Sources of emittance growth - CERN · Massimo Giovannozzi CAS - 27/09-09/10 October 2009 11. Scattering through thin foil - VI. The solution of the system is given by This can be

Massimo GiovannozziMassimo Giovannozzi CAS CAS --

27/0927/09--09/10 October 200909/10 October 2009 11

Sources of emittance Sources of emittance growth (Hadrons)growth (Hadrons)

M. GiovannozziM. GiovannozziCERN CERN --

BE DepartmentBE Department

Summary:Summary:

Introduction Introduction

EEmittance growth in singlemittance growth in single-- passage systemspassage systems

Scattering through thin foilsScattering through thin foils

Emittance growth in multiEmittance growth in multi-- passage systemspassage systems

Injection processInjection process

Scattering processesScattering processes

OthersOthers

Emittance manipulationEmittance manipulation

LongitudinalLongitudinal

TransverseTransverse

Acknowledgements:

D. Brandt and D. Möhl

Acknowledgements: Acknowledgements:

D. Brandt and D. MD. Brandt and D. Mööhlhl


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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

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

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

22 2 DDDDA


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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:Emittance: value of the CourantEmittance: value of the Courant--Snyder Snyder invariant corresponding to a given fraction of invariant corresponding to a given fraction of particles.particles.Example: rms emittance for Gaussian beams.Example: rms emittance for Gaussian beams.

Why emittance can grow?Why emittance can grow?

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


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IIIIII

Why emittance growth is an issue?Why emittance growth is an issue?

Machine performance is limited or reduced, Machine performance is limited or reduced, e.g. e.g.

Beam losses can generated Beam losses can generated In the case of a collider the In the case of a collider the luminosityluminosity

(i.e. (i.e.

the rate of collisions per unit time) is reduced.the rate of collisions per unit time) is reduced.


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IVIV

Filamentation is one of the key concepts for Filamentation is one of the key concepts for computing emittance growthcomputing emittance growth

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.


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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 particles receive an angular The particles receive an angular kickkick


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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 rms means of the rms scattering angle:scattering angle:

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

ixiixixi

iiii

ApppAxx

)cos()sin(

0

0

xxp xxx

corrrad

pp

rms LLq

pcMeV

1/14

Normalised coordinateNormalised coordinate

= 0 at the = 0 at the location of the foillocation of the foil


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IIIIII

By assuming that:By assuming that:Scattering angle and betatronic phase are uncorrelatedScattering angle and betatronic phase are uncorrelatedAveraging over betatronic phase (due to filamentation) is Averaging over betatronic phase (due to filamentation) is possiblepossible

Using the relationUsing the relation

The final result readsThe final result reads

xrmsrms 22

2220

222ixixiii ApxA

2

2Arms


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IVIV

Few remarksFew remarksThe special case with The special case with

= 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 emittance using Evaluate the new optical parameters and emittance using the statistical definitionthe statistical definition

The emittance growth depends on the betaThe emittance growth 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


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VV

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

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

Using the relationUsing the relation

)sin(/11)sin(/1

)cos()cos(

11

111

xoxoox

xoxoo

AAp

AAx

1

212

0

202

12

102

02

22

22

xrms

x

xx

xx

AAp

AAx

2

2Arms


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VIVI

The solution of the The solution of the system is given bysystem is given by

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

rms

rmsxx

xrmsrms

0

2

01

02

41

4

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

Downstream of the foil the transfer line should be matched using the new Twiss parameters x1

, x1

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

Downstream of the foil the transfer Downstream of the foil the transfer line should be matched using the new line should be matched using the new Twiss parameters Twiss parameters x1x1

, , x1x1

21

20

0

1

202

0

220

221

2

A

A

A

AA

x

x

rmsx


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-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)

A stripping foil, A stripping foil, 0.8 mm thick 0.8 mm thick AlAl, is located in , is located in the lowthe low--beta beta insertioninsertion


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ExercisesExercises

Compute the Twiss parameters and emittance Compute the Twiss parameters and emittance growth for a growth for a THICKTHICK

foilfoil

Hint: slice the foil assuming a sequence of drifts Hint: slice the foil assuming a sequence of drifts and thin scatterers.and thin scatterers.

Compute the Twiss parameters and emittance Compute the Twiss parameters and emittance growth for a growth for a THICKTHICK

foil in a foil in a quadrupolarquadrupolar

fieldfield

Hint: same as before, but now the drifts should be Hint: same as before, but now the drifts should be replaced by quadrupoles.replaced by quadrupoles.


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Injection process Injection process --

II

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

In case the incoming beam has an energy In case the incoming beam has an energy error, then the effect will be a combination error, then the effect will be a combination of the two.of the two.

In all cases filamentation, i.e. nonlinear In all cases filamentation, i.e. nonlinear imperfections in the ring, is the source of imperfections in the ring, is the source of emittance growth.emittance growth.


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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 betatron oscillations around the The beam performs betatron oscillations around the closed orbit. The emittance grows due to closed orbit. The emittance grows due to 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 lecturelecture

on feedback systems)on feedback systems)

is the best solution.is the best solution.


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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 filamentationAfter filamentation

sinsincoscosrrp

rrx

iimx

iim

222

222

rrr

pxr

im

mxmm

2222. 2

121

21 rrrx imfilafter 2. 2

rrmsfilafter

rms


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IVIV

Example of beam Example of beam distribution distribution generated by generated by steering errors and steering errors and filamentation.filamentation.The beam core is The beam core is displaced displaced --> large > large effect on emittanceeffect on emittance


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VV

Dispersion mismatch: analysis is similar to that for Dispersion mismatch: analysis is similar to that for steering errors.steering errors.A particle with momentum offset A particle with momentum offset p/p will have p/p will have injection conditions given by injection conditions given by

The vector The vector r is obtained by: taking difference of r is obtained by: taking difference of injection conditions; transforming in normalised phase injection conditions; transforming in normalised phase space. Then after squaring and averaging over the space. Then after squaring and averaging over the beam distribution the final result isbeam distribution the final result is

2222 pDDDr

ppDp

ppDx

xiix

xii

/

/'

While the machine While the machine requires injection requires injection conditions given byconditions given by ppDp

ppDx

xmmx

xmm

/

/'


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VIVI

Example of beam Example of beam distribution generated distribution generated by dispersion by dispersion mismatch and mismatch and filamentation.filamentation.The effect is on the The effect is on the tails of the beam tails of the beam distribution.distribution.


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VIIVII

Optics errors:Optics errors:Optical parameters of the transfer line at the injection Optical parameters of the transfer line at the injection point are different from those of the ring.point are different from those of the ring.

Consequences:Consequences:The beam performs quadrupolar oscillations (size changes The beam performs quadrupolar oscillations (size changes on a turnon a turn--byby--turn basis). The emittance grows due to turn basis). The emittance grows due to filamentation.filamentation.

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


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VIIIVIII

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


27/0927/09--09/10 October 200909/10 October 2009 2222


IXIX

imi

i

m

m

i

m

m

i

rmsfilafter

rms

F

F

2

21

.

The computation of the emittance blowThe computation of the emittance blow--up due to optics up due to optics errors is very similar to previous cases.errors is very similar to previous cases.

The final result reads:The final result reads:

NB: in this case the emittance growth is proportional to the initial value of emittance

NB: in this case the emittance growth is NB: in this case the emittance growth is proportional to the initial value of emittanceproportional to the initial value of emittance


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XX

Example of beam Example of beam distribution generated distribution generated by optics mismatch by optics mismatch and filamentation.and filamentation.The beam core is also The beam core is also affected as well as affected as well as the tails.the tails.


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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 foil)on 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.

betaaveragetheiswhereL

tcp

cMeVqkrad

p

ppk

2

22 /142

NB: p

ct

represents the scatterer length until time t.

That is why good vacuum is necessary!

NB: NB: pp

ctct

represents the scatterer length until represents the scatterer length until time time t.t.

That is why good vacuum is necessary!That is why good vacuum is necessary!


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IIII

IntraIntra--beam scatteringbeam scatteringMultiple (small angle)Multiple (small angle)

Coulomb scatteringCoulomb scattering between charged between charged

particles.particles.Single scatteringSingle scattering

events lead to events lead to Touscheck effect.Touscheck effect.All three degrees of All three degrees of freedom are affected.freedom are affected.


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IIIIII

Features of IBSFeatures of IBS

For 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 emittances is constant.emittances is constant.Above transition the sum of the emittances Above transition the sum of the emittances always grows.always grows.In any strong focusing lattice the sum of the In any strong focusing lattice the sum of the emittances always grows.emittances always grows.Even though the sum of emittances grows, Even though the sum of emittances grows, emittance reduction in one plane is predicted emittance reduction in one plane is predicted by simulations, but never observed in real by simulations, but never observed in real machines.machines.


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IVIV

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

lyxlyx

F ,,0,,

11

Strong dependence on charge

x,y,l

* are normalised emittances

Nb

of particles/bunch

r0

classical proton radius

Strong dependence on Strong dependence on chargecharge

x,y,lx,y,l

* are normalised * are normalised emittancesemittances

NNbb

of particles/bunchof particles/bunch

rr00

classical proton radiusclassical proton radius


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Others Others

Diffusive phenomena:Diffusive phenomena:Resonance crossingsResonance crossings

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..


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Emittance manipulationEmittance manipulation

Emittance is normally preserved.Emittance is normally preserved.Sometimes, however, it is necessary to Sometimes, however, it is necessary to manipulate the beam so to reduce its manipulate the beam so to reduce its emittance.emittance.

Standard techniques: electron cooling, stochastic Standard techniques: electron cooling, stochastic cooling cooling --> > covered by a specific lecturecovered by a specific lecture..Less standard techniques: longitudinal or transverse Less standard techniques: longitudinal or transverse beam splitting.beam splitting.


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

II

Courtesy R. Garoby -

CERNCourtesy R. Garoby Courtesy R. Garoby --

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


27/0927/09--09/10 October 200909/10 October 2009 3131


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

IIII

Courtesy R. Garoby -

CERNCourtesy R. Garoby Courtesy R. Garoby --

CERNCERN

Measurement results obtained at the CERN Proton Synchrotron

Measurement results obtained Measurement results obtained at the CERN Proton at the CERN Proton SynchrotronSynchrotron


27/0927/09--09/10 October 200909/10 October 2009 3232


manipulation: CERN manipulation: CERN PS multiPS multi--turn extractionturn extraction

––

II

The main ingredients are:The main ingredients are:The beam is split in the transverse phase space The beam is split 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.


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manipulation: manipulation: crossing thirdcrossing third--order resonanceorder resonance


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manipulation: manipulation: crossing fourthcrossing fourth--order resonanceorder resonance

A series of A series of horizontal beam horizontal beam profiles have been profiles have been taken when crossing taken when crossing the fourththe fourth--order order resonance.resonance.

Measurement result

s obtained at the CERN Proton Synchrotron

Measurement results Measurement result

s obtained at the CERN obtained at the CERN Proton SynchrotronProton Synchrotron

Massimo Massimo GiovannozziGiovannozzi CAS CAS --

27/0927/09--09/10 October 200909/10 October 2009 3535

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, Jyvaskyla, Finland , 1992).10 (CAS, Jyvaskyla, 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, Aarhus, 10 (CAS, Aarhus, Denmark 1986).Denmark 1986).

Sources of emittance growth (Hadrons)Introduction - 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 - VISlide Number 12ExercisesInjection process - IInjection process - IIInjection process - IIIInjection process - IVInjection process - VInjection process - VIInjection process - VIIInjection process - VIIIInjection process - IXInjection process - XScattering processes - I Scattering processes - IIScattering processes - IIIScattering processes - IVOthers Emittance manipulationLongitudinal manipulation: LHC beam in PS machine - ILongitudinal manipulation: LHC beam in PS machine - IITransverse manipulation: CERN PS multi-turn extraction – ITransverse manipulation: crossing third-order resonanceTransverse manipulation: crossing fourth-order resonanceSome references

Documents

Sources of emittance growth - CERN · Massimo Giovannozzi CAS - 27/09-09/10 October 2009 11. Scattering through thin foil - VI. The solution of the system is given by This can be