Sources of emittance growth - CERN · Massimo Giovannozzi CAS - 2-14 October 2005 9 Scattering through thin foil - IV Few remarks A special case with α = 0 at the location of the

Massimo GiovannozziMassimo Giovannozzi CAS CAS -- 22--14 October 2005 14 October 2005 11

Sources of Sources of emittanceemittancegrowthgrowth

M. M. GiovannozziGiovannozziCERNCERN-- AB DepartmentAB Department

Summary:Summary:Introduction Introduction

EEmittancemittance growth in singlegrowth in single--passage systemspassage systems

Scattering through thin foilsScattering through thin foils

EmittanceEmittance growth in multigrowth in multi--passage systemspassage systems

Injection processInjection process

Scattering processesScattering processes

OthersOthers

EmittanceEmittance manipulationmanipulationLongitudinalLongitudinal

TransverseTransverse

Acknowledgements:

D. Brandt and D. Möhl

Acknowledgements: Acknowledgements:

D. Brandt and D. D. Brandt and D. MMööhlhl


Introduction Introduction -- II

The starting point is the wellThe starting point is the well--known Hillknown Hill’’s s equationequation

Such an equation has an invariant (the soSuch an equation has an invariant (the so--called Courantcalled Courant--Snyder invariant)Snyder invariant)

0(s)(s)(s) xKx =+″

22 2 xxxxA ′+′+= βαγ

Parenthetically: in a bending-free region the following dispersion invariant existsParenthetically: in a bendingParenthetically: in a bending--free region free region the following dispersion invariant existsthe following dispersion invariant exists

22 2 DDDDA ′+′+= βαγ


Introduction Introduction -- IIII

In the case of a beam, i.e. an ensemble of In the case of a beam, i.e. an ensemble of particles:particles:EmittanceEmittance: value of the Courant: value of the Courant--Snyder Snyder invariant corresponding to a given fraction of invariant corresponding to a given fraction of particles.particles.Example: Example: rmsrms emittanceemittance for Gaussian beams.for Gaussian beams.

Why Why emittanceemittance can grow?can grow?

Hill equation is linear Hill equation is linear --> in the presence of > in the presence of nonlinear effects nonlinear effects emittanceemittance is no more is no more conserved.conserved.


Introduction Introduction -- IIIIII

Why Why emittanceemittance growth is an issue?growth is an issue?

Machine performance is reduced, e.g. in the Machine performance is reduced, e.g. in the case of a case of a collidercollider the the luminosityluminosity (i.e. the rate (i.e. the rate of collisions per unit time) is reduced.of collisions per unit time) is reduced.It can lead to beam losses.It can lead to beam losses.


Introduction Introduction -- IVIV

FilamentationFilamentation is one of the key concepts for is one of the key concepts for computing computing emittanceemittance growthgrowth

Due to the presence of nonlinear imperfections, Due to the presence of nonlinear imperfections, the rotation frequency in phase space is amplitudethe rotation frequency in phase space is amplitude--dependent.dependent.After a certain time the initial beam distribution is After a certain time the initial beam distribution is smeared out to fill a phase space ellipse.smeared out to fill a phase space ellipse.


Scattering through thin foil Scattering through thin foil -- II

Typical situation: Typical situation: Vacuum window between the transfer line and a target (in Vacuum window between the transfer line and a target (in case of fixed target physics)case of fixed target physics)Vacuum window to separate standard vacuum in transfer line Vacuum window to separate standard vacuum in transfer line from high vacuum in circular machinefrom high vacuum in circular machine

Incident Incident beambeam

Emerging Emerging beambeam

The beam receives an The beam receives an angular kickangular kick


Scattering through thin foil Scattering through thin foil -- IIII

Multiple Coulomb scattering Multiple Coulomb scattering due to beamdue to beam--matter matter interaction is described by interaction is described by means of the means of the rmsrmsscattering angle:scattering angle:

Downstream of the foil the Downstream of the foil the transformed coordinates transformed coordinates are given byare given by

θβψψ

+=Δ+→

=→

)cos()sin(

iioxixi

iioii

ApppAxx

xxpx ′+= βα

( )corrrad

p

p

rms LLq

pcMeV ε

βθ += 1/14

Normalised coordinateNormalised coordinate αα = 0 at the = 0 at the location of the foillocation of the foil


Scattering through thin foil Scattering through thin foil --IIIIII

By assuming that:By assuming that:Scattering angle and Scattering angle and betatronicbetatronic phase are uncorrelatedphase are uncorrelatedAveraging of the phase, i.e. assuming Averaging of the phase, i.e. assuming filamentationfilamentation in the in the transfer line or the subsequent machinetransfer line or the subsequent machine

Therefore the final result isTherefore the final result is

Taking into account the statistical definition of Taking into account the statistical definition of emittanceemittance

βθπε 2)(2 rmsrms

=Δ

2220

222iixiii ApxA θβ+=+=

2220

2 22 rmsxx θβσσ +=


Scattering through thin foil Scattering through thin foil --IVIV

Few remarksFew remarksA special case with A special case with αα = 0 at the location of the = 0 at the location of the thin foil is discussed thin foil is discussed --> it can be generalised. > it can be generalised. The correct way of treating this problem is (see The correct way of treating this problem is (see next slides):next slides):

Compute all three secondCompute all three second--order moments of the beam order moments of the beam distribution downstream of the foildistribution downstream of the foilEvaluate the new optical parameters and Evaluate the new optical parameters and emittanceemittance using using the statistical definitionthe statistical definition

The The emittanceemittance growth depends on the betagrowth depends on the beta--function!function!

THE SMALLER THE VALUE OF THE BETATHE SMALLER THE VALUE OF THE BETA--FUNCTION AT THE LOCATION OF THE FOIL FUNCTION AT THE LOCATION OF THE FOIL

THE SMALLER THE EMITTANCE GROWTHTHE SMALLER THE EMITTANCE GROWTH


Scattering through thin foil Scattering through thin foil --VV

Correct computation (always for Correct computation (always for αα=0):=0):

By squaring and averaging over the beam distributionBy squaring and averaging over the beam distribution

Using the relationUsing the relation

)1

sin(1

/11

)sin(/1

)1

cos(11

)cos(

ψβθψβ

ψβψβ

Aooo

Ax

Aooo

Ax

−=+−=′

==

1

12

0

0

10

2

2

2

22

22222

10

ββθ

ββ

>′<

><=><

+

=

AAx

AAx

rms

2

2Arms πε =


Scattering through thin foil Scattering through thin foil --VIVI

The solution of the system is The solution of the system is

This can be solved exactly, or by assuming that the This can be solved exactly, or by assuming that the relative relative emittanceemittance growth is small, thengrowth is small, then

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−=

=Δ

rmsrms

rmsrms

εθπββ

πε βθ

2

41

4

01

02NB: the emittance growth is now only half

of the previous estimate!

Downstream of the foil the transfer line should be matched using the new Twissparameters α1, β1

NB: the NB: the emittanceemittance growth is now only half growth is now only half of the previous estimate!of the previous estimate!

Downstream of the foil the transfer line Downstream of the foil the transfer line should be matched using the new should be matched using the new TwissTwissparameters parameters αα1, 1, ββ11

21

20

0

1

202

0

220

221

2

A

A

A

AArms

=

=−

ββ

θβ


-10

10

30

50

70

0 50 100 150 200 250 300

Example: ion stripping for LHC Example: ion stripping for LHC lead beam between PS and SPSlead beam between PS and SPS

ββHH

ββVV

DDHH

DDVV

Stripper locationStripper location

Courtesy M. Martini - CERNCourtesy M. Martini Courtesy M. Martini -- CERNCERN

PbPb54+54+ --> Pb> Pb82+82+

A lowA low--beta beta insertion is insertion is designed (beta designed (beta reduced by a reduced by a factor of 5)factor of 5)

Stripping foil, Stripping foil, 0.8 mm thick 0.8 mm thick Al, is located in Al, is located in the lowthe low--beta beta insertioninsertion


ExercisesExercises……

Compute the Compute the TwissTwiss parameters and parameters and emittanceemittancegrowth for a growth for a THICKTHICK foilfoilCompute the Compute the TwissTwiss parameters and parameters and emittanceemittancegrowth for a growth for a THICKTHICK foil in a foil in a quadrupolarquadrupolar fieldfield


Injection process Injection process -- II

Two sources of errors:Two sources of errors:Steering errors.Steering errors.Optics errors (Optics errors (TwissTwiss parameters and dispersion).parameters and dispersion).

In case the incoming beam has an energy In case the incoming beam has an energy error, then the next effect will be a error, then the next effect will be a combination of the two.combination of the two.In all cases In all cases filamentationfilamentation, i.e. nonlinear , i.e. nonlinear imperfection in the ring, is the source of imperfection in the ring, is the source of emittanceemittance growth.growth.


Injection process Injection process -- IIII

Steering errors:Steering errors:Injection conditions, i.e. position and angle, do not Injection conditions, i.e. position and angle, do not match position and angle of the closed orbit.match position and angle of the closed orbit.

Consequences:Consequences:The beam performs The beam performs betatronbetatron oscillations around the oscillations around the closed orbit. The closed orbit. The emittanceemittance grows due to the grows due to the filamentationfilamentation

Solution:Solution:Change the injection conditions, either by steering Change the injection conditions, either by steering in the transfer line or using the septum and the in the transfer line or using the septum and the kicker.kicker.In practice, slow drifts of settings may require In practice, slow drifts of settings may require regular tuning. In this case a regular tuning. In this case a damper (see damper (see lecturelectureon feedback systems)on feedback systems) is the best solution.is the best solution.


Injection process Injection process -- IIIIII

Analysis in normalised phase Analysis in normalised phase space (space (ii stands for injection stands for injection mm for machine):for machine):

Squaring and averaging givesSquaring and averaging gives

After After filamentationfilamentation

ψψψψ

cossincoscos

rrprrx

iim

iim

Δ+=

Δ+=

222

222

rrr

pxr

im

mmm

Δ+><

>+<

rrrx im Δ+>< 212

212

212 21

222 ri Δ+=σσ


Injection process Injection process -- IVIV

Example of beam Example of beam distribution distribution generated by generated by steering errors steering errors and and filamentationfilamentation..The beam core is The beam core is displaced displaced --> large > large effect on effect on emittanceemittance growthgrowth


Injection process Injection process -- VV

Dispersion mismatch: Dispersion mismatch: analysis is similar to analysis is similar to that for steering errors.that for steering errors.In this case In this case

The final result isThe final result is

( )2

222

21

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦⎤

⎢⎣⎡ Δ+′Δ+Δ=Δ

pDDDr p

σαβ

( )[ ]⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛Δ+′Δ+Δ+= 20

222. /

211 σ

σαβεε

pDDD prms

filafterrms


Injection process Injection process -- VIVI

Optics errors:Optics errors:Optical parameters (Optical parameters (TwissTwiss and and dispersion) of the transfer line dispersion) of the transfer line at the injection point are at the injection point are different from those of the ringdifferent from those of the ring

Consequences:Consequences:The beam performs The beam performs quadrupolarquadrupolaroscillations (size changes on a oscillations (size changes on a turnturn--byby--turn basis). The turn basis). The emittanceemittance grows due to the grows due to the filamentationfilamentation

Solution:Solution:Tune transfer line to match Tune transfer line to match optics of the ringoptics of the ring

xx

xx’’Ring ellipseRing ellipse

Transfer Transfer line ellipseline ellipse


Injection process Injection process -- VIIVII

In normalised phase space (that of the ring) the In normalised phase space (that of the ring) the injected beam will fill an ellipse due to the injected beam will fill an ellipse due to the mismatch of the opticsmismatch of the optics……


Injection process Injection process -- VIIIVIII

iiliil baApabAx ψψ cos/sin/ , ==

( )2

//22 baabAr ii+

>=<

2//

.⎟⎠⎞⎜

⎝⎛

=+ baab

rmsfilafter

rms εε

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−++=

=

lml

l

m

m

l

m

m

l

rmsfilafter

rms

FF

βββα

βα

ββ

ββ

εε2

.

21

Given the initial condition of the beam end of the transfer Given the initial condition of the beam end of the transfer lineline

By squaring and averaging over the particlesBy squaring and averaging over the particles’’ distributiondistribution

The The emittanceemittance after after filamentationfilamentation is given byis given by


Scattering processes Scattering processes -- I I

Two main categories considered:Two main categories considered:Scattering on residual gas Scattering on residual gas --> similar to scattering > similar to scattering on a thin foil (the gas replaces the foilon a thin foil (the gas replaces the foil……))

IntraIntra--beam scattering, i.e. Coulomb scattering beam scattering, i.e. Coulomb scattering between charged particles in the beam.between charged particles in the beam.

betaaveragewhereL

tcp

cMeVqkrad

p

ppk β

ββ

βε πσ

2/1422

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛=Δ

NB: βpct represents the scatterer length until time tNB: NB: ββppctct represents the represents the scattererscatterer length until time length until time tt

This is why good vacuum is necessary!This is why good This is why good vacuum is necessary!vacuum is necessary!


Scattering processes Scattering processes -- IIII

IntraIntra--beam scatteringbeam scatteringMultiple (small angle)Multiple (small angle)Coulomb scatteringCoulomb scatteringbetween charged between charged particles.particles.Single scatteringSingle scatteringevents lead to events lead to TouscheckTouscheck effect.effect.All three degrees of All three degrees of freedom are affected.freedom are affected.


Scattering processes Scattering processes -- IIIIIIHow to compute IBS?How to compute IBS?

Transform Transform momentamomenta of colliding particles into their centre of colliding particles into their centre of mass system.of mass system.Rutherford crossRutherford cross--section is used to compute change of section is used to compute change of momentamomenta..Transform the new Transform the new momentamomenta back to the laboratory back to the laboratory system.system.Calculate the change of the Calculate the change of the emittancesemittances due to the change due to the change of of momentamomenta at the given location of the collision.at the given location of the collision.

1.1. For each scattering event compute average over all For each scattering event compute average over all possible scattering angles (impact parameters from the possible scattering angles (impact parameters from the size of the nucleus to the beam radius).size of the nucleus to the beam radius).

2.2. Take the average over Take the average over momentamomenta and transverse position of and transverse position of the particles at the given location on the ring the particles at the given location on the ring circumference (assuming Gaussian distribution in all phase circumference (assuming Gaussian distribution in all phase space planes.space planes.

3.3. Compute the average around the circumference, including Compute the average around the circumference, including lattice functions, to determine the change per turn.lattice functions, to determine the change per turn.


Scattering processes Scattering processes -- IVIV

Features of IBSFeatures of IBSFor constant lattice functions and below For constant lattice functions and below transition energy, the sum of the three transition energy, the sum of the three emittancesemittances is constant.is constant.Above transition the sum of the Above transition the sum of the emittancesemittancesalways grows.always grows.In any strong focusing lattice the sum of the In any strong focusing lattice the sum of the emittancesemittances always grows.always grows.Even though the sum grows from the Even though the sum grows from the theoretical point of view theoretical point of view emittanceemittance reduction reduction in one plane predicted by simulations, but in one plane predicted by simulations, but were never observed.were never observed.


Scattering processes Scattering processes -- VV

Scaling laws of IBSScaling laws of IBSAccurate computations can be performed only with Accurate computations can be performed only with numerical tools.numerical tools.However, scaling laws can be derived.However, scaling laws can be derived.AssumingAssuming

ThenThen

***

22

0***2

222

0

0 /)/4(/1

lyx

b

lyx

b AqN

EAqrN

εεεγεεεγπτ

⎟⎟⎠

⎞⎜⎜⎝

⎛

∝⎟⎟⎠

⎞⎜⎜⎝

⎛

=

lyxlyx

F ,,0,,

11ττ

=

Strong dependence on charge

εx,y,l* are normalisedemittances

Nb of particles/bunch

r0 classical proton radius

Strong dependence on Strong dependence on chargecharge

εεx,y,lx,y,l* are * are normalisednormalisedemittancesemittances

NNbb of particles/bunchof particles/bunch

rr00 classical proton radiusclassical proton radius


Others Others

Diffusive phenomena:Diffusive phenomena:Resonance crossingsResonance crossingsRipple in the power convertersRipple in the power converters

Collective effectsCollective effectsSpace charge (soft part of Coulomb interactions Space charge (soft part of Coulomb interactions between charged particles in the beam) between charged particles in the beam) --> > covered covered by a specific lectureby a specific lecture..BeamBeam--beam beam --> > covered by a specific lecturecovered by a specific lecture..Instabilities Instabilities --> > covered by a specific lecturecovered by a specific lecture..


EmittanceEmittance manipulationmanipulation

EmittanceEmittance is (hopefully) conservedis (hopefully) conserved……Sometimes, it is necessary to manipulate the Sometimes, it is necessary to manipulate the beam so to reduce its beam so to reduce its emittanceemittance..

Standard techniques: electron cooling, stochastic Standard techniques: electron cooling, stochastic cooling.cooling.Less standard techniques: longitudinal or transverse Less standard techniques: longitudinal or transverse emittanceemittance splitting.splitting.


LongitudinalLongitudinal manipulation: manipulation: LHC beam in PS machine LHC beam in PS machine -- II

Courtesy R. Garoby - CERNCourtesy R. Courtesy R. GarobyGaroby -- CERNCERN

0.35 eVs (bunch)1.1 × 1011 ppb

1 eVs (bunch)4.4 × 1011 ppb

1.4 eVs (bunch)13.2 × 1011 ppb

40 %Blow-up

(pessimistic)

115 %Blow-up

(voluntary)

Triple splittingat 1.4 GeV

Quadruple splitting at 25 GeV

PS injection:2+4 bunches in 2 batches

Empt

ybu

cket

Accelerationto 25 GeV

PS ejection:72 bunches

in 1 turn

320 ns beam gap

6 buncheson h=7

18 buncheson h=21

72 buncheson h=84


LongitudinalLongitudinal manipulation: manipulation: LHC beam in PS machine LHC beam in PS machine --IIII

Courtesy R. Garoby - CERNCourtesy R. Courtesy R. GarobyGaroby -- CERNCERN

Measurement results obtained at the CERN Proton Synchrotron

Measurement results Measurement results obtained at the CERN obtained at the CERN Proton SynchrotronProton Synchrotron


TransverseTransverse manipulation: manipulation: novel multinovel multi--turn extractionturn extraction ––

II

The main ingredients of the novel extraction:The main ingredients of the novel extraction:The beam splitting is not performed using a The beam splitting is not performed using a

mechanical device, thus avoiding losses. Indeed, the mechanical device, thus avoiding losses. Indeed, the beam is separated in the transverse phase space beam is separated in the transverse phase space using using

Nonlinear magnetic elementsNonlinear magnetic elements (sextupoles ad (sextupoles ad octupoles) to create octupoles) to create stable islandsstable islands..Slow (Slow (adiabaticadiabatic) tune) tune--variation to cross an variation to cross an appropriate resonance.appropriate resonance.

NB: third-order extraction is based on separatrices (see O. Brüning lecture)NB: thirdNB: third--order extraction is based on order extraction is based on separatricesseparatrices (see O. (see O. BrBrüüningning lecture)lecture)


TransverseTransverse manipulation: novel manipulation: novel multimulti--turn extractionturn extraction –– IIII

-1 1-1

1

X

X’

(b)

Right: intermediate Right: intermediate phase space topology. phase space topology. Islands are created Islands are created near the centre.near the centre.

-1 1-1

1

X

X’

(c)

Bottom: final phase Bottom: final phase space topology. space topology. Islands are separated Islands are separated to allow extraction.to allow extraction.

-1 1-1

1

X

X’

(a)Left: initial phase Left: initial phase space topology. No space topology. No islands.islands.

Four islands

require

tune = 0.25

Four islands Four islands

requirerequire

tune = 0.25tune = 0.25


TransverseTransverse manipulation: manipulation: novel multinovel multi--turn extractionturn extraction ––IIIIII

Tune variationTune variation

Phase space Phase space portraitportrait

Simulation Simulation parameters:parameters:

HHéénonnon--like like map (i.e. 2D map (i.e. 2D polynomial polynomial ––degree 3degree 3 --mapping) mapping) representing representing a FODO cell a FODO cell with with sextupole and sextupole and octupoleoctupole


A movie to show the evolution of A movie to show the evolution of beam distributionbeam distribution

A series of A series of horizontal beam horizontal beam profiles have been profiles have been taken during the taken during the capture process.capture process.

Measurement results obtained at the CERN Proton Synchrotron

Measurement results Measurement results obtained at the CERN obtained at the CERN Proton SynchrotronProton Synchrotron


Some referencesSome references

D. Edwards, M. D. Edwards, M. SyphersSyphers: An introduction to the : An introduction to the physics of high energy accelerators, J. Wiley & Sons, physics of high energy accelerators, J. Wiley & Sons, NY 1993.NY 1993.P. Bryant, K. P. Bryant, K. JohnsenJohnsen: The principles of circular : The principles of circular accelerators and storage rings, Cambridge University accelerators and storage rings, Cambridge University press, 1993.press, 1993.P. Bryant: P. Bryant: Beam transfer lines, CERN yellow rep. 94Beam transfer lines, CERN yellow rep. 94--10 (CAS, 10 (CAS, JyvaskylaJyvaskyla, Finland , 1992)., Finland , 1992).J. J. BuonBuon: Beam phase space and : Beam phase space and emittanceemittance, CERN , CERN yellow rep. 91yellow rep. 91--04 (CAS, 04 (CAS, JulichJulich, Germany, 1990)., Germany, 1990).A. A. PiwinskiPiwinski: Intra: Intra--beam scattering, CERN yellow rep. beam scattering, CERN yellow rep. 8585--19 (CAS, Gif19 (CAS, Gif--sursur--Yvette, France 1984).Yvette, France 1984).A. H. SorensenA. H. Sorensen: Introduction to intra: Introduction to intra--beam beam scattering, CERN yellow rep. 87scattering, CERN yellow rep. 87--10 (CAS, 10 (CAS, AarhusAarhus, , Denmark 1986).Denmark 1986).

Sources of emittance growthIntroduction - IIntroduction - IIIntroduction - IIIIntroduction - IVScattering through thin foil - IScattering through thin foil - IIScattering through thin foil - IIIScattering through thin foil - IVScattering through thin foil - VScattering through thin foil - VIExercises…Injection process - IInjection process - IIInjection process - IIIInjection process - IVInjection process - VInjection process - VIInjection process - VIIInjection process - VIIIScattering processes - I Scattering processes - IIScattering processes - IIIScattering processes - IVScattering processes - VOthers Emittance manipulationLongitudinal manipulation: LHC beam in PS machine - ILongitudinal manipulation: LHC beam in PS machine - IITransverse manipulation: novel multi-turn extraction – ITransverse manipulation: novel multi-turn extraction – IITransverse manipulation: novel multi-turn extraction – IIIA movie to show the evolution of beam distributionSome references

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Sources of emittance growth - CERN · Massimo Giovannozzi CAS - 2-14 October 2005 9 Scattering through thin foil - IV Few remarks A special case with α = 0 at the location of the