Massimo GiovannozziCAS - 27/09-09/10 October 20091 Sources of emittance growth (Hadrons) M....

Massimo GiovannozziMassimo Giovannozzi CAS - 27/09-09/10 October 2009CAS - 27/09-09/10 October 2009 11

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

M. GiovannozziM. GiovannozziCERN - BE DepartmentCERN - BE Department

Summary:Summary:

Introduction Introduction

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

Scattering through thin Scattering through thin foilsfoils

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

Injection processInjection process

Scattering processesScattering processes

OthersOthers

Emittance Emittance manipulationmanipulation

LongitudinalLongitudinal

TransverseTransverse

Acknowledgements: Acknowledgements:

D. Brandt and D. D. Brandt and D. MöhlMöhl

Acknowledgements: Acknowledgements:

D. Brandt and D. D. Brandt and D. MöhlMöhl


Introduction - IIntroduction - I

The starting point is the well-known The starting point is the well-known Hill’s equationHill’s equation

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

0(s)(s)(s) xKx

22 2 xxxxA

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

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

22 2 DDDDA


Introduction - IIIntroduction - II

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

Why emittance can grow?Why emittance can grow?

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


Introduction - IIIIntroduction - III

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

Machine performance is limited or Machine performance is limited or reduced, e.g. reduced, 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. the rate of collisions per unit time) (i.e. the rate of collisions per unit time) is reduced.is reduced.


Introduction - IVIntroduction - IV

Filamentation is one of the key Filamentation is one of the key concepts for computing emittance concepts for computing emittance growthgrowth

Due to the presence of nonlinear Due to the presence of nonlinear imperfections, the rotation frequency in imperfections, the rotation frequency in phase space is amplitude-dependent.phase space is amplitude-dependent.

After a certain time the initial beam After a certain time the initial beam distribution is smeared out to fill a phase distribution is smeared out to fill a phase space ellipse.space ellipse.


Scattering through thin foil Scattering through thin foil - I- I

Typical situation: Typical situation: Vacuum window between the transfer line and a Vacuum window between the transfer line and a target (in case of fixed target physics)target (in case of fixed target physics)

Vacuum window to separate standard vacuum in Vacuum window to separate standard vacuum in transfer line from high vacuum in circular machinetransfer line from high vacuum in circular machine

IncidenIncident beamt beam

EmergiEmerging ng

beambeam

The particles receive an The particles receive an angular kickangular kick


Scattering through thin foil Scattering through thin foil - II- II

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

Downstream of the foil Downstream of the foil the transformed the transformed coordinates are given coordinates are given byby

ixiixixi

iiii

Appp

Axx

)cos(

)sin(

0

0

xxp xxx

corrrad

pp

rms L

Lq

p

cMeV

1/14

Normalised Normalised coordinatecoordinate

= 0 at the = 0 at the location of the location of the

foilfoil


Scattering through thin foil - Scattering through thin foil - IIIIII

By assuming that:By assuming that:Scattering angle and betatronic phase are Scattering angle and betatronic phase are uncorrelateduncorrelated

Averaging over betatronic phase (due to Averaging over betatronic phase (due to filamentation) is possiblefilamentation) is possible

Using the relationUsing the relation

The final result readsThe final result reads

xrmsrms 2

2

222

0

222

ixixiii ApxA

2

2Arms


Scattering through thin foil - Scattering through thin foil - IVIV

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

Compute all three second-order moments of the Compute all three second-order moments of the beam distribution downstream of the foilbeam distribution downstream of the foilEvaluate the new optical parameters and Evaluate the new optical parameters and emittance using the statistical definitionemittance using the statistical definition

The emittance growth depends on the beta-The emittance growth depends on the beta-function!function!

THE SMALLER THE VALUE OF THE BETA-THE 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 =0):=0):

By squaring and averaging over the beam By squaring and averaging over the beam distributiondistribution

Using the relationUsing the relation

)sin(/11

)sin(/1

)cos()cos(

11

111

xoxoox

xoxoo

AAp

AAx

1

2

12

0

202

12

102

02

22

22

xrms

x

xx

xx

AAp

AAx

2

2Arms


Scattering through thin foil - Scattering through thin foil - VIVI

The solution of the The solution of the

system is given bysystem is given by

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

rms

rmsxx

xrmsrms

0

2

01

0

2

41

4

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

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

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

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

2

1

2

0

0

1

2

02

0

22

0

22

12

A

A

A

AA

x

x

rmsx


-10

10

30

50

70

0 50 100 150 200 250 300

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

ββHH

ββVV

DDHH

DDVV

Stripper locationStripper location

Courtesy M. Martini - Courtesy M. Martini - CERNCERN

Courtesy M. Martini - Courtesy M. Martini - CERNCERN

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

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

A stripping A stripping foil, foil, 0.8 mm 0.8 mm thick Althick Al, is , is located in the located in the low-beta low-beta insertioninsertion


ExercisesExercises

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

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

Compute the Twiss parameters and Compute the Twiss parameters and emittance growth for a emittance growth for a THICKTHICK foil in a foil in a quadrupolarquadrupolar field field

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


Injection process - IInjection process - I

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.


Injection process - IIInjection process - II

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

Consequences:Consequences:The beam performs betatron oscillations The beam performs betatron oscillations around the closed orbit. The emittance grows around the closed orbit. The emittance grows due to filamentationdue to filamentation

Solution:Solution:Change the injection conditions, either by Change the injection conditions, either by steering in the transfer line or using the steering in the transfer line or using the septum and the kicker.septum and the 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 is the best solution.solution.


Injection process - IIIInjection process - III

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

Squaring and averaging Squaring and averaging givesgives

After filamentationAfter filamentation

sinsin

coscos

rrp

rrx

iimx

iim

222

222

rrr

pxr

im

mxmm

2222. 2

121

21 rrrx imfilafter 2.

2rrms

filafterrms


Injection process - IVInjection process - IV

Example of beam Example of beam distribution distribution generated by generated by steering errors steering errors and and filamentation.filamentation.

The beam core is The beam core is displaced -> displaced -> large effect on large effect on emittanceemittance


Injection process - VInjection process - V

Dispersion mismatch: analysis is similar to that Dispersion mismatch: analysis is similar to that for steering errors.for 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 injection conditions; transforming in normalised phase space. Then after squaring and averaging phase space. Then after squaring and averaging over the beam distribution the final result isover the beam distribution the final result is

2222pDDDr

ppDp

ppDx

xiix

xii

/

/'

While the While the machine machine requires requires injection injection conditions given conditions given byby

ppDp

ppDx

xmmx

xmm

/

/'


Injection process - VIInjection process - VI

Example of beam Example of beam distribution distribution generated by generated by dispersion dispersion mismatch and mismatch and filamentation.filamentation.

The effect is on The effect is on the tails of the the tails of the beam distribution.beam distribution.


Injection process - VIIInjection process - VII

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

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

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


Injection process - VIIIInjection process - VIII

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


Injection process - IXInjection process - IX

imi

i

m

m

i

m

m

i

rms

filafter

rms

F

F

2

2

1

.

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

The final result reads:The final result reads:

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

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


Injection process - XInjection process - X

Example of beam Example of beam distribution distribution generated by optics generated by optics mismatch and mismatch and filamentation.filamentation.

The beam core is The beam core is also affected as well also affected as well as the tails.as the tails.


Scattering processes - I Scattering processes - I

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

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

betaaveragetheiswhereL

tc

p

cMeVqk

rad

p

ppk

2

22 /142

NB: NB: ppctct represents the scatterer length represents the scatterer length until time until time t.t.

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

NB: NB: ppctct represents the scatterer length represents the scatterer length until time until time t.t.

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


Scattering processes - IIScattering processes - II

Intra-beam Intra-beam scatteringscattering

Multiple (small Multiple (small angle)angle) Coulomb Coulomb scatteringscattering between between charged particles.charged particles.

Single scatteringSingle scattering events lead to events lead to Touscheck effect.Touscheck effect.

All three degrees of All three degrees of freedom are freedom are affected.affected.


Scattering processes - IIIScattering processes - III

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


Scattering processes - IVScattering processes - IV

Scaling laws of IBSScaling laws of IBSAccurate computations can be performed only Accurate computations can be performed only with numerical tools.with 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 A

qN

E

A

qrN

lyxlyx

F ,,0,,

11

Strong dependence Strong dependence on chargeon charge

x,y,lx,y,l* are normalised * are normalised emittancesemittances

NNbb of particles/bunch of particles/bunch

rr00 classical proton classical proton radiusradius

Strong dependence Strong dependence on chargeon charge

x,y,lx,y,l* are normalised * are normalised emittancesemittances

NNbb of particles/bunch of particles/bunch

rr00 classical proton classical proton radiusradius


Others Others

Diffusive phenomena:Diffusive phenomena:Resonance crossingsResonance crossings

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

Beam-beam -> Beam-beam -> covered by a specific lecturecovered by a specific lecture..

Instabilities -> Instabilities -> covered by a specific lecturecovered by a specific lecture..


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, Standard techniques: electron cooling, stochastic cooling -> stochastic cooling -> covered by a specific covered by a specific lecturelecture..

Less standard techniques: longitudinal or Less standard techniques: longitudinal or transverse beam splitting.transverse beam splitting.


LongitudinalLongitudinal manipulation: manipulation: LHC beam in PS machine - ILHC beam in PS machine - I

Courtesy R. Garoby - CERNCourtesy R. Garoby - CERNCourtesy R. Garoby - CERNCourtesy R. Garoby - CERN

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

Empty

bucket

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

Courtesy R. Garoby - CERNCourtesy R. Garoby - CERNCourtesy R. Garoby - CERNCourtesy R. Garoby - CERN

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

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


TransverseTransverse manipulation: manipulation: CERN PS multi-turn CERN PS multi-turn

extractionextraction – I – IThe main ingredients are:The main ingredients are:

The beam is split in the transverse phase The beam is split in the transverse phase space using space using

Nonlinear magnetic elementsNonlinear magnetic elements (sextupoles ad (sextupoles ad octupoles) to create octupoles) to create stable islandsstable islands..

Slow (Slow (adiabaticadiabatic) tune-variation to cross an ) tune-variation to cross an appropriate resonance.appropriate resonance.


TransverseTransverse manipulation: manipulation: crossing third-order crossing third-order

resonanceresonance


TransverseTransverse manipulation: manipulation: crossing fourth-order crossing fourth-order resonanceresonance

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

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

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


Some referencesSome references

D. Edwards, M. SyphersD. Edwards, M. Syphers: An introduction to the : An introduction to the physics of high energy accelerators, J. Wiley & physics of high energy accelerators, J. Wiley & Sons, NY 1993.Sons, NY 1993.P. Bryant, K. JohnsenP. Bryant, K. Johnsen: The principles of circular : The principles of circular accelerators and storage rings, Cambridge accelerators and storage rings, Cambridge University press, 1993.University press, 1993.P. Bryant: P. Bryant: Beam transfer lines, CERN yellow Beam transfer lines, CERN yellow rep. 94-10 (CAS, Jyvaskyla, Finland , 1992).rep. 94-10 (CAS, Jyvaskyla, Finland , 1992).J. BuonJ. Buon: Beam phase space and emittance, : Beam phase space and emittance, CERN yellow rep. 91-04 (CAS, Julich, Germany, CERN yellow rep. 91-04 (CAS, Julich, Germany, 1990).1990).A. PiwinskiA. Piwinski: Intra-beam scattering, CERN yellow : Intra-beam scattering, CERN yellow rep. 85-19 (CAS, Gif-sur-Yvette, France 1984).rep. 85-19 (CAS, Gif-sur-Yvette, France 1984).A. H. SorensenA. H. Sorensen: Introduction to intra-beam : Introduction to intra-beam scattering, CERN yellow rep. 87-10 (CAS, scattering, CERN yellow rep. 87-10 (CAS, Aarhus, Denmark 1986).Aarhus, Denmark 1986).

Massimo GiovannozziCAS - 27/09-09/10 October 20091 Sources of emittance growth (Hadrons) M....

Documents

Transcript of Massimo GiovannozziCAS - 27/09-09/10 October 20091 Sources of emittance growth (Hadrons) M....