point or on time interval between points (replace Qn by wtn)
the constant n0 is computed to eliminate the phase of the originalharmonic; since the phase dependence is removed, Lomb’s methodis more accurate than the FFT
A.-S. Muller
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BPM in a dispersive arc region
BPM in a straight sectionwithout dispersion
Lomb normalized periodogram for previous measurements
(A.-S. Muller)
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comparison Lomb-FFT for CERN PS simulation with two spectral lines [A.-S.Muller, ’04]]
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still anactive area of research…
most of the above“improvements”rely onharmonic motion
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swept frequency excitation
t sin
tAxb sin
excitation
measure response
amplitude phasevary w in steps
180o phase jump
“beam transferfunction”
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beam transfer functiontransverse impedanceradiation damping
at betatron tune: A zero slopeq maximum slope!
phase can be monitored by phase-locked loop
if beam is excited by VCO lock-in amplifier
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phase locked loop (for continuous tune control)
“locked”in frequency
VCO frequency = betatron frequency“lock-in amplifier”
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xxxx xxp /'
xx /
xxxx xxp /'
xx /
frev 2frev 3frev
2frev
frev/2
2frev/2
fx frev-fx
single bunch
2 bunches phase space frequency spectrum
revolutionharmonics
p mode
s mode
0
0at high currentfractional tune canbe different for s and p modes!
bumches in phase spacefor p mode
multiple bunches
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n n
ititi xx eTnTtenTttx )2/()()(
bunch 1 bunch 2 f=0: s modef=p: p mode
nb bunches nb multibunch modes
measuring spectrum from 0 to nb frev/2 suffices!
at low current
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the tune, or not the tune, that is the question
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modifications of tune signal
• damping• filamentation• chromaticity • linear coupling
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W. Fischer and F. Schmidt, CERN SL/Note 93-64 (AP)
fast decay, ripple& “noise”
many lines
tune measurements for proton beams in the CERN-SPS
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W. Fischer and F. Schmidt, CERN SL/Note 93-64 (AP)
tune measurements for proton beams in the CERN-SPS
phase-space plot suggests filamentation
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coherent oscillations, damping & filamentation
response to a kick: coherent damping
2
' 22 xxxJ
...111
HTSR
)(Re1
zeffbHT
ZN
2)(cos0
tJexxt
exp. decay
amplitude-dependentoscillation frequency
head-tail damping
synchrotron-radiation damping
bunch population
chromaticity
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1/t
1/(100 turns)
1/tSR
Ib
Q’=2.7
Q’=14
LEP 45.63 GeV,damping rate1/ t vs. Ibunch
for different chromaticities[A.-S. Muller]
xrev
SR
JE
fU
0
0
2
11
horizontal damping partition number
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LEP: tune change during dampingQ
turns[A.-S. Muller]
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LEP: detuning with amplitude from single kickQ
2 J[A.-S. Muller]
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another response to a kick: filamentation
21
1~ :/1
:1 21
tt
et tc
)(cos1
2
22
122
ttet
Zx t
tZ
x
R. Meller et al.,
SSC-N -360, 1987
both different from e-t/t
Z: kick in sa =(2mw0)2
Q=Q0-ma2
amplitude in s
ex.:
turns1000after
1 :10 4
t
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amplitude decoherence factor vs. turn number
5s kick
1/5 s kick
oscillationamplitude
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chromaticity
pp
pp
'
QQ /'
normalized unnormalized
relation
chromaticity describes the change of focusing and tune with particle energy
usually 2 or more families of sextupoles are used to compensateand control the chromatcity
small chromaticity is desired to minimize tune spread and amountof synchrobetatron coupling (maximize dynamic aperture)
but large positive chromaticity is often employed to dampinstabilities (ESRF, Tevatron, SPS,…)
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response to a kick: decoherence due to chromaticity
d>0d<0
(Q’>0)
t=0 t=Ts/2 t=Ts
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)(cos
2/sin'2 2
2
22
tteZxt
Q
Q
x
ss
R. Meller et al.,SSC-N -360, 1987
x
time
Ts
large x
small x
chromaticity
FFT over small number of turns → widening of tune peakFFT over several synchrotron periods
→ synchotron sidebands around betatron tune
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rfrf
yxC
yxyx
ff
Q
pp
/
1
/'
,
2
,,
another method to determine total chromaticitytune shift as a function of rf frequency
horizontal tune vs changein rf frequency measuredat LEP;the dashed line showsthe linear chromaticityas determined by measurements at+/- 50 kHz
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measuring the natural chromaticity (Q’ w/o sextupoles)from tune shift vs. dipole field
B
B
p
p
BB
QQ yxnat
yx /' ,
,
B
B
rf
rf
2
1
for e-, the orbit is unchanged(determined by rf!)
for p, simultaneouschange in rf frequencyrequired to keep thesame orbit:
electronring
natural chromaticitymeasured at PEP-II HER
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xyQd
yd
yxQd
xd
202
2
202
2
2
y-x v,
2
yxu
0
0
202
2
202
2
vQd
vd
uQd
ud
20
2
20
2
v
u
222 vu QQ
linear coupling: model of 2 coupled oscillators
k: coupling strength
normal-mode coordinates:
decoupled equations new eigen-frequencies
frequency split:measure of strength of coupling
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closest tune approach
near the difference resonance 0qQQ yx
22_, 2
1 qQQqQQQ yxyxIII
the tunes of the two eigenmodes, in the vertical plane, are
uncoupled tunes
tunes can approach each otheronly up to distance |k_|
correction strategy;use two skew quadrupoles(ideally with (D fx-fy)~p/2) tominimize |k_|, namelythe distance of closest tune approach
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closest tune approach in the PEP-II HER before final correction; shown are the measured fractional tunes as a function of the horizontal tune knob; the minimum tune distance is equal to the driving term |k_| of the difference resonance
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another way to measure |k_|: kick response over many turns
envelopes ofhorizontal andvertical oscillationsexhibit beating
plane of kick
orthogonalplane
beating period
2max
2min
x
xS
brevTf
S1
define
one can show that ! exampleATF
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coupling transfer function
excite beam in x detect coherent y motion
used for continuous monitoring of coupling at the CERN ISR in the 1970s;is considered for LHC coupling control
r
i
ir
c
c
ccA
arctg
22
amplitude and phaseof vertical response;complex value of k_
ISRcouplingtransferfunction
J.-P. Koutchouk,1980
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RHIC-Tevatron-LHC2005
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RHIC-Tevatron-LHC2005
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Courtesy B. Goddard
many other complications and challenges, for example:• space charge• ionized gas molecules• electron cloud• beam-beam interaction• radiation damping• .. etc etc … all these phenomena affect beam response to excitation
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some “complications”:
colliding beam tune spectraspace charge tune spreadmeasuring the incoherent tune
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transverse tune measurement (swept-frequency excitation) with 2 colliding bunches at TRISTAN. Vertical axis: 10 dB/div., horizontal axis: 1 kHz/div [K. Hirata, T. Ieiri]
Hysteresis:2 fixed pointsdue to nonlinearbeam-beam force
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p mode s mode
continuumincoherent spectrum
equal intensity intensity ratio 0.55
simulated tune spectrum for two colliding beams
p mode not “Landau damped” p mode “Landau damped”M.P. Zorzano & F.Z., PRST-AB 3, 044401 (2000)W. Herr, M.P. Zorzano, F. Jones, PRST-AB 4, 054402 (2002)
if the coherent tune lies outside the continuum “Landau damping” is lost and the beam can be unstable; prediction for the LHC
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evidence for coherent beam-beam modes@RHIC
RHIC BTF amplitude response at 250 GeV
without collisions with pp collisions
26. February 2009, Michiko Minty
H & V signals for beam 1H & V signals for beam 1
H & V signals for beam 2 H & V signals for beam 2
p mode? s mode?
continuum?
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space charge
example for space-chargelimited synchrotron:betatron tune diagram andareas covered by direct tunespread at injection, intermediate energy,and extraction, for the CERN Proton Synchrotron Booster.During acceleration,acceleration gets weakerand the “necktie” areashrinks, enabling the externalmachine tunes to move the“necktie” to a region clearof betatron resonances(up to 4th order)
K.H. Schindl
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how to measure the incoherent tune shift/spread?
K.H. Schindl
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Schottky monitordirectly measures “incoherent” tune(oscillation frequency of individual particles)
w/o centroid motion
incoherent signal fN
emittance #particles
frequency bandwidth
“Schottky monitors” stochastic cooling
FNALslotted
waveguide1.7-GHzSchottky
pickupdesign
fabricatedFNAL
Schottkypickupdesign
R. Pasquinelli et al
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applications
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tune measurements are useful
to determine: • beta functions , b coupling strength |
_|k• chromaticity x, Q’• transverse impedance • nonlinear fields an, bn
to improve:• dynamic aperture A• emittance e• lifetime t• instability thresholds Ithr
Z
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example applications
- chromaticity- betatron coupling- damping & decoherence- tune shift with amplitude- high-order resonances- tune scans- b function measurement- measurement of nonlinear field errors
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...2
1 22
2
0
I
I
QI
I
QQQ
cos2
sin2
'
cos2
IIx
Ix
some applications of tune measurements
1. tune shift with amplitude
due to nonlinear fields(octupoles, 12-poles,sextupoles,…)
e.g., use FFT with data windowing
accurate tune evaluation over 32 turns
action-angle-coordinatesI, y
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can measure entire curve Q(I) after single injection by calculating Q&I for each time interval (making use of radiation damping)
Measurement of tune shift with amplitude in LEP at 20 GeV using a high-precision FFT tune analysis
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2. higher-order resonances
plQkQ yx k,l,p integer
excited by nonlinear fields
distortion in phase spacechaos, dynamic aperture,particles loss …
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dynamic aperture
separatrix
linearmotion
islands of 3rd
order resonance3Qx=p
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tune vs. amplitude
resonance
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phase-space distortionadditional lines in Fourier spectrum
line at Qx lines at Qx,at 2Qx, at 4 Qx
no line at 3 Qx!plQkQ yx
yx
yx
QlkQ
lQQk
)1(
)1(
resonance
in x signal
in y signal reconstruct nonlinearities from FFT SUSSIX
cos))3cos((~ AAx
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turn
x [mm]
Dynamic aperture experiment in the CERN SPS (~1995).Beam was kicked at about turn 100. Change in offset related to (1,0) resonance or (0,0) line [Courtesy F. Schmidt, 2000].
change in apparent closedorbit after ‘kick’
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FFT amplitude
fractional tuneFFT of beam position, evidencing nonlinear resonance lines[Courtesy F. Schmidt, 2000]
many more lines in frequency spectrum!
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amplitude of (-2,0) line vs. longitudinal position for a hypothetical ringwith three sextupoles – the strength of the nonlinear line depends on s
Ph.D. thesisR. Tomas,2003
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Ph.D. thesisR. Tomas,2003
measurement compared with simulation for the SPS;locations of 7 strong sextupoles are indicated
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example - checking the coupling strengthLHC beam commissioning 11 September 2008
R. Tomas
beating confirms x-y tune split by 5 integers!
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3. tune scans
beam lifetime dynamic aperture
beam-beam interactionsensitive to tunes!
measure ),( yx QQ
tune diagramhigher-order resonances appear as stripes with reduced lifetime
• identify (&compensate) harmful resonances• find optimum working point• compare with simulations
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dynamicaperturesimulation;tune scanaround24.709 (x),23.634 (y)
measuredbeam lifetimearound sameworking point
PEP-II HER [Y. Cai, 1998]
different slope attributed tocalibration error of tune knobs
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if
QQQK
2sin2cos12cot2
0
1
12cot 0
Q
Q
K
Q
4
(tune not near integer or half integer)
and
4. measuring the b function with “K modulation”
vary quadrupole strength DK, detect tune change DQ,obtain b at quadrupole
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quality of approximation
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Optics test in Fermilab Recycler Ring, March 2000. Betatron tunes vs. strength of quadrupole QT601.
measurement
prediction including “exact”nonlineardependence
prediction of linear approximation
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5. tune based reconstruction of local nonlinear field errors
Deformedclosed orbit
Betatronicmotion
Nonlinear error
General tune response for sextupolar and octupolar errors when deforming CO with steerers
The element of the nonlinear tune response matrix are
G.Franchetti,A.Parfenova,I.Hofmann PRTAB 11, 094001 (2008)
Feed-down
G. Franchetti
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Tune measurements in SIS
From the ``slopes’’ xQxi the
probing sextupolar errors can be reconstructed
A. Parfenova, PhD Thesis 2008, Frankfurt UniversityA.Parenova, G.Franchetti,I.Hofmann Proc. EPAC08, THPC066, p. 3137
G. Franchetti
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synchrotron tune
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determined with 10-3
precisionvoltagecalibration energy loss
due to SR andimpedance
from localizationof rf cavities(computed)
synchrotron tuneas a function of total rf voltagein LEP at 60.6 GeV;the two curves arefits to the 640 mA and10 mA data;tfe difference due tocurrent-dependentparasitic modes isclearly visible
(A.-S. Muller)
LEP modelfor synchrotrontune
2/1
222
44
2
22222 1ˆˆ
2
1
UEE
VMg
E
VeghQ Cs
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if the energy is known at one point, i.e., on a spin resonance, the rf voltage can be calibrated from the Qs vs Vc curve
voltage calibration factor g
fittedbeam energy
energyknown from resonantdepolarization
(A.-S. Muller)
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display of complex tune signals
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3D-“BBQ”-Measurement in CERN SPS WITHOUT ANY EXTERNAL EXCITATION
M. Gasior
fixed target beamfixed target beam
LHC beamLHC beam
studies of the transverse damper influence on the tune path width
SPS spectrum 1,
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bizarre resonance occurred when one bunch LHC beam was going coast, measured during LHC collimator studies on 12/10/04
M. GasiorSPS continuous tune measurement example
SPS spectrum 2
zoom on low frequency synchrotron sidebands of the revolution frequency
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continuous BBQ tune diagnostics at the CERN SPS
M. Gasior
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summaryintroduction
tune, coherent & incoherent tune, detectors
integer betatron tunefractional betatron tune
precision measurement, tune tracking, multiple bunches
modifications of tune signaldamping, filamentation, chromaticity, linear coupling
some “complications”colliding beams, space charge, measuring incoherent tune
applicationstune shift with amplitude,high-order resonances,tune
scans, b function measurement, nonlinear field errors
synchrotron tunedisplay of complex tune signals
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more examples and othertypes of measurements may be found in this book
further literature
CARE-HHH-ABI workshop on Schottky, Tune and Chromaticity Diagnostics (with Real-Time Feedback), Chamonix, France, 11-13 December 2007, Proceedings CARE-Conf-08-003-HHH (editor Kay Wittenburg)
Measurement and Control of Charged Particle BeamsM.G. Minty, F. Zimmermann, Springer Verlag, Berlin, N.Y., Tokyo, 2003.