NPAC2010-2011 S2

Advanced Concept for AcceleratorsAdvanced Concept for Accelerators

Lecture n°2a

Beam Emittance

Monday 17 January 2011

Olivier NapolyCEA-Saclay, France

2nd Version

10 April 2011

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10 April 2011

References

Books:

An Introduction to the Physics of High Energy AcceleratorsD. A. Edwards, M. J. Syphers (Wiley Series in Beam Physics and Accelerator Technology)

Principles of Charged Particle AccelerationStanley Humphries (John Wiley and Sons, New York, 1986)

Web based lectures: US Particle Accelerator Schools : http://www.lns.cornell.edu/~dugan/USPAS/

CERN Particle Accelerator Schools: http://cas.web.cern.ch/cas/

Nicolas Pichoff SFP : http://nicolas pichoff perso sfr fr/index fichiers/slide0001 htmNicolas Pichoff SFP : http://nicolas.pichoff.perso.sfr.fr/index_fichiers/slide0001.htm

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Collider Parameters, Physical Constants and Notations

E , energy and p , momentum

B , magnetic rigidity (B = p/e)c = 299 792 458 m/se = 1.602 177 33 10-19 C

L , luminosity

Q , bunch charge

mec2 = 510 999 eVµ0

= 4 10-7 N A-2 permeability = 1/µ c2 permittivity

N , number of particles in the bunch (N = Q/e)

nb , number of bunches in the traine

ecm2

2

4

0= 1/µ0c permittivity

re = 2.818 10-15 m

withfrep , pulse repetition rate

x and y , horizontal and vertical emittances

er04

x* and y

*, rms horizontal and vertical beam sizes at the IP

z , bunch length

,, , Twiss parameters

, tune

h d

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, phase advance,

Single Particle Dynamics: Summary

• The motion of individual trajectories in an external electromagnetic field derives from Newton’s law with the Lorentz Force:

pd

),( BE

xdwith , and particle momentum.

• It obeys the laws of Hamiltonian Mechanics:

)( BvEqdtpd

vmp dtxdv

(Hamilton-Jacobi equations)QH

dtPd

PH

dtQd

,

with

d i th d

)( , xAqpPxQ

222)()()( cmAqPcQqVPQH

and is the conserved energy,

where is the electromagnetic potential 4-vector.),( AcV

)()(),( cmAqPcQqVPQH

• Hamilton equations derives from a Least Action Principle based on the Action:

22

)(tt

dtHQPdtL S

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11 tt

Hamiltonian Mechanics and Beam PropertiesHamiltonian Mechanics is valid as long asHamiltonian Mechanics is valid as long as 1. There is no collective effects (space charge, wake fields)2. Synchrotron Radiation is ignored3. There is no beam collimation and no beam losses3. There is no beam collimation and no beam lossesThese conditions are realized, as a first approximation, along accelerators.

Hamiltonian Mechanics allows a powerful description of the beam dynamics using p p y gSymplectic Transformations of the beam particle 6-dimensional Phase Space

Coordinates .

PQ

X

Introducing the 6D Skew Matrix , with the 3D unit matrix,

0

0

3

3

11

J 31

P

Hamilton-Jacobi equations read: Hdtd

XJX

S l i thi t f ti t t d i i th ti l ti f ti lSolving this system of equations amounts to deriving the time evolution of particle trajectories, and therefore the expressions of the transformation which maps the coordinates at the initial time , into at a later time .)( 1tX )( 2tX

)(XX2t1t

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This map is noted , for convenience.)( 12 XX

Some Symplectic Algebra in Even Dimensions D=2,4,6,… 2 1J• The skew matrix is anti-symmetric and such that .

• The Symplectic Group Sp(D) is composed of the symplectic matrices S such

JJ D

2 1J

that :

In other words (and by analogy with orthogonal matrices), they leaves the anti-

JSJS

symmetric quadratic form invariant.

• The matrix itself is symplectic: .

XJX

J )D(SpJy p

• If , then . N.B:

)(p

)d (

)D(SpS )D(SpS

S )()( SlS

JSJS 1

• If , then ; in other words .

• The group Sp(D) is generated by the matrices such that T is a )exp( TJ

1)det( S)D(SpS )D()D( SlSp

symmetric matrix . (3 for D=2,10 for D=4, 21 for D=6)

• In 2 dimensions: Sp(2) Sl (2).

2)1D(D)D(dim Sp

1

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pIndeed, for any regular 2D matrix: )det(1 MJMJM

Symplectic Mapst1st property: Symplecticity of Maps

The transformation which maps the particle coordinates at t0 , into the particle coordinates at t , is symplectic in the sense that its Jacobian

t i i l ti t i

)( 0XX 0XX

)(Xmatrix is a symplectic matrix.

Proof:Let us note the Jacobian matrix of the map .)()( ttM

XJ

)( 00 tX)(tX

0tt

Let us note the Jacobian matrix of the map .

Since is a symplectic matrix,

)()(0

M X

d60 )( 1J tM

0

.00 JJJJ )(t)(t MΤM

HHHdd JJXJJXJ 22

If we show that , then

Proof:

0)( MMdt

d JJJ .)()()()( 00 JJJJJJJ tttt MMMM

MM HHHdtdt

JJX

JJX

XX

J XXX2

0

2

00

)( 222 HHd JJJJJJJJJJ )(

22

MMMM

MMMMMM

HH

HHdt

JJJJ

JJJJJJJJJJ

XX

XX

H2

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0 since is symmetric.H2X

Liouville Theoremd2nd property: the Phase Space Volume occupied by a beam population is

invariant of motion

06

066 )(det XXJX ddtd M

3rd property: Particle Densities are invariant of motionGi ti l d it di t ib ti h th t th

Population 0Population 0Population)(det XXJX ddtd M

XX 6)( ddN )(XGiven a particle density distribution such that , then following the motion of the particles from t0 to t :

XX )( ddN )(X

)()()()( 06

06

0 XXXX ddtdNtdNdN

)( 0tdN)(tdN

t)()(

)()()()(

0

000

XX

P0 P P

0tt

P0 P PYES NO

t0 t0

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Q0 Q Q

Liouville TheoremGlobal Phase Space Volumes (‘areas’) are invariant of Motion:Global Phase Space Volumes ( areas ) are invariant of Motion:

P0 P Plinear motion non-linear motionGaussian-like ellipsoid

t0 t

Q Q Q

P0 P P

Q0 Q Q

Rectangular collimated P0 P P

t0 t

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Q0 Q Q

The Particle Density and its Moments)(XThe beam is modelled by a particle density distribution which generates a

set of characteristic real numbers, its Moments.

Th fi t th t

)(XP

The first three moments 0th order moment, beam charge (or population):

Nd )(6 XX Q

1st order moment, centre of mass (or 6-vector beam centroid):

Nd )(XX

1 6

2nd order moment, beam matrix (6x6 matrix):

XXXXC )(1 6 dN

1

)()( CCXXCXCXΣ

are widely used to characterize the beam transport

)())((1 6 CXCXXX dN

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are widely used to characterize the beam transport.

The Gaussian Particle Density• The Gaussian density distribution in even D dimensions (D=2 4 6)• The Gaussian density distribution, in even D dimensions (D=2,4,6)

)()(

21exp

)d ()2()( 1

DG CXΣCXΣ

X N

is fully characterized by its 3 first moments .

2)det()2( D ΣΣC ,,N

P

• It simplifies to the ‘usual’ Gaussian distribution, assuminga centred beam at a waist (upright ellipse):

2 0 Q

in D=2 dimensions.

Then

20

0 and 0

P

Q

ΣC

22 PQN

P

Q

.

22G 22

exp2

),(PQPQ

PQNPQ

Q

• It is the beam distribution mostly used in Beam Tracking simulation codesto probe the beam transport properties

Example of 2D phase space distribution in the transverse motion variables (x,x’).

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to probe the beam transport properties. courtesy D. Uriot, N. Pichoff

The Uniform (or Water-bag) Particle Density 10for1 u

• Introducing the uniform function P

elsewhere 0

10for 1 uu

the ellipsoidal Uniform density distribution, in even D dimensions

)()()( 121U CXΣCXX cNc

Qis also fully characterized by its 3 first moments .

)()()( 21U

ΣC ,,NQ

• It simplifies to the ‘usual’ Uniform distribution, assuminga centred beam at a waist (upright ellipse):

2 0

P

Q

in D=2 dimensions.

Then

20

0 and 0

P

Q

ΣC

.

2

2

2

2

U 444),(

PQPQ

PQNPQ

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QQ

Emittance represents the phase-space volume occupied by beamThe Concept of Beam Emittance

Emittance represents the phase-space volume occupied by beam particles. The phase-space can be either: 2D [(x,x’), (y,y’), or (z,Energy)] when one concentrates on the beam motion in a 1 Dimensional space [horizontal vertical or longitudinal];motion in a 1 Dimensional space [horizontal, vertical, or longitudinal]; 4D [(x,x’,y,y’), or (x,x’,z,Energy)] when one concentrates on the beam motion in a 2 Dimensional physical space [(hor., vert.), or (hor., long.)];

Example of 4D phase space representation in the transverse motion variables (x,x’,y,y’).courtesy D Uriot N Pichoffcourtesy D. Uriot, N. Pichoff

6D [(x,x’,y,y’,z,Energy)] when one considers the beam motion in the full

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3 Dimensional physical space [(x,y,z)].

Emittance for the Uniform (or Water-bag) Particle Density• For the ellipsoidal Uniform density distribution in 2 dimensions• For the ellipsoidal Uniform density distribution, in 2 dimensions

22

U )( PQNPQ

PQ

,

the full emittance is given by the yellow area: Q 4

22U 444),(

PQPQ

PQ

the full emittance is given by the yellow area: .

Another (preferred) definition is the 1- emittance, or RMS emittance, given by the area enclosed by the 1- dotted ellipse:

PQ 4

22 PQy p

PQ 1

12

2

2

2

PQ

PQ

• These definitions generalize to even D dimensions

, )()()( 121U CXΣCXX cNc

P

with

21U

Q2/

2)det()Volume( DD cS

)d t()V l ( S S

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and with the D-sphere.)det()Volume(1 DS DS

• For the Gaussian density distribution in 2 dimensions

Emittance for the Gaussian Particle Density• For the Gaussian density distribution, in 2 dimensions P

22

exp)( PQNPQ

the full emittance is infinite because the beam distribution extends to infinity and its

Q

22G 22exp

2),(

PQPQ

PQ

the full emittance is infinite because the beam distribution extends to infinity and its area is not compact.Another (practical) definition is the 1- emittance, or RMS emittance, given by the area enclosed by the 1 ellipse: 22area enclosed by the 1- ellipse:

.PQ 1

12

2

2

2

PQ

PQ

P• These definitions generalize to even D dimensions

)()(1exp)( 1 CXΣCXX N

P

with .)det(2/1 D

)()(2

exp)det()2(

)(DG CXΣCX

ΣX

Q

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t )det(1

Emittance represents the phase-space volume occupied by beamThe Concept of Beam Emittance (bis)

Emittance represents the phase-space volume occupied by beam particles. It is one of the most important concept of Accelerator Physics, because:1 Emittance is an invariant of motion in Hamiltonian systems (no1. Emittance is an invariant of motion in Hamiltonian systems (no

collimators, no synchrotron radiation)2. Emittance characterizes the “order” (or “temperature”) of the beam

di t ib tidistribution.The lower the emittance, the easier it is to manipulate the beam through

beam pipe apertures, collimators, focus points, RF buckets, etc...p p p , , p , ,Beams with too large transverse emittance must be “cooled” either with

“cooling rings” with synchrotron radiation, or “electron cooling” using intra beam scattering with an external low emittance electron beamsintra-beam scattering with an external low emittance electron beams.

Example of 2D phase space p p prepresentations in the transverse motion variables (x,x’) and (y,y’).courtesy D. Uriot, N. Pichoff

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Some Algebra about the Beam Matrix1st propert is a s mmetric definite positi e matriΣ1st property: is a symmetric definite-positive matrixΣ

)()( CXCXΣ

2nd property: is strictly positive (except for the point like distribution) and can beΣ 0)( and 2

000 CXXXΣXΣΣ2nd property: is strictly positive (except for the point-like distribution) and can be inverted.

3rd property: Normal form of :

Σ

Σp p y

,symplectic with

SSΕSΣ

0 , 0000

and 2

13

i

ε

ε00ε

Ε00 3

3

ε0

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Some Algebra about the Beam Matrix3rd propert Normal form of Σ3rd property: Normal form of

Proof:

Introducing the Square Root Matrix such that and the

Σ

21Σ 2121 ΣΣΣ Introducing the Square Root Matrix such that , and the anti-symmetric matrix , one can find an orthogonal matrix Osuch that

2121 ΣJΣA Σ ΣΣΣ

~ith~

ε0

AOAOA(kind of Jordan normal form for anti-symmetric matrices)

0ε

with

0εAOAOA

Then, introducing

one can show that

ε00ε

ΕΣOΕS with 2121

1)

2) 6D generalizedΕOΣJΣOΕSJS 21212121ΕΕOΣΣΣOΕSΣS 21212121

Twiss parametersPhase AdvanceEmittance

JΕAΕ

ΕOAOΕ

21121

21121

~

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JΕAΕ

Linearized Motion• An accelerator is designed around a reference trajectory (closed orbit ina circular accelerator), which is, if ibl l ti

)(ref tX

if possible, a planar curve connectingstraight lines (zero field on axis) andcircle arcs (uniform B field normal to the plane)

• For small deviations about the accelerator reference trajectory , the

transformation map can be linearized into:

)(ref tX)( 0XX

)))((())((),()( 20ref00ref00ref tOtttt XXXXRXX

Xwith a symplectic matrix

• At the first order, the beam centre vector and the beam matrix transform as:

))((),( 0ref0

0 ttt XXXR

At the first order, the beam centre vector and the beam matrix transform as:

)()(

))((),()( 0ref00ref

tttt

tttt

RΣRΣ

XCRXC

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),(),( 000 tttt RΣRΣ

Intrinsic Emittances• The 3 quantities are called the Intrinsic Emittances : they are invariant of the motion, in the linearized motion approximation.

T l l t th i t i i itt th 2 ti

321 ,,

• To calculate these intrinsic emittances, one uses the 2 properties:

)()det( 2321Σ

D=6 dimensions: ???

)tr()1(2)tr()tr()( 2221 nnnn εJEJΣSJΕSJΣ

D=6 dimensions: ???

D=4 dimensions:2

)det(16)(tr)tr(1 222 ΣJΣJΣ

)(2)tr(

)()det(22

21

2

221

JΣ

Σ

)det(16)(tr)tr(1

)det(16)(tr)tr(2

2222

1

ΣJΣJΣ

ΣJΣJΣ

D=2 dimensions:

where A is the area of the 2D ellipse A )det(Σ

)()()(

22

11 XΣX

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where A is the area of the 2D ellipse )( 1XΣX

Switching BasisWe ( ill) s itch to a ne coordinates basis

1QWe (will) switch to a new coordinates basis

2

1

1

QPQ

Q

3

2

2

QPQ

PQ XX

000010

in which is the skew matrix,

3

3

PQ

001000000001

Jin which is the skew matrix,

010000100000000100

J

1

0000000000

and the normal form of the beam matrix is

010000

2

1

000000000000000

Ε

3

2

000000000000000

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300000

Projected EmittancesCase of Unco pled Beam Matri• Case of Uncoupled Beam Matrix

In some ideal locations along perfect beam lines, the beam matrix is uncoupled in the 3 physical dimensions (x,y z):

x 00Σ

Then the 3 emittances are given by

z

y

x

Σ000Σ0Σ

Then, the 3 emittances are given by

• Case of 4D Transverse Coupled Matrix

z

)det( ,)det( ,)det( zzyyxx ΣΣΣ • Case of 4D Transverse Coupled MatrixIn reality, either by concept or by the effect of misalignment errors, the beam matrix is x-y coupled

xyx ΣΣ

Σ .

Accelerator instrumentation usually built in horizontal and vertical frames allows to

yxy

y

ΣΣΣ

Accelerator instrumentation, usually built in horizontal and vertical frames, allows to measure the projected emittances

)det(~ ,)det(~yyxx ΣΣ

~

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which are not invariant of motion, and such that yxyx ,,

The Beta Functions of the Beam (D=2) • The 2D beam matrix is parameterized as follows:• The 2D beam matrix is parameterized as follows:

with

xx

xx

Σ 1det 2

xxx

xx

By definition , if .

xx xx

xxxxxxxx pxpx 22 , , 0C

• The 2D beam ellipse can then be parameterized by11 XΣX x

xxxxxx pxpx 22 2‘Courant-Snyder Invariant’

• The remarkable points can be• The remarkable points can be recovered from the above equation, and also from the search for extrema:

0)()( dd e.g.

0)()( xxxxxxx dppxdxpx

xxx pxdx 2

max 0)(0

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xxx 2max

Normalized Variables (D=2) The 2D beam matri takes the normal form• The 2D beam matrix takes the normal form:

with

SSSSΣ x

xx

0

0)1)det((

101

Sx

S

This suggests introducing the Normalized Variables

x0 1 xx

xxxx

x

x px

px

vu

011S

such that

11 1SΣSΣuuuu

21SΣSΣ xxu vvvv

• In these variables, the particle motion is described by simple rotations around a circle in (u,v) ‘phase space’ (cf. Beam Optics section).But most of the information is retained in the -functions.

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The General Form of the Beam Matrix (D=6)In Final Focus Systems several simplifications are in force :In Final Focus Systems, several simplifications are in force :1. the longitudinal motion is frozen: the arc length differences between trajectories

are negligible and the bunch length is constant.2 the transverse (x y) motions are not coupled2. the transverse (x,y) motions are not coupled.

The resulting 6D beam matrix is parametrized as follows:

2000 xxxxx

2

2

000000

x

yyyyy

pxxxx

xxxxx

2

2

00000000

y

z

pyyyy

yyyyyΣ

with the dispersion coefficients measure the correlations

22222 0

00000

yx pypx

z

),,,( pypx with the dispersion coefficients measure the correlations between transverse position, momentum and energy, e.g.

.xpx

x

pr

xr

22262216 ,

),,,(yx pypx

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x

x

xp

p

pxx

x

226226

Sources of Emittance GrowthLinear Collider beams are experiencing many sources of emittance growth (orLinear Collider beams are experiencing many sources of emittance growth (or emittance degradation), in particular through the Beam Delivery Systems :

• Non Hamiltonian Mechanics :S (S ) Synchrotron Radiation (SR) : an interesting case because it works both sides:

1. at low energy, big injected emittances (transverse and longitudinal) are damped (i.e. reduced) down to the equilibrium emittances of Damping Rings;

2 at high energy the longitudinal emittance growth dominates due to the strong energy2. at high energy, the longitudinal emittance growth dominates due to the strong energy dependence of SR, and couples to the transverse motion through magnet chromaticity.

Collective Effects, e.g. beam-beam interactions

N li ti

cavitiestail performs

• Non-linear motion a source of the Linear Emittance growth but does not violate Liouville Theorem

tailaccelerator axis

tail performsoscillation• Coupling

a source of 4D projected emittance growth but not of 4D intrinsic emittance growth

head

head

tail

y

• Transverse Wakefields a source of 4D projected emittance growth but not of 4D intrinsic emittance growth

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5 km 10 km but not of 4D intrinsic emittance growth.

Beam Line Coordinate SystemParticle trajectories are parameteri ed ith respect to a Reference TrajectorParticle trajectories are parameterized with respect to a Reference Trajectory

associated to a Reference Momentum .)(ref tR

)()( refref tRmtP

One introduces the curvilinear coordinate system defined by the Moving Frame of (planar) Reference Trajectory, such that :)ˆ,ˆ,ˆ( syx

R

dtRds ref

arc length

R

Rsref

ref

tangent vector

xd

sd ˆˆ

sxd

dsˆˆ

TRANSPORTt di

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ds K. Brown et al = curvature radius

Particle Motion CoordinatesThen the particle motion is parameteri ed b thro gh))(( syx Then, the particle motion is parameterized by through))(,,( syx

))()(()(ˆ)(ˆ)()()( ref ysyxsxsRsR

where are the timeand position of the particle when

))()(()( ref sstst )(),( sRst

and position of the particle whencrossing the plane normal to thereference trajectory at .)(sRref

Note 1: for particlesahead of the reference particle.

N t 2 f l f

0

Note 2: for a planar reference trajectory:

ydyxdxsdsxhRd ˆˆˆ)1(

with 1

h

ydyxdxsdsxhRd )1(

TRANSPORTt di

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K. Brown et al = curvature radius

The Resulting Hamiltonian To st d the e ol tion of Particle Motion Beam Ph sicists like to trade the TimeTo study the evolution of Particle Motion, Beam Physicists like to trade the Time variable t with the Arc Length variable s defined by :

t

After some Canonical manipulations this leads to a new Hamiltonian

t

duuRs0

)(ref

)( PQG

After some Canonical manipulations, this leads to a new Hamiltonian

))(/)()(1(),( 22222 AqPcmcqVPqAhQPQG tsx

),( PQG

in terms of the conjugate variables:

qApx xx

H

qApPyQ yy

,

Hamiltonian mechanics and Symplecticity provide powerful tools to study beam dynamics over very long times, like the beam orbits in circular colliders.

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y y gThey are of lesser importance for the design of linacs and transfer beam lines.

TRANSPORT Coordinate SystemFor transfer lines like BDS and linear accelerators we turn to more familiar but• For transfer lines, like BDS and linear accelerators, we turn to more familiar, but

not symplectic conjugate variables, the TRANSPORT coordinates:

x

ith

yx'

X

)1(' vvxhdsdxx

cz

sx

a time variable !

transverse ‘angles’with

zyy'

X

refref /)(

'

ppp

vvdsdyy sy

transverse angles

relative momentum deviation

refref )( ppp deviation

These coordinates are to the first order proportional to symplectic conjugate• These coordinates are to the first order proportional to symplectic conjugate variables:

pxmcpyxovvx ')),(1(' 22

y

x

yy

xx

pypx

mcpyxovvy

mcpyxovvx')),(1('

)),(1(22

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Geometric and Normalized EmittancesGeometric emittances of nco pled motion• Geometric emittances of uncoupled motion

In TRANSPORT coordinates, the ‘geometric’ emittances calculated from the beam matrixmatrix

with)det( ,)det( yyxx ΣΣ

y

x

ΣΣ

Σ0

0

is not invariant under acceleration. y

• Normalized emittances

Normalized emittance invariant under linearized motion are given byNormalized emittance, invariant under linearized motion, are given by

)det( ,)det( ,, yyNxxN ΣΣ

For high energy beam lines and

)det()det( ΣΣ1

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yyyNxxxN )det( ,)det( ,, ΣΣ