“tune”Frank Zimmermann

DITANET School, Royal Holloway,2 April 2009

02/04/09 DITANET School

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

introduction

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)

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

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

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

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

integer betatron tune

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

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

fractional betatron tune

• precision measurement• tune tracking• multiple bunches

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

two categories:

• precision tune measurements

• tune tracking to monitor & control fast changes e.g. during acceleration

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

N

j

inQj

jeQnx1

2)()(

FFT signal = expansion coefficient

Q error of FFT: N

Q2

1

due to discreteness of steps

50010 3 NQ

checking the fractional tuneLHC beam commissioning 10 September 2008

signal decay(due to de-bunching)

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)

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

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

)(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

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

QQ

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

example: refined FFTs (M. Giovannozzi et al.)

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

BPM in a dispersive arc region

BPM in a straight sectionwithout dispersion

Lomb normalized periodogram for previous measurements

(A.-S. Muller)

comparison Lomb-FFT for CERN PS simulation with two spectral lines [A.-S.Muller, ’04]]

still anactive area of research…

most of the above“improvements”rely onharmonic motion

swept frequency excitation

t sin

tAxb sin

excitation

measure response

amplitude phasevary w in steps

180o phase jump

“beam transferfunction”

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

phase locked loop (for continuous tune control)

“locked”in frequency

VCO frequency = betatron frequency“lock-in amplifier”

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

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

the tune, or not the tune, that is the question

modifications of tune signal

• damping• filamentation• chromaticity • linear coupling

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

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

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

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

LEP: tune change during dampingQ

turns[A.-S. Muller]

LEP: detuning with amplitude from single kickQ

2 J[A.-S. Muller]

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

amplitude decoherence factor vs. turn number

5s kick

1/5 s kick

oscillationamplitude

chromaticity

pp

QQ

pp

QQ

'

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,…)

response to a kick: decoherence due to chromaticity

d>0d<0

(Q’>0)

t=0 t=Ts/2 t=Ts

)(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

rfrf

yxC

yxyx

ff

Q

pp

QQ

/

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

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

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

QQ

QQ

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

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

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

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

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

RHIC-Tevatron-LHC2005

RHIC-Tevatron-LHC2005

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

some “complications”:

colliding beam tune spectraspace charge tune spreadmeasuring the incoherent tune

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

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

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?

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

how to measure the incoherent tune shift/spread?

K.H. Schindl

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

applications

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

example applications

- chromaticity- betatron coupling- damping & decoherence- tune shift with amplitude- high-order resonances- tune scans- b function measurement- measurement of nonlinear field errors

...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

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

2. higher-order resonances

plQkQ yx k,l,p integer

excited by nonlinear fields

distortion in phase spacechaos, dynamic aperture,particles loss …

dynamic aperture

separatrix

linearmotion

islands of 3rd

order resonance3Qx=p

tune vs. amplitude

resonance

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

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’

FFT amplitude

fractional tuneFFT of beam position, evidencing nonlinear resonance lines[Courtesy F. Schmidt, 2000]

many more lines in frequency spectrum!

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

Ph.D. thesisR. Tomas,2003

measurement compared with simulation for the SPS;locations of 7 strong sextupoles are indicated

example - checking the coupling strengthLHC beam commissioning 11 September 2008

R. Tomas

beating confirms x-y tune split by 5 integers!

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

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

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

quality of approximation

Optics test in Fermilab Recycler Ring, March 2000. Betatron tunes vs. strength of quadrupole QT601.

measurement

prediction including “exact”nonlineardependence

prediction of linear approximation

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

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

synchrotron tune

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

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)

display of complex tune signals

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,

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

continuous BBQ tune diagnostics at the CERN SPS

M. Gasior

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

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.