Massimo GiovannozziCAS - 2-14 October 20051 Sources of emittance growth M. Giovannozzi CERN- AB...

Massimo GiovanMassimo Giovannozzinozzi

CAS - 2-14 October 2005 CAS - 2-14 October 2005 11

Sources of emittance Sources of emittance growthgrowth

M. GiovannozziM. GiovannozziCERN- AB DepartmentCERN- AB 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 reduced, e.g. in Machine performance is reduced, e.g. in the case of a collider the the case of a collider the luminosityluminosity (i.e. (i.e. the rate of collisions per unit time) is the rate of collisions per unit time) is reduced.reduced.

It can lead to beam losses.It can lead to beam losses.



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 beam receives an The beam receives 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

)cos(

)sin(

iioxixi

iioii

Appp

Axx

xxpx

corr

rad

p

p

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 - III- III

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

Averaging of the phase, i.e. assuming filamentation Averaging of the phase, i.e. assuming filamentation in the transfer line or the subsequent machinein the transfer 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 2)(

2 rmsrms

2220

222iixiii ApxA

2220

2 22 rmsxx



Scattering through thin foil Scattering through thin foil - IV- IV

Few remarksFew remarksA special case with A 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 - V- V

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

)1

sin(1

/11

)sin(/1

)1

cos(11

)cos(

Aooo

Ax

Aooo

Ax

1

12

0

0

10

2

2

2

22

222

2210

AAx

AAx

rms

2

2Arms



Scattering through thin foil Scattering through thin foil - VI- VI

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

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

rms

rmsrms

2

41

4

01

02NB: 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 1, 1, 11

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 1, 1, 11

21

20

0

1

202

0

220

221

2

A

A

A

AArms



-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)

Stripping foil, Stripping foil, 0.8 mm thick 0.8 mm thick Al, is located Al, is located in the low-in the low-beta insertionbeta insertion



Exercises…Exercises…

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

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



Injection process - IInjection process - I

Two sources of errors:Two sources of errors:Steering errors.Steering errors.

Optics errors (Twiss parameters and Optics errors (Twiss parameters and dispersion).dispersion).

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

In all cases filamentation, i.e. nonlinear In all cases filamentation, i.e. nonlinear imperfection in the ring, is the source of imperfection 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 the filamentationdue to the 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

cossin

coscos

rrp

rrx

iim

iim

222

222

rrr

pxr

im

mmm

rrrx im 212

212

212 21

222 ri



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 emittance emittance growthgrowth



Injection process - VInjection process - V

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

In this case In this case

The final result isThe final result is

2

222

2

1

pDDDr p

2

0

222. /

2

11

pDDD p

rmsfilafter

rms



Injection process - VIInjection process - VI

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

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

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

xx

x’x’Ring Ring ellipseellipse

Transfer Transfer line line ellipseellipse



Injection process - VIIInjection process - VII

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 optics…due to the mismatch of the optics…



Injection process - VIIIInjection process - VIII

iiliil baApabAx cos/sin/ ,

2

//22 baabAr ii

2

//.

baab

rmsfilafter

rms

lml

l

m

m

l

m

m

l

rmsfilafter

rms

F

F

2

.

2

1

Given the initial condition of the beam end of the Given the initial condition of the beam end of the transfer linetransfer line

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

The emittance after filamentation is given byThe emittance after filamentation is given by



Scattering processes - I Scattering processes - I

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

betaaveragewhereL

tc

p

cMeVqk

rad

p

ppk

2/1422

2

NB: NB: ppctct represents the represents the scatterer length until scatterer length until time time tt

NB: NB: ppctct represents the represents the scatterer length until scatterer length until time time tt

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

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



Scattering processes - Scattering processes - IIII

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

Transform momenta of colliding particles into their Transform momenta of colliding particles into their centre of mass system.centre of mass system.Rutherford cross-section is used to compute change Rutherford cross-section is used to compute change of momenta.of momenta.Transform the new momenta back to the laboratory Transform the new momenta back to the laboratory system.system.Calculate the change of the emittances due to the Calculate the change of the emittances due to the change of momenta at the given location of the change of momenta at the given location of the collision.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 possible scattering angles (impact parameters from the size of the nucleus to the beam radius).the size of the nucleus to the beam radius).

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

3.3. Compute the average around the circumference, Compute the average around the circumference, including lattice functions, to determine the change including lattice functions, to determine the change per turn.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 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 grows from the Even though the sum grows from the theoretical point of view emittance theoretical point of view emittance reduction in one plane predicted by reduction in one plane predicted by simulations, but were never observed.simulations, but were never observed.



Scattering processes - Scattering processes - VV

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

Ripple in the power convertersRipple in the power converters

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 (hopefully) conserved…Emittance is (hopefully) conserved…

Sometimes, it is necessary to Sometimes, 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.

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



LongitudinalLongitudinal manipulation: LHC beam manipulation: LHC beam in PS machine - Iin 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: novel multi-turn novel multi-turn

extractionextraction – I – I

The main ingredients of the novel The main ingredients of the novel extraction:extraction:

The beam splitting is not performed using a The beam splitting is not performed using a mechanical device, thus avoiding losses. mechanical device, thus avoiding losses. Indeed, the beam is separated in the Indeed, the beam is separated in the transverse phase space using transverse phase 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.

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

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



TransverseTransverse manipulation: manipulation: novel multi-turn extractionnovel multi-turn extraction

– II– II

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

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

Left: initial phase Left: initial phase space topology. space topology. No islands.No islands.

Four Four islands islands

requirerequire

tune = tune = 0.250.25

Four Four islands islands

requirerequire

tune = tune = 0.250.25



TransverseTransverse manipulation: novel manipulation: novel multi-turn extractionmulti-turn extraction – – IIIIII

Tune Tune variationvariation

Phase Phase space space portraitportrait

Simulation Simulation parametersparameters::

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



A movie to show the A movie to show the evolution of beam evolution of beam distributiondistribution

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

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 - 2-14 October 20051 Sources of emittance growth M. Giovannozzi CERN- AB...

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