• 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

QQ

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

QQ

pp

QQ

'

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

QQ

/

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