synchronous phase synchrotron tune dispersion momentum compaction chromaticity
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• synchronous phase• synchrotron tune • dispersion• momentum compaction• chromaticity
Longitudinal Optics Measurement and Correction
[MCCPB, Chapter 7]
1. synchrotron tune & synchronous phase
tp
pprf ,
0
0
rev
rfC
ET
eV
dt
d
dt
d
2
2
)(
1
longitudinal coordinates:
equations of motion
CC
C
momentum compaction factor
linearize around synchronousphase
s ~
0 ,0~
~2
2
22
2
2
ss dt
d
dt
d
rev
scs TE
Ve2
21
synchrotron frequency
smooth approximation (not correct forrings with strong rf focusing, which requiredifference equations for localized rf; examples: LEP, DAFNE-II)
tVV rfc cosusually sinusoidal rf voltage:
synchrotron tune
sidebands of revolution frequency due to modulation of arrival time and for nonzero Dalso modulation of transverse position
revrfh harmonic number
cs eV
U1cos
pararadsc UUUUeV homcossynchronous phase
from synchrotron tuneand/or quantum lifetime,or direct measurement (?)
E
heV
f
fQ scc
rev
ss 2
2
2
sin)1(
illustration of synchronous phase
measurement of multiple synchrotron sidebands at injection into the SLAC electron damping ring; the synchrotron frequency is given by the difference frequency between the fundamental and the nearest synchrotron sideband
2. dispersion
)()()()( .. sDsxsxsx xoc
)()( CsDsD xx
)()( 16 sRsx
dispersion
betatron motionclosed orbit
in a storage ring
in a transport line or linac
(1,6) transport matrix element from point wheremomentum error is introduced
measuring the dispersiona) rf frequency shift
rfrf
Cy
rfrfCx
ff
ysD
ff
xsD
2
2
)(
)(
rf
rf
C f
f
21
1
momentum
deviation
detect change in horizontal and vertical orbit
‘static’ dispersion measurement in the PEP-II HER; orbit change induced by a 2-kHz shift in rf frequency; nominal frequency is 476 MHz, harmonic number h=3492, and the momentum compaction factor c~0.0024
‘static’ dispersion measurement at the KEK/ATF Damping Ring before (top)and after applying a correction (bottom) based on exciting steering magnets; the vertical dispersion was measured by a 5-kHz shift in rf frequency; a dispersion of D=10 mm corresponds to x~30 m.
measuring the dispersionb) rf modulation
rf is modulated at synchrotron frequency; induced orbit variation atthis frequency dispersion
non-vanishing D in cavities ‘spurious’ dispersion
Borer et al., LEP
CERN SL/91-38 (AP)‘Effect of residual dispersion at the RF cavities on the dynamicmeasurement of dispersion in LEP”by Francesco Ruggiero
QQHD
s
s
2cos2cos
2sin2sinmax
arc function
dispersion invariant in cavities
22 '1
DDDH
measuring the dispersionc) rf amplitude or phase jump
SLC damping rings (V)ATF damping ring ()
also these give spurious results, if there is dispersion in the cavities
x=-D
0'' kxx skxx sinˆ
d) resonant dispersion growth & resonant correction
betatron oscillation
constant amplitude
dispersion
skxkkDD sinˆ1
''
dipole feed-down due totrajectory offset in quadrupoleleads to resonant excitation
skksskx
D cossin2
ˆ
D increases linearly
FFT around ring of normalized Dy peaks near Qy!(F. Ruggiero, A. Zholents)
correct dispersion by special orbit bumps across arcs
these produce dispersion
with amplification factor
)()(sin 0ssYyco
)()(sin)(
)(0ssQAY
s
sDy
ycellcell
Q
QNA
sin
'Q’cell: chromaticity of single FODO cellNcell: number of cells covered by bump
bumps across various arcs can be combined in a symmetricor anti-symmetric manner to control either D or D’ at the collision points
e) higher-order dispersion in transport line or linac
several energy steps are made by variation of rf amplitude and phase
...)()()()('
...)()()()(3
26662
26626
31666
216616
sUsTsRsx
sUsTsRsx
...33
2210 i
xix
ix
ii DDDxx
or, alternatively, write at ith BPM
determine initial conditions of nth order dispersion and correct with linear combination of multipole magnets
):0(120,
):0(110, ' ii
xnii
xnixn RDRDD
then, for set of M BPMs fit oscillation
algorithm developed for SLAC North and South Ring-to-Linac transfer line (P. Emma)
evidence of 3rd order dispersion in the SLC ring-to-linac transfer line
BPM x
%1
largecubiccomponent!
3rd order dispersion for all BPMs in the RTL and in the early linac
U1666
s [m]
3rd orderdispersion in linac is fitted to compute U1666 and U2666
matrix elements
large 3rd order dispersion led to irrecoverable emittance growth
BPM data
22222
11121 2
1
2
1'cos'' DxKDxKxxx
2121 , DDxx
2212221
2212 2' DKKxDKKxKKx
T211 geometric (insignificant)
T216 chromaticityK1=K2
T266 2nd order dispersionK1=-K2
multiknobs forcorrrecting2nd order dispersion,chromaticity, …
at the SLC a pair of octupoles was installed tocancel the U1666 and U2666 terms; emittance was minimized by scanning octupole-pair setting:
the octupolestrength forwhich the emittanceis minimizedagrees withthe fit fromthe BPM data
3. momentum compaction
dss
sD
CCC
C
)(
)(1
measuring the momentum compaction factora) synchrotron tune
CscC
s vp
VheQ
~2
cosˆ1
0
2
)sinˆ since )!ˆ(:(note ˆ vs. measure UVeVfVQ sccscs
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 A and10 A data;tfe difference due tocurrent-dependentparasitic modes isclearly visible
(A.-S. Muller)
LEP model 2/1
222
44
2
22222 1ˆˆ
2
1
UEE
VMg
E
VeghQ Cs
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)
measuring the momentum compaction factorb) bunch length
revs
Cz fQ
c
2
1 2
partition damping longit. ring,over average ...
,/1 m, 1084.3 with
13
2
23
2
J
GxCGJ
GCq
q
either measured from decoherence due to nonzero chromaticity(see last week) or calculated from optics:
can be verified by measuring horizontal emittance ~/(3-J) or longitudinal dampingtime ~1/J
plot bunch lengthvs. inverse synchrotrontune
rms bunch length measured by streak camera in the PEP-II HER as a function of the inverse synchrotron tune
fitted slope determines the momentum compaction factor, if the rms energy spread is known
2
2max0
2 where
e
Q
T
sq
measuring the momentum compaction factorc) quantum lifetime (for electron storage rings)
)(02max qFhE
eUC
qqqF
U
Veq c 1
cos12)( ,ˆ
12
0
),,(
),(ˆ
sCz
Csc
Qf
QfV
momentum acceptance
with
note:
using previous equations
revs
Cz fQ
c
2
1 2
0
2
2
cosˆ1
vp
VheQ scCs
recipe:measure Qs. z and q for different rf voltagesand fit for C!
(assumes lifetime limited byquantum fluctuations)
measuring the momentum compaction factord) direct measurement using streak camera
R56 measurementfor the asynchronousarc of the KEKB linacbefore and after correction;a streak camera was used to measure arrival timeas a function ofbeam energy
streak camera triggerwas locked to thelinac rf frequencyupstream of the arc
correction was doneby changing a few quad strengths
measuring the momentum compaction factore) beam energy via resonant depolarization
[MeV] 440.6486
]MeV[0
Eae
rf
rf
Crf
rf
C f
f
f
f
p
p
1
1
12
spin tune
if radially oscillation field is in resonance with the fractional partof the spin tune, the effect of the field adds up over many turnsand the beam depolarizes; the exact value of the resonant frequencydetermines the beam energy via the above equation
recipe: measure energy change caused by shift in rf frequency
slope is momentum compaction factor
change of beam energy E as a function of rf frequency frf in LEP
only last 4 digits of frequency are displayed (nominal value is 352 254 170 Hz);several strong spin resonances are indicated by the dotted lines; from this measurement the momentum compaction was determined to be 0.000186+/-0.000002, consistent with theoretical value 0.0001859
(R. Assmann)
measuring the momentum compaction factorf) change in field strength for unbunched proton beam
energy of unbunched proton beam is constant, neglecting SR losses
if strength of all magnets (dipoles and quadrupoles) is increased bya factor B/B, the orbit moves inwards and the revolution time isreduced; this change in revolution time can be detected by a Schottkymonitor
the momentum compaction factor follows from change in revolution period:
B
B
T
TC
2
1
this change in revolution period can be detected by a Schottky monitor
4. chromaticity
pp
pp
'
QQ /'
normalized unnormalized
relation
chromaticity describes the change of focusing 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,…)
rfrf
yxC
yxyx
ff
Q
pp
/
1
/'
,
2
,,
measuring the total chromaticitya) tune 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
measuring the total chromaticityb) head-tail phase shift deflect bunch transversely and measure
the oscillation of head and tail over Ts
1)2cos(
)(1
,
2
,
srfyx
Cyx nQQ
n
chromaticityinferred fromthe measurementsof the head-tailphase shift at theCERN SPStop left: head oscillationafter kicktop right: center oscillationafter kickbottom left: phase of headand center and differencebottom right:chromaticity inferred foreach turn
measuring the total chromaticityc) from de- and recoherence after kick
tQ
Qts
eetA
cos1
'
2
2
2
2
)(
chromaticity
measuring the total chromaticityd) from difference in sideband width on Schottky pick up
Qff
ffn
p
pQn
f
f
21
21
0
)(
Tevatron, 2004
measuring the natural chromaticity (Q’ w/o sextupoles)e) 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
measuring the local chromaticityf) from tune shift vs. K at different rf frequencies
K
Q yxyx
,
, 4
alternatively and faster, by measuring betatron phaseadvance around the ring at different rf frequencies
g) chromaticity control in s.c. proton rings
variation of chromaticity Q’ intime at injection in HERA due topersistent current decay
same picture with automaticcorrection based on continuoussextupole-field measurements in two reference magnets
variation of chromaticity during acceleration in HERAa) measured w/o correctionb) variation expected from reference magnetsc) measured with correction
variation of chromaticity at the start of cycle in HERA
h) application: measuring the central frequency
measure tune vs. rf frequencdy for different sextupole strengths; find intersection -> central frequency; used for monitoring energy changes rf
rf
C f
f
p
p
2
1
LEP
Summary• synchronous phase• synchrotron tune• dispersion frequency change, phase modulation, amplitude or phase jump, transport line, higher-order dispersion• momentum compaction factor synchrotron tune, bunch length, lifetime, path length vs. energy, beam energy vs. frequency, change in field + Schottky monitor• chromaticity tune shift due to frequency change, head-tail phase shift, decoherence, Schottky monitor, natural chromaticity, chromaticity in s.c. storage rings, central frequency
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