An Introduction to

Beam DynamicsBeam DynamicsC. Zhang

6th OCPA Accelerator School

July29 - August 7, 2010 • Beijing

G1BG1B

PrefacePrefaceBeam dynamics is the study of particle beams,

their motion in environments, involving external electro-magnetic fields and their interactions, including the interaction of beams with matters, of beams with beams, and of particle beams with radiation. Evolving from concepts and ideas derived from classical mechanics, electromagnetism, statistical physics, and quantum physics. The study of beams is opening up a very rich field, with new effects being discovered and new types of beams with novel characteristics being realized. Basic knowledge of the beam physics is briefly introduced in this lecture for the students who are preparing to work in the field of Particle accelerators.

ContentsContentsBasic ConceptsTransverse MotionLongitudinal MotionCollective EffectsLepton and Hadron

1. Basic Concepts1. Basic ConceptsConstants & Relations

Motion in E-M Fields

Linear accelerators

Synchrotrons

1.1 Constants & RelationsSpeed of lightRelativistic energyRelativistic momentum

E-p relationship

Kinetic energyEquation of motion under Lorentz force ( ) ( )BvEqv

dtdmf

dtpd rrrrrr

×+=⇒= γ0

c=2.99792458×1010 cm/sec

E=mc2=m0γc2

p=mv=m0γβc

β=v/c γ=(1-β2)-1/2

E2/c2=p2+m02c2

Ultra-relativistic particles: β≈ 1, E≈ pc

T=E-m0c2=m0c2(γ −1)

Constants & RelationsConstants & Relations (cont.)(cont.)

Planck constant

Electron charge

Electron volts

Energy in eV

Energy and rest mass

Electron

Proton

[ ]e

cme

mcE2

02

eV γ==

e=1.6021773×10-19 Coulumbs

1eV=1.6021773×10-19 Joule

1eV/c2=1.78×10-36 kg

m0, e=0.51099906 MeV/c2=9.1093897×10-31 kg

m0, p=938.2723 MeV/c2=1.6726231×10-27 kg

h=6.626075×10-34 J s

1.2 1.2 Motion in EMotion in E--M FieldsM FieldsGoverned by Lorentz force

Acceleration along a uniform electric field (B=0)

[ ]BvEqdtpd rrrr

×+=

( ) EpE

qcBvEpE

qcdtdE

dtpdpc

dtdEE

cmcpE

rrrrrr

rr

r

⋅=×+⋅=⇒

⋅=⇒

+=

22

2

420

222

cvtmeEx

vtz<<

⎪⎭

⎪⎬⎫

≈

≈for path parabolic

22

0γ

A magnetic field does not alter a particle’s energy. Only an electric field can do this.

C.R. Prior: The Physics of Accelerators

1.3 1.3 Linear Linear AcceleratorsAcceleratorsSimplest example is a vacuum chamber with one or more DC accelerating structures with the E-field aligned in the direction of motion.

Limited to a few MeVTo achieve energies higher than the highest voltage in the system, the E-fields are alternating at RF cavities.

Avoids expensive magnetsNo loss of energy from synchrotron radiation (q.v.)But requires many structures, limited energy gain/metreLarge energy increase requires a long accelerator

BEPC electron-positron linac

SNS Linac, Oak Ridge

Structure 1:Structure 1:Travelling wave structure: particles keep in phase with the accelerating waveform. Phase velocity in the waveguide is greater than c and needs to be reduced to the particle velocity with a series of irises inside the tube whose polarity changes with time. In order to match the phase of the particles with the polarity of the irises, the distance between the irises increases farther down the structure where the particle is moving faster. But note that electrons at 3 MeV are already at 0.99c.

Structure 2:Structure 2:

A series of drift tubes alternately connected to high frequency oscillator.Particles accelerated in gaps, drift inside tubes .For constant frequency generator, drift tubes increase in length as velocity increases.Beam has pulsed structure.

1.4 1.4 SynchrotronSynchrotronssPrinciple of frequency modulation but in addition variation in time of B-field to match increase in energy and keep revolution radius constant.

Magnetic field produced by several bending magnets (dipoles), increases linearly with momentum. For q=e and high energies:

Practical limitations for magnetic fields => high energies only at large radius

e.g. LHC: E = 7 TeV, B = 8.36 T, ρ = 2.7 km

ωnf =

ρω v

EqBc

==2

qBp

=ρ

[m] [T] 0.3[GeV]so ρBEceE

epBρ ≈≈=

Types of Types of SynchrotronSynchrotronssStorage rings: accumulate particles and keep circulating for long periods; used for high intensity beams to inject into more powerful machines or synchrotron radiation factories.Colliders: two beams circulating in opposite directions, made tointersect; maximises energy in centre of mass frame.

2. Transverse Motion2. Transverse Motion(Prof. S.Y.Lee)(Prof. S.Y.Lee)

Motion Description

Bending

Focusing

Hill’s Equation

Phase Space

Lattice

Dispersion

Orbit Distortion

Coupling

Non-linearity

2.12.1 Motion DescriptionMotion Description

x- horizontaly- verticals- longitudinal

Transverse

2.2 Bending2.2 Bending(Prof. (Prof. C.ChC.Ch. Wang). Wang)

In a synchrotron, the confining magnetic field comes from a system of several magnetic dipoles forming a closed arc. Dipoles are mounted apart, separated by straight sections/vacuum chambers including equipment for focusing, acceleration, injection, extraction, collimation, experimental areas, vacuum pumps.

qBp

=ρ

BEPCII dipole

x

By

By increasing E (hence p) and B together in a synchrotron, it is possible to maintain a constant radius and accelerate a beam of particles.

A sequence of focusing-defocusing fields provides a stronger net focusing force.

x

By

BEPCII quadrupole

Quadrupoles focus horizontally, defocus vertically or vice versa. Forces are linearly proportional to displacement from axis.A succession of opposed elements enable particles to follow stable trajectories, making small (betatron) oscillations about the design orbit.

2.3 Focusing 2.3 Focusing (Prof. (Prof. C.ChC.Ch. Wang). Wang)

2.4 Hill2.4 Hill’’s Equations EquationEquation of transverse motion

Drift:

Quadrupole:

Dipole:

Sextupole:

Hill’s Equation:

0,0 =−′′=+′′ ykyxkx

02,0)( 22 =−′′=−+′′ xykyyxkx ss

0,0 =′′=′′ yx

0,012 =′′=+′′ yxx

ρ

0)(,0)( =+′′=+′′ yskyxskx yx

Solution of the HillSolution of the Hill’’s Equations EquationHill’s equation (linear-periodic coefficients)

Property of machine

β(s) – envelope function

Property of the beam

ε − EmittancePhysical meaning (H or V planes)

))(sin()()( 0φφεβ += sssu

0)(2

2=+ usk

dsud

dxdB

Bsk y

ρ1)( −=

∫=)(

)(s

dssβ

φ

)()( ssu εβ=env

)(/)( ssu βε=′

like restoring constant in harmonic motion, the solution is

ConditionMaximum excursions

Envelope

and u denotes x and ywhere

General equation of ellipse is

2.5 Phase Space2.5 Phase SpaceUnder linear forces, any particle moves on an ellipse in phase space (x, x´).

Ellipse rotates in magnets and shears between magnets, but its area is preserved:

Emittance

x

x´

x

x´

εγαβ =+′+′ 22 2 xxxx

222 xxxxrms ′−′=εAcceptance: Ax,y > εx,y

max

2

max

2

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎟⎠

⎞⎜⎜⎝

⎛=

yy

xx

YAXAββ

,

α, β, γ are functions of distance (Twiss parameters), and ε is a constant. Area = πε.

For non-linear beams can use 95% emittance ellipse or RMS emittance

(statistical definition)

2.6 2.6 LatticeLattice(Prof. C.C. Kuo)(Prof. C.C. Kuo)

The pattern of focusing magnets, bending magnets and straight sections connecting in between;It has a strong influence on aperture of vacuum chambers and thus other systems of the accelerator.

Matrix formalismMatrix formalism

[ ]⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

Δ−ΔΔ++Δ−

−

ΔΔ+Δ=

φαφββ

ββφααφαα

φββφαφβ

β

sin)(cos)()(

sin))(1(cos))((

sin)()sin(cos)(

)/(0

0

00

000

12

sss

ss

ss

ssM

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

+=

μαμμγμβμαμsincossin

sinsincos)/(

0

0000 ssM

As a consequence of the linearity of Hill’s equations, we can describe the evolution of the trajectories in a lattice by means of linear transformations

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛)(')(

)/()(')(

)(')()(')(

)(')(

0

012

0

0

sysy

ssMsysy

sSsSsCsC

sysy

In terms of the amplitude and phase function the transfer matrix is

where α(s)= dβ/ds .

βαγ

21 +=

For a periodic machine the transfer matrix over a turn reduces to

Example: FODO LatticeExample: FODO LatticeThe matrix for one period between mid-planes of F magnet in thin lens approximation is

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+=

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−±−

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛±⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

μαμμγμβμαμsincossin

sinsincos

2/1)2/1(2/2/1(22/1

12/101

101

1/101

101

12/101

222

22

,

fLfLfLfLLfL

fL

fL

fM yx

m

mm

0

1 sin

)2/sin(12

)2

1(cos

,

,,

2

21

,

=

=±

=

−= −

yx

yxyx

yx

L

fL

α

γμμβ

μ

πμν 2/)( cN=

2.7 2.7 DispersionDispersionThe dispersion has its origin that a particle of higher momentum is deflected through a lesser angle in a bending magnet.

The solutions of the equation can be written in terms of the optics functions

02

1)(1''ppxskx Δ

=⎟⎟⎠

⎞⎜⎜⎝

⎛−+

ρρ

)())(cos()()(0

0 sDppsssx Δ

+−= φφεβ

It was shown that the equations of motion of a charged particle is a linear Hill’s equation

[ ] sdsssss

D )()(cos)()(

sin2)(

φφπνρβ

πνβ

−−= ∫

2.8 2.8 OrbitOrbit DistortionDistortionDipole errors may cause orbit distortion of particle beam.

Element Source Field Error Deflection Direction

Drift space (l=d) Stray field ⟨ΔBs⟩ ⟨ΔBs⟩/Bρ x, y

Field error ⟨ΔB/B ⟩ θ⋅⟨ΔB/B ⟩ x Dipole (angle=θ )

Tilt ⟨Δθz ⟩ θ⋅⟨Δθz ⟩ y

Quadrupole (Kl) Displacement ⟨Δx,y ⟩ Kl⋅⟨Δx,y ⟩ x, y

Similar to dispersion case, the equation of motion is written as

BsB

Bkyy )(1'' Δ

=+ρ

The solutions are )())(cos()()( 0 sysssy cod+−= φφεβ

[ ] sdsssB

ssBsyCOD )()(cos

)()()(

sin2)(

φφπνρ

βπν

β−−

Δ= ∫

2.9 Linear 2.9 Linear CouplingCouplingCoupling is the phenomena that energy exchange between two oscillators. In accelerators, the horizontal and vertical motions are also coupled:

xMxMkykyyMyMkxkx

y

x′−′−−=+′′

′−′+−=+′′

)2/()2/(

, 2

1

0⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−

∂∂

−=y

Bx

BB

k yxρ ρB

BM s−=Where

x-y coupling can be compensated with skew quadrupoles and anti-solenoids.Measured with tune approaching.

22

2

)(2)()(

νννν

εε

κΔ+−

Δ==

yxx

y

2.10 2.10 NonNon--linearitylinearity

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

+=⎟

⎟⎠

⎞⎜⎜⎝

⎛−+ ∑

=

M

n

nnn izxn

sijskxsksds

xd

2122

2)(

!)()(Re)(

)(1

ρ

Hill’s equation with higher order terms in the expansion of magnetic field

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

+−=+ ∑

=

M

n

nnn izxn

sijskzskds

zd

212

2)(

!)()(Im)(

The Hill’s equations acquire additional nonlinear terms:

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

+=+ ∑

=

M

n

nnnxz izx

nsijskBiBB

100 )(

!)()(

ρ )0,0(00

1n

yn

n x

BB

k∂

∂=

ρ Normal multipoles

)0,0(00

1nx

n

n xB

Bj

∂

∂=

ρSkew multipoles

There is no analytical solution available in general, and the equations have to be solved by tracking or perturbative analysis.

Analytical and Tracking StudiesAnalytical and Tracking StudiesStable and unstable fixed points appears which are connected by separatrices.Islands enclose the stable fixed points.On a resonance the particle jumps from one island to next and the tune is locked at the resonance value.Region of chaotic motion appear.The region of stable motion, called dynamic aperture, is limited by the appearance of unstable fixed points.

Qx = 1/5

3. Longitudinal Motion 3. Longitudinal Motion (Prof. S.Y.Lee)(Prof. S.Y.Lee)

RF Cavities & Acceleration

Synchrotron Oscillation

Bunches and Buckets

γ-Transition

3.1 RF Cavities & Acceleration3.1 RF Cavities & Acceleration(Prof. R.F. Wang)(Prof. R.F. Wang)

Necessary conditions for acceleration.Both linear and circular accelerators use electromagnetic fields oscillating in resonant cavities to apply the accelerating force.Linac– particles follow straight path through series of cavities.Circular accelerators – particles follow circular path in B-field and particles return to same accelerating cavity each time around.

sqVE ϕsin0=Δ

3.2 Synchrotron Oscillation3.2 Synchrotron Oscillation

( )sEfhV

φφγβ

ηπφ sinsin2

20

200 −−=&&

02 22

0

200 =Δ+=Δ+ φφφ

γβηπφ sΩ

EfhV &&&&

2

cos 02

0

0 fE

hVf ss γβπ

φη=

γβπφη

ν 20

0

2 cos

EhV

ff ss

s ==

This is a biased rigid pendulum

Synchrotron tune

For small amplitudes

Synchrotron frequency

3.3 Bunches and Buckets3.3 Bunches and Buckets

( )ss

s φφφ

φ sinsincos

2−

Ω−=&&

( ) .sincoscos2

22consts

s

s =+Ω

− φφφφ

φ&

( ) ( ) ( )[ ].sincoscos

sincoscos2

222

ssss

ss

s

s φφπφπφ

φφφφ

φ−+−

Ω−=+

Ω−

&

For large amplitudes

Integrated becomes an invariant

The second term is the potential energy function, and the equation of each separatrix is

And the half height when is

( ) ( )21

0max

⎭⎬⎫

⎩⎨⎧

±=Δ ss

s GEh

eVEE φηπ

β ( ) ( )[ ]ssssG φφπφφ sin2cos2 −−=

0=φ&&

Bucket size should be larger than bunch size (ΔE/Es and Δφ (Δτ, Δz))

3.4 3.4 γγ--TransitionTransitionRecall synchrotron focusing strength:

γβφπη

20

2002 cos 2

EfhVΩ s

s =

02

1//

RD

dpdR

Rp

dpdp

pdpfdf

−=+==γ

ββ

η

If η⋅cosφs<0, then Ω2<0, the oscillation gets unstable, where

ppDRR o

Δ⋅+=

The η changes from positive to negative at transition γ :

DR

RD

tr0

02 or 01

==−= γγ

η

The motion is stable for φs∈(0, π/2) , and unstable φs∈(0, π) for γ<γtr, and vice versa for γ>γtr.To cross γtr should be avoided in design or apply the phase-jump technique.

High EnergyLow energy

φs

4. Collective Effects4. Collective Effects(Prof. A.W Chao)(Prof. A.W Chao)

Space ChargeWake Fields and ImpedanceCoherent InstabilitiesBeam-beam EffectsLandau Damping

rEE =→

φBB =→

η…charge density in C/m3

λ... constant line charge π a2ηI…constant current λβc = π a2ηβca…beam radius

Electric Magnetic

0div

εηE =

→ →→= JμB 0curl

→→→→

∫∫∫ = AdBcurldφrB→→→

∫∫∫∫∫ = SdEdVE div

Volume element

Current density (βcη)

cylinder radius r cross sectionlength l radius r

πβcηrμrπBφ2

02 =rππ Eεηπlr r 20

2 =

202 a

rβcπε

IEr = 2202 a

rcπε

IBφ =

s Apply these integrals over

cross section

rx

y

φ

ErBφ

aAdr

φdrI

s

4.1 Space Charge4.1 Space Charge

For β 1 (γ>>1) FE=qEr =-FB=-q⋅Bφ⋅v so ΣF=0

Incoherent Tune Shift in a SynchrotronIncoherent Tune Shift in a Synchrotron

( ) 0)()( =++′′ xsKsKx SC

(s)dsβsKπ

dssβskπ

Δν x

πR

x

πR

xx )()()( SC∫∫ ==2

0

2

0 41

41

νx0 (external) + Δνx (space charge)

For small “gradient errors” kx

x

xxπR

x cγeβRIr

(s)a(s)β

cγeβRIrds

a(s)β

cγeβIr

πΔ

εν 33

0233

02

2

033

0241

−=−=−= ∫

Using I = (Neβc)/(2πR) with N…number of particles in ring εx,yare x, y emittance containing 100% of particles32

0

2 γβπNrΔ

x,yx,y ε

ν −=

“Direct” space charge, unbunched beam in a synchrotronVanishes for γ >>1Important for low-energy machinesIndependent of machine size 2πR for a given N

Beam not bunched (so no acceleration)Uniform density in the circular x-y cross section (not very realistic)

4.2 Wake Fields and Impedance4.2 Wake Fields and Impedance

θθ

θθθθ

mrzWeIzrF

mmrmrzWeIzrFm

mm

mmm

cos)(),,(

)sinˆcosˆ()(),,(

||

1

′−=

−−= −⊥r

Impedance Z = Zr + iZi

Induced voltage V ~ Iw Z = –IB ZV acts back on the beam

Intensity dependant

Wake field will be driven by a beam when there is a discontinuity in the vacuum chamber:

Impedances are just Fourier transforms of wake functions:

)()( , )(/

)( /||/ zWevdzZzWe

vdz

cviZ m

vzimm

vzim ′== −∞

∞−−⊥ ∫∫ ωω ωω

4.3 Instabilities4.3 Instabilities LinacsLinacsEnergy variation along the bunch length

Induced by longitudinal wake fieldsIt can be compensated for by properly phasing.

Transverse beam break-upInduced by transverse wake fields

zdszxzzWzL

NeszxsksEszxdsdsE

dsd

z′′−′′=+ ∫

∞ ),()()(),()()()],()([ 1

22 λ

It can be cured by the BNS damping: applying stronger focusing on tail than head of the bunch to balance the wake.

i

f

f EE

LEklWNe

kk ln

4)(

21

2−≈

Δ

InstabilitiesInstabilities Circular acceleratorsCircular acceleratorsNegative Mass InstabilityRobinson instability

Caused by fundamental cavity modeBeam will be stabilized by properly tuning the cavity (hω0> ωrf for γ>γt).

Head-tail InstabilityCaused by transverse wake fieldBeam will be stabilized by properly setting chromaticity (ξ> 0 for γ>γt).

Strong head-tail InstabilityCaused by transverse wake as counterpart of beam break-up in a linac.Once the threshold is exceeded the instability tends to grow very fast.

Microwave instability – increasing energy spread and bunch lengthPotential-well distortion – increasing bunch length

Coupled bunch instability multi-bunch effects caused by long-range transverse & longitudinal wakesCure: control of impedance and applying feedback systems

seE

∂∂

−=λ

γπε 204

4.4 Landau Damping4.4 Landau DampingTwo oscillators excited together become incoherent and give zero centre of charge motion after a number of turns comparable to the reciprocal of their frequency difference.

External excitation is outside the frequency range of the oscillators: No damping

External excitation is inside the frequency range of the oscillators The integral has a pole at Ω = ω: Landau damping

Excitation with ω

4.5 Beam4.5 Beam--beam Effectsbeam EffectsW = Energy available in center-of-mass for making new particles For fixed target :

Ec.m. ≅ 2mTEB... and we rapidly run out of money trying to gain a factor 10 in c.m. energy

But a storage ring , colliding two beams, gives:

E c .m . ≅ 2 E B

Problem: Smaller probability that accelerated particles collide .... "Luminosity" of a collider

L = N1N 21A

βc2πR

≈ 1029 .. .1034 cm −2s −1

Beam-beam interactions are non-linear and collective effects22 2/

22)( σ

πσρ rener −= ( )( ) 11

222 2/2

0

2σβ

επr

r er

neF −−±−=

For small amplitude particles, the beam-beam force can be approximated as a focusing lens in both x and y planes with focal length f given by

)(

2)(,

1,

yxyx

eyx

Nrfσσγσ +

=−

The incoherent linear beam-beam tune shifts are

Exhaustive study can be carried out with beam-beam simulation.

( ) Nr

f yxyxyx

yxeyxyxyx ,

,

,,1,, )(24

ξσσπγσ

βπ

βν =

+==Δ

∗∗−

( )∗

+=

y

yxxc

r

RfL

β

ξξεπ2

0

21

With the maximum beam-beam tune shift, luminosity is

5. Leptons and Hadrons5. Leptons and HadronsLeptons vs. HadronsSynchrotron radiationRadiation dampingRadiation excitationRings vs. Linacs

e+ e-

5.1 Leptons vs. Hadrons5.1 Leptons vs. HadronsHadron Collider (p, ions):

Composite nature of protonsCan only use pt conservationHuge QCD background

LEP：Z0 → 3 jets

LHC: H → ZZ → 4μ

p p

Lepton Collider:

Elementary particlesWell defined initial stateBeam polarizationProduces particles democraticallyMomentum conservation eases decay product analysis

RHIC: Au+Au →……

ppeff

eeeff EE ⋅≈

−+

10

5.2 Synchrotron radiation5.2 Synchrotron radiationThe electromagnetic radiation emitted when the charged particles are accelerated；

ρεγ

0

42

0 3eU =

The particle loses the energy of U0 in one turn；

)()(46.88)(4

0 mGeVEkeVUρ

=For electrons:

For protons:

e.g. LEP2: E=100GeV，ρ=3100 m, U0=1.85 GeV

e.g. LHC: E=7000GeV，ρ= 2804 m, U0= 6.66 keV

)()(1078.7)(4

120 m

GeVEkeVUρ

−×=

5.3 Radiation damping5.3 Radiation dampingSynchrotron radiation loss is compensated by the RF fields:

)sin(00 seVU ϕ=

Longitudinal: higher the energy of particle more SR loses;

This will cause radiation damping.

02 22

2=++ εωε

τε

ss dt

ddtd

Vertical: radiated momentum is with P⊥, while RF compensates P|| ;Horizontal: After the photon emission

δxβ + δxε = 0

5.4 Radiation excitation5.4 Radiation excitation

γρ

σ ⋅⎟⎟⎠

⎞⎜⎜⎝

⎛=

2/1

36455

mcEh

2

32

/1

/

33255

ρ

ργεH

Jmc xx

h=

><

><= 2

3

/1

/

36455

ρ

ρβε

y

yy Jmc

h

With pure damping, the emittance will be zero in x, y and z planes.The radiated energy is emitted in quanta: each quantum carries an energy u = ħω;

22 ''2)( xxxx DDDDsH βαγ ++=Ji are damping partition numbers, H is dispersion invariant

The emission process is instantaneous and the time of emission of individual quanta is statistically independent; Radiation damping combined with radiation excitation determine the equilibrium beam distribution and therefore emittance, beam size, energy spread and bunch length.

Es

z Ωc σασ =

Diffusion effect off

Diffusion effect on

5.5 Rings vs. Linacs 5.5 Rings vs. Linacs (electrons)(electrons)

RingsRingsAccelerate + collide every turn‘Re-use’ RF + ‘re-use’ particles

⇒ efficientFinite beam emittance. Cost:

Linear costs(magnets, tunnel, etc.) : $lin∝ ρRF costs: $RF ∝ U0 ∝ E4/ ρOptimum when: $lin ∝ $RF ⇒ ρ ∝ E2

The size and the optimized cost scale as E2

Also the energy loss per turn scales as E2

⇒ get inefficient when E

LinacsLinacsOne-pass acceleration + collision⇒ need high gradientNO bending magnets ⇒ NO synchrotron radiationSmall beam size to reach a high LuminosityMuch less limited by beam-beam effect A lot of accelerating structuresCost scaling linear with E

e+ e-

Source

Damping ring

Main linac

Beam delivery

Bibliography Bibliography 1. E. Wilson, An Introduction to Particle Accelerators,

OUP, (2001)2. D. Edwards & M. Syphers: An Introduction to the

Physics of High Energy Accelerators3. S.Y. Lee: Accelerator Physics4. A.W.Chao and M.Tigner, Handbook of Accelerator

Physics and Engineering5. M. Conte & W. MacKay: An Introduction to the

Physics of Particle Accelerators6. M. Livingston & J. Blewett: Particle Accelerators7. M. Reiser: Theory and Design of Charged Particle

Beams

QuestionsQuestions(1) A 50 GeV (kinetic energy) proton

synchrotron will receive an injected beam at 4 GeV (kinetic energy) from an injector. The 2 T magnetic field of the bending magnets is excited at top energy of 50GeV. Given that the rest mass of the proton E0 is 0.93827 GeV:a) What is the Bρ at 4 GeV and at 50 GeV?b) What is the bending radius ρ?

(2) A quadrupole doublet (half of the FODO cell from the mid-point of the F to the mid point D) consists of two lenses of focal length f1 and f2 separated by a drift length of l.

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+−

−=

2*

1

11

1

fl

f

lfl

M where2121

11*

1ff

lfff

+−=

f f

l

1 2

s

Please show, by writing the three matrices for the lenses, the product matrix is: