Basic Accelerator physics for Linear Collider

June 17 2005

The 5th HEP Summer School

@ 경북대학교 고에너지 물리 연구소

포항 가속기 연구소김 은 산

1. Introduction

This lecture provides an introduction to accelerator physics required to understand and study a linear collider.

The lecture begins with a basic beam dynamics and then progresses into more detailed discussions of important subtopics.

2. Contents 2.1 Beam description

2.1.1 Coordinates

2.1.2 Beam moments and emittances

2.1.3 Luminosity

2.1.4 Bunch evolution in free space – the need for focusing

2.1.5 Beam-envelope function ( beta-function )

2.2 Transverse motion

2.2.1 Dipole

2.2.2 Equation of motion in quadrupoles

2.2.3 Constant focusing

2.2.4 Strong focusing principle

2.2.5 Betatron oscillation and phase advance

2.2.6 Equation of motion including dipole field and energy error

2.3 Longitudinal motion

2.3.1 Cylindrical cavity

2.3.2 Acceleration in linear accelerators

2.3.3 Adiabatic damping

2.3.4 Wakefield and beam breakup

3.1 Introduction of International linear collider (ILC)

3.1.1 Beam parameters

3.1.2 Damping ring

3.1.3 Parameters of dog-bone damping ring

2.1.1 Coordinates

We consider high-energy electron beams: E=mc2 (=5x105 for E=500 GeV), p=mc ~1 A bunch consists of electrons specified by the ph

ase space coordinates (x,x,y,y,z,)

zx

x

s

S = position of the bunch center

along the accelerator axis

x,y = transverse coordinates

x = dx/ds, y = dy/ds

Z = position of a particle relative

to beam center

E-Eo/Eo

2.1.2 beam moments and emittances

Beam distribution in phase space is often of Gaussian shape

and is completely described by second order beam moments:

x = <x2> : rms beam size in x-direction

y = <y2> : rms beam size in y-direction

x = <x2> : rms beam angular divergence in x-direction

y = <y2> : rms beam angular divergence in y-direction

z = <z2> : rms bunch length

= <2> : rms momentum spread

There are also correlation moments <xx>, <xy>, <z>, etc.

Beams are represented by phase space ellipses:

No correlation With x-x correlation

The phase space area is refereed to as emittances. In the absence of correlations, rms emittance is given by

x = xx y = yy z = z

xx

x x

2.1.3 Luminosity

Consider a bunch of electron beams colliding with a bunch of positions moving the opposite direction:

electron positron

Let e+ex be the cross section that an e+e- collision produces a particular state. Event rate = N- e+ex N+ A f /A N+ N- : Number of positrons ( electron) in each bunch A : transverse area of the beam f : repetition rate

Luminosity : L = N+ N- f / A More accurate calculation with Gaussian beams yield : L = N+ N- f / (4 x y )

2.1.4 Bunch evolution in free space – the need for focusing

A bunch may be in a tight, Gaussian shape at a certain location such as the collision point. What happens to the bunch if we let it evolve freely without any focusing device?

Let’s consider a bunch at s=0. An i-th electron in the bunch has the

transverse coordinates xi(0) and angle xi(0). Moving to a distance s, the coordinate becomes

xi(0) = xi(0) + s xi(0), xi(0) = xi(0). Thus beam moments becomes

<x2>s = < xi2(s)> = <xi

2(0) + 2xi(0) xi(0) + s2xi2(0)>

= <x2>0+ s2<x2>0 (assuming no correlation at s=0) Thus beam size increases due to the angular spread.

s

xi

xi

xi = xi(s) /s

xi(s)

At large s, the angle and coordinate becomes correlated < xi(s) xi(s) > = s < xi

2(0) >.

x

x

S=0S0

xi=sxi

In the presence of correlation, the rms emittance is defined to be

x(s) = (<x2>s <x2>s - <xx>2s )

2.1.5 Beam-envelope function ( beta-function)

S-dependence of the rms beam size can be parameterized by introducing a function x(s):

x(s) = (xx(s)) = (x2

(0) + x2(0)s2 )

Since x(s) = x(0) x(0), we have x(s) = x(0) /x (0) + s2 /x(0) /x(0) = x+ s2 / xxxcollision point )

For y0.2 mm, beam size at first quadrupole 1 m away y(1m) = y(0) (1+(1m/y =y(0) x 500.

The beta function is the property of the external focusing arrangement. In a linear collider, one normally requires

xz. If this condition is violated, the beam density changes significantly during collisions leading to degradation in the luminosity. “ Hourglass effect”.

2.2 Transvese motion

2.2.1 Dipoles In a dipole field B, the particle trajectory is a circle of radius

=p/eB.

Magnetic rigidity : B[Tm] = P[GeV] / 0.3.

L /

L

2.2.2 Equation of motion in quadrupoles

In a quadrupole, four poles of alternating polarities are placed symmetrically about beam center.

The field vanishes at origin; Bx=By=0 at x=y=0. Near the origin, By=(By/x)ox, Bx=(Bx/y)oy.

From Maxwell’s equation, x B=0, G= =(By/x)o=(Bx/y)o.

The equation for transverse momentum componets (px,py)=p , dp/dt = e(vxB)

The eq. of motion in quadrupoles becomes than

d2x/ds2 = -Kx and d2y/ds2 = -Ky. where K=eG/p

2.2.3 Constant focusing

For K > 0 and constant, the x-motion is sinusoidal.

d2x/ds2 = - Kx, x = Acos( Ks+), x = - KAsin(Ks+)

For a random distribution of A and the beam is a collection of simusoidal trajectories:

Constant envelope

The beam envelope is constant: x = 1/2<A2> = const.

x = K x

xx

x = 1/2<A2> K xx

/x = 1/K

<A2> = 2 xx and K =1/ x

This is well-focused beam in the x-direction. However, it is defocusing in y-direction.

2.2.4 Strong focusing principle We make the thin lens approximation, that is, particles are

deflected without changing displacement.

d2x/ds2 = - K x, x = xf - xi = -x / F, x = xf - xi = 0.

Here F = 1/Ks is the focal length.

s

F

x x = - x/F

The quadrupole is focusing in x-direction if F>0. Periodic arrangement of focusing quadrupoles will keep the beam focused in x-direction. However, the same quadrupole in y-direction will be defocusing.

x

xy

If we place a quadrupole of equal strength but opposite sign at waist locations?

We see that the focusing properties in the y-direction are identical to the x- direction. The beam envelopes in the x and y directions will look as follows: The beam is focused in both directions.

The trajectory of en electron in FODO lattice is pseudo-sinusoidal

with a period 4d. The pseudo-sinusoidal motion is referred to as betatron motion.

d

2.2.5 Betatron oscillation and phase advance

Eq. of motion : x+K(s)x = 0, where K(s) = K(s+L)

General solution is

x= (2xx(s))cos((s)+), (s)=ds1/x(s)

The envelope function x is periodic solution of

½ ¼ 2+= For K=0, the solution is s=ss

Phase advance per period is =ds1/(s) = is defined as tune.

2.2.6 Equation of motion including dipole field and energy error

• The motion in a dipole is circular. Transverse displacement of a displaced circle measured from the reference circle will be sinusoidal with a period of 2.

The eq. of motion in dipole for small displacement is x+x/2 = 0.

The eq. of motion in both dipoles and quadrupoles is x+ (1/+ K) x =0.y - Ky =0.

X

X

2

Consider a particle with a larger momentum p than reference momentum po. = (p-po)/po. Quadrupole strength is reduced to (1-)K. Momentum error produces an orbit displacement in dipole. Thus x-displacement becomes x=x+ x . Function x is called dispersion.

x + (1/+K) x= 1/ x = 1 /K+1

Quadrupoles displaced transversely produce dipole fields and generate dispersion. Thus quadrupole displacement in a linac must be tightly controlled to minimize residual dispersion and beam size increases due to momentum spread.

2.3 Longitudinal motion2.3.1 Cylindrical cavity

The simplest mode useful for acceleration is TM010 mode (TM : transverse magnetic, 0->no -variation, 1->first radial mod

e, 0->no z-variation), with frequency w=2.405 c/. The z-component electric field is z= oJo(2.405r/)cos(t+ ) T

he energy gain of a particle passing the center of the cavity

at t=0 is E = e zdz = eo cos(zk/v+) dz

= eV (sin/ ) cos =dv Phase should be 0 for maximum acceleration.

Ez

H Beam axis

d

To maintain accelerating field the cavity must be fed with rf power to balance the ohmic loss at the cavity surface from the oscillating current. Ploss = V2/Rs

We want a large Rs t so that required power for a given acceleration voltage is small.

Power per unit distance Ploss /L= (V/L)2 / (Rs /L)

Rs /L ~ It is advantageous to employ higher frequency rf such as x-band

(GHz). The drawback is that the structure becomes small and wakefield effect becomes more severe.

Superconducting rf at 2K is attractive becauese shunt impedance, being proportional to Q, is about 106 (~1010/104) times larger compared to normal rf structures.But cryogenic system is complexity and cost.

2.3.2 Acceleration in linear accelerators

In a linac with multi-cell cavities, accelerating field is represented by a traveling sinusoidal wave

z= ocos(t-kz)

The energy gain in a length L is

E = e zdz = eoL cos(to )

(L)=(0)+ eoL/(mc2) cos(to )

0 to

2.3.3 Adiabatic damping

Emittance is conserved for transverse motion when there is no acceleration. With acceleration transverse angle becomes smaller: Thus, transverse emittance will not be conserved. However, phase space (x px) will be conserved. Since px= m x, normalized emittance

nx= x= xis conserved. As the energy increases due to acceleration unnormalized emittance decreases as

x= nx This phenomenon is referred as adiabatic damping.

pz

x p x /pz

x

2.3.4 Wakefield and instability

Passage of charged particle beams induce electromagnetic field in rf cavities and other structures in linac. The beam-induced fields, wakefield, act back on the beams and may cause instability.

Longitudinal wakefield may lead to energy spread and transverse wakefield may cause a beam breakup(BBU).

Wakefields are characterized by a wakefunction which give the force on a test charge following a charge at a distance z.

q

Q

z x1 Force on a test charge = qQ W1(z) x1

x1x2

z

Ne/2Ne/2

Head particle undergoes a free betatron oscillation. Assuming a constant focusing k

2, x1(s) = xx1cos(ks) The eq. For the displacement x2 of an electron in trailing part is d2x2/ds2 + k2x2 = Ne2 W1(z)x1 cos(ks) / 2E x2 (s) = Ne2 W1(z)x1s sin(ks) / 4kE The betatron amplitude of the electrons in the trailing part grows linearly and will break out of bunch. Amplification factor : =Ne2 W1(z) L / 4kE

For SLAC linac, taking z=1mm, W1(z)=1.8V/(pC)(mm)(m), k=6x10-5(mm-1), Ne=8nC,E=1GeV, s=3km, =

Transverse BBU can be suppressed by arranging focusing of the trailing part to be slightly stronger,I.e.,by replacing

k2 by (k+k)2 with k= Ne2 W1(z)/4Ek

Under this condition both parts of the bunch move together

and BBU is suppressed. : BNS damping

3.1 Introduction of ILC3.1.1 Beam parameters

Beam and IP parameters for 1 TeV cms. E_cms (GeV) 1000 N 2.00E+10 Nb 2820 T_sep (ns) 336.9 Buckets @ 1.3 GHz 438 I_ave (A) 0.0095 Gradient 35.00 MV/m BetaX 2.44E-02 m BetaY 4.00E-04 m SigX 4.89E-07 m SigY 4.0E-09 m SigZ 3.00E-04 m Luminosity (m-2s-1) 3.81E+38

3.1.2 Damping ring

Damping rings are necessary to reduce the emittances produced by the particle sources to the small values required for the linear collider. Emittance reduction is achieved via the process of radiation damping, i.e. the combination of synchrotron radiation in bending fields with energy gain in RF cavities.

The design of the damping ring has to ensure a small emittance and a sufficient damping rate.

One of the main design criteria for the damping ring comes from the long beam pulse: a 1ms pulse containing 2820 bunches.

Designs of damping rings are determined by upstream and downstream systems

• source

pre-linac

Damping ring

Bunch compressor

linac

Beam delivery

Interaction region

Design choice are based on

- injection/extraction scheme

- beam dynamics

- reliability and flexibility for operation

Beam has a high bunch charge (2*1010) and low emittance

- collective instabilities is important

Damping ring requires sufficient acceptance in transverse and

longitudinal directions.

- dynamic aperture is a key issue.

3.1.3 Parameters of dog-bone damping ring

Energy 5GeVCircumference 17 kmHor. extracted emittance 8 x 10−6 mVer. extracted emittance 0.02 x 10−6 mInjected emittance (x/y) 0.01m (10−5 m) Damping time 28ms Number of bunches 2820Bunch spacing 20 x 10−9 sNumber of particles per bunch 2 x 1010

Current 160mAEnergy loss/turn 21MeV Total radiated power 3.2MW Tunes 72.28 , 44.18Chromaticities −125, −68Momentum compaction 0.12 x 10−3

Equilibrium bunch length 6mmEquilibrium momentum spread 0.13% Momentum acceptance 1%

Summary

The design requirements of the linear collider are very challenging: acceleration of high-current electron beam to several hundred GeV, damping ring issues, low-emittance transport beam dynamics, focusing to a few-nanometer beam size and collision with similarly prepared opposing positron beams.

Works of ILC accelerator design are opened to everybody who has an interesting and concern. Please join! Coming workshops for the ILC ILC BDIR and Europe ILC ( 20th June – 24th June , UK) Snowmass ( 15th Aug. – 19th Aug. , USA)