Transcript 2.1 Coordinates - The Center for High Energy Physics

Basic Accelerator physics
for Linear Collider
June 17 2005
The 5th HEP Summer School
@ 경북대학교 고에너지 물리 연구소
포항 가속기 연구소
김
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 g (g=5x105 for E=500 GeV), p=mcbg, b~1
 A bunch consists of electrons specified by the
phase space coordinates (x,x,y,y,z,d)
x
x
z
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
d = 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:
sx =  <x2>
sy =  <y2>
sx =  <x2>
sy =  <y2>
sz =  <z2>
sd =  <d2>
: rms beam size in x-direction
: rms beam size in y-direction
: rms beam angular divergence in x-direction
: rms beam angular divergence in y-direction
: rms bunch length
: rms momentum spread
There are also correlation moments <xx>, <xy>, <zd>, etc.
 Beams are represented by phase space ellipses:
x
x
x
No correlation
x
With x-x correlation
 The phase space area is refereed to as emittances. In the
absence of correlations, rms emittance is given by
e x = s xs x
ey = sysy
e z = s zs d
2.1.3 Luminosity
 Consider a bunch of electron beams colliding with a bunch
of positions moving the opposite direction:
electron
positron
 Let s e+ex be the cross section that an e+e- collision produces a particular state.
Event rate = N- s e+ex 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 p sx sy )
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).
xi
xi
Thus beam moments becomes
s
<x2>s = < xi2(s)> = <xi2(0) + 2xi(0) xi(0) + s2xi2(0)>
= <x2>0+ s2<x2>0 (assuming no correlation at s=0)
Thus beam size increases due to the angular spread.
x
xi = xi(s) /s
Dxi=sxi
xi(s)
x
S0
At large s, the angle and coordinate becomes correlated
< xi(s) xi(s) > = s < xi2(0) >.
In the presence of correlation, the rms emittance is defined to be
ex(s) =  (<x2>s <x2>s - <xx>2s )
S=0
2.1.5 Beam-envelope function ( beta-function)
 S-dependence of the rms beam size can be parameterized by
introducing a function bx(s):
sx(s) = (exbx(s)) =  (sx2 (0) + sx2(0)s2 )
Since ex(s) = sx(0) sx(0), we have
bx(s) = sx(0) /sx (0) + s2 /sx(0) /sx(0)
= bx* + s2 / bx* (bx*= bx(0), * : collision point )
 For by* = 0.2 mm, beam size at first quadrupole 1 m away
sy(1m) = sy(0)  (1+(1m/by*)2 ) = sy(0) x 500.
The beta function is the property of the external focusing
arrangement. In a linear collider, one normally requires
bx*  sz. 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
r=p/eB.
Magnetic rigidity : Br[Tm] = P[GeV] / 0.3.
L
Dq=L / r
r
Dq
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+f), x = - KAsin(Ks+f)
For a random distribution of A and f, the beam is a collection of
simusoidal trajectories:
Constant envelope
The beam envelope is constant: sx2 = 1/2<A2> = const.
sx 2 = K sx2 ex = sx sx = 1/2<A2> K bx = sx2 / ex = 1/K
 <A2> = 2 exbx and K =1/ bx2
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, Dx = xf - xi = -x / F, Dx = xf - xi = 0.
Here F = 1/KDs is the focal length.
bx
Dx = - x/F
x
F
Ds
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.
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 xdirection. The beam envelopes in the x and y directions will look as follows:
The beam is focused in both directions.
by
bx
d
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.
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= (2exbx(s))cos(f(s)+fo), f(s)=ds1/bx(s)
 The envelope function bx is periodic solution of
½ bb - ¼ b2+b2 K = 1.
For K=0, the solution is b(s) = bo+ (s-so)2/bo
Phase advance per period is m=ds1/b(s)
u = m/2p 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 2pr.
The eq. of motion in dipole for small displacement is
x+x/r2 = 0.
X
r
2pr
X
The eq. of motion in both dipoles and quadrupoles is
x+ (1/r2 + K) x =0. y - Ky =0.
 Consider a particle with a larger momentum p than
reference momentum po.
d = (p-po)/po. Quadrupole
strength is reduced to (1-d)K. Momentum error produces
an orbit displacement in dipole. Thus x-displacement
becomes x=xb+ hx d. Function hx is called dispersion.
hx + (1/r2 +K) hx = 1/ r, hx = 1 /r(K+1/r2)
 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 motion
2.3.1 Cylindrical cavity
Ez
r
Hf
Beam axis
d
 The simplest mode useful for acceleration is TM010 mode
(TM : transverse magnetic, 0->no f-variation, 1->first radial
mode, 0->no z-variation), with frequency w=2.405 c/r.
 The z-component electric field is ez= eoJo(2.405r/r)cos(wt+ fo)
The energy gain of a particle passing the center of the cavity
DE = e ezdz = eeo cos(wzk/v+fo) dz
= eV (sinq / q) cos fo, q = wd/2v
Phase fo should be 0 for maximum acceleration.
at t=0 is
 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 ~ w
 It is advantageous to employ higher frequency rf such as x-band
(w=11.4 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
ez= eocos(wt-kz)
The energy gain in a length L is
DE = e ezdz = eeoL cos(wto )
 g(L)=g(0)+ eeoL/(mc2) cos(wto )
0
p
2p
wto
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 (Dx Dpx) will be
conserved. Since Dpx= mgb Dx, normalized emittance
enx= gb ex = g ex is conserved. As the energy increases due
to acceleration unnormalized emittance decreases as
ex = enx / g
This phenomenon is referred as adiabatic damping.
Dpz
x
x
Dp x /p
z
2.3.4 Wakefield and instability
 Passage of charged particle beams induce electromagnetic
field in rf cavities and other structures in linac. The beaminduced 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
Ne/2
Ne/2
x1
x2
sz
Head particle undergoes a free betatron oscillation.
Assuming a constant focusing kb2, x1(s) = x1cos(kbs)
The eq. For the displacement x2 of an electron in trailing part is
d2x2/ds2 + kb2x2 = Ne2 W1(z)x1 cos(kbs) / 2E
x2 (s) = Ne2 W1(z)x1s sin(kbs) / 4kbE
The betatron amplitude of the electrons in the trailing part
grows linearly and will break out of bunch.
Amplification factor : Y = Ne2 W1(z) L / 4kbE
For SLAC linac, taking z=1mm, W1(z)=1.8V/(pC)(mm)(m),
kb =6x10-5(mm-1), Ne=8nC,E=1GeV, s=3km, Y = 180!
 Transverse BBU can be suppressed by arranging focusing of
the trailing part to be slightly stronger,I.e.,by replacing
kb2 by (kb+Dkb)2 with Dkb = Ne2 W1(z)/4E kb
Under this condition both parts of the bunch move together
and BBU is suppressed. : BNS damping
3.1 Introduction of ILC
3.1.1 Beam parameters
Beam and IP parameters for 1 TeV cms.
E_cms (GeV)
N
Nb
T_sep (ns)
Buckets @ 1.3 GHz
I_ave (A)
Gradient
1000
2.00E+10
2820
336.9
438
0.0095
35.00 MV/m
BetaX
BetaY
SigX
SigY
SigZ
Luminosity (m-2s-1)
2.44E-02 m
4.00E-04 m
4.89E-07 m
4.0E-09 m
3.00E-04 m
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
 Damping ring requires sufficient acceptance in transverse and
longitudinal directions.
- dynamic aperture is a key issue.
pre-linac
 Design choice are based on
Damping ring
- injection/extraction scheme
- beam dynamics
Bunch compressor
linac
- reliability and flexibility for operation
 Beam has a high bunch charge (2*1010) and low emittance
- collective instabilities is important
Beam delivery
Interaction region
3.1.3 Parameters of dog-bone damping ring
Energy
Circumference
Hor. extracted emittance
Ver. extracted emittance
Injected emittance (x/y)
5GeV
17 km
8 x 10−6 m
0.02 x 10−6 m
0.01m (10−5 m)
Damping time
Number of bunches
Bunch spacing
Number of particles per bunch
Current
Energy loss/turn
Total radiated power
Tunes
Chromaticities
Momentum compaction
Equilibrium bunch length
Equilibrium momentum spread
Momentum acceptance
28ms
2820
20 x 10−9 s
2 x 1010
160mA
21MeV
3.2MW
72.28 , 44.18
−125, −68
0.12 x 10−3
6mm
0.13%
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)