A. Transverse (Betatron) Motion1. Linear betatron motion2. Effects of linear magnet imperfection3. Dispersion function of off momentum particle, momentum

compaction, phase slip factor and phase stability; Simple Lattice design considerations

4. Chromatic aberration

S.Y. Lee, IU, July 2012

B. Longitudinal synchrotron Motion1. Synchrotron Equation of motion, stable and fixed points,

bucket area, bucket height, and synchrotron tune 2. Adiabatic synchrotron motion3. Non-adiabatic synchrotron motion at transition energy and

imaginary γT lattice.4. Topics related to Betatron and synchrotorn motion in

accelerators Update (after 6/30/2012) is available at:http://physics.indiana.edu/~shylee/ocpa12/

Dipole

θ=ℓ/ρ=Bℓ/BρBqm

2

The first course to learn is how to optimize the arrangement of beam control elements in achieving the wanted beam properties.

quadrupole

f>0, if focusing, f<0 if defocusing

f

BB

f

11

)( BEedtPd

sxzdssdx

dsrds

, ,0 zzxxrr

0

xszsx , 1

222 )( AePcmceH

ABtAE

,

Frenet-Serret coordinate system: We assume that there exists a closed orbit r0(s). The coordinates around the reference orbit is defined by

Bqm

2

Bending radius

What is the momentum rigidity of proton beam at momentum 3 GeV/c?

Frenet-Serret coordinate system:

1. The tangential vector points toward the direction of beam motion

2. We define the normal vector points outward from the curve. So that ̂ ̂

http://en.wikipedia.org/wiki/Frenet-Serretx˄

s˄z˄

x˄

s˄y˄

3. Some physicists use the coordinates shown at the bottom right with ̂

How to transform from the original coordinate system onto the Frenet-Serret coordinate system? Generating function!

, , ,)1(

)(),,,(

333

03

zPzFpxP

xFpsPx

sFp

zzxxrPzsxPF

zxs

2/122

2

222 )()(

)/1()(

zzxxss eApeAp

xeApcmceH

, , ,)1( zAAxAAsAxA zxs

The phase space coordinates are (x,s,z) with independent coordinate t. In one revolution, the time advances T0, called the orbital period. In one orbital period, the orbit length is the circumference C.

Accelerator components repeat in each orbital period. It would be nice to use s as the independent coordinate. How to exchange coordinates t and s?

dx

ds

These equations indicate that –ps becomes the new Hamiltonian with the (x,px,z,pz,t,-H) and s as the independent coordinate.

szzxx eAeApeApp

xxpH

])()[(2

/1)1(~ 22

BBzsKz

BBxsKx x

zz

x

)( ,)( Hill’s equation

Synchrotron motion12112 ),sin(sin nnnsnnn E

EeVEE

.~

' ,~

' ,~

' ,~

' ,~

' ,~

't

HHHHt

zHp

pHz

xHp

pHx z

zx

x

tAE

2/122

2

222 )()(

)/1()(

zzxxss eApeAp

xeApcmceH

Solving for –ps, we find the new Hamiltonian: P2=(E/c)2−m2c2

szx eAppp

xxpH

][2

/1)1( 22

Transverse magnetic field: A=B, B=0. For 2D magnetic field, B can be represented by either one component of the vector potential As, or by a scalerpotential Φ, i.e. Bx=-As/z, Bz=As/x, or Bx=xΦ, Bz=zΦ. Although the field can be represented two ways, only the vector potential serves as the “potential” in the betatron Hamiltonian. For two dimensional magnetic field, one can expand the magnetic field using Beth representation:

n

nnnxz jzxjabBjBB 0

, , quad, : , dipole, :

10101

000

zbBBxbBBbbBBb

xz

z

sextupole; skew : sextupole, :, , quad; skew :

, dipole; (vertical) skew :

22

10101

000

abxaBBzaBBa

aBBa

xz

x

n

nnns jzxjab

nBA 1

0 11Re

.)1(

,)1(

20

202

xpp

BBz

xpp

BBxx

x

z

BBzsKz

BBxsKx

xz

zx

)(

,)(

szx eAppp

xxpH

][2

/1)1( 22

00 qBqBmp

Momentum Rigidity (magnetic rigidity)

∓ ∆0 ∆Notation: e=q

BBsK

BBsK

z

x

1

12

)(

1)(

For p=p0, we have

For transverse magnetic field, we can choose Ax=Az=0. The Hamiltonian becomes

xBAg

wNLgNIB

s 0

200

0 ,

221

2

220

20

1

2

8 ,2

zxBA

axNL

aNIB

s

c

BBsK

BBsK

BBzsKz

BBxsKx

zx

xz

zx

112 )( ,1)(

)( ,)(

The upper/lower sign for positive/negative charge particles

n

n

n xBB

How to solve the Hill’s equation?

0)( ,0)( zsKzxsKx zx

Let y represent either x or z: ,0)( ysKy

The focusing function is piecewise constant!

||cosh||sinh

cossin

0

00

)(

BsAsKBsKA

sKBsKA

yKK

sK

)('

)(),(

)(')(

0

00 sy

syssM

sysy

... ,)(cos

)(sin

)(sin

)(cos),(

0

01

0

00

ssK

ssK

ssKK

ssKssM K

Det(M(s2,s1))=1

Example: For a focusing quadrupole, we find

,1)( ,11)( ,0)( 2 xB

BsK

xB

BsKysKy z

zz

x

1 1/0 1

cos

sin

sin

cos),(

1

0 fK

K

KK

KssM K

1. focusing quadrupole:

1 1/0 1

||cosh

||sinh

||sinh||

||cosh),( ||

1

0

fK

K

KK

KssM K

2. de-focusing quadrupole:

Thin lens approximation: Let |K|ℓ→1/f as ℓ→0.

3. Drift space: K=0

1 0 1

),( 0

ssM

f

4. Sector Dipole: Kx(s)=1/ρ2

1 0 1

cos

sin

sin

cos),(

010

ssM

a. When the bend angle is small ℓ/ρ<<1, the transfer matrix of a dipole is equivalent to that of a drift space.

b. When the bending radius ρ is small, dipoles can provide strong horizontal focusing with Kx=1/ρ2. The vertical focusing is adjusted by trimming the edge angle.

5. Rectangular Dipole: δ=θ/2

1 0

sin 1

1 2tan

0 1

cossin

cos

1 2tan

0 1

),( sin0

ssM

δ

cossin

sin

cos),( 10ssM x

1sin

01

),( 0

ssM x

1 0

1 ),( 00

ssMSmall angle approximation:

,0)( ysKy

)('

)(),()...,(),(),(

)(')(

0

001211 sy

syssMssMssMssM

sysy

nnnnn

The transfer matrices can be used to phase space evolution of Hill’s equation:

mr0.200

mm10

0

0

0

0

zzxx

Particle Trajectories

RED: x(s); GREEN: y(s); BLUE: dispersion*10z-axis (m)

1614121086420

x(m

m),

y(m

m),

10*d

isp(

m)

16141210

86420

-2-4-6-8

-10-12-14

Particle Trajectories

RED: x(s); GREEN: y(s); BLUE: dispersion*10z-axis (m)

4035302520151050

x(m

m),

y(m

m),

10*d

isp(

m)

25

20

15

10

5

0

-5

-10

-15

-20

-25

Lq=1m, Kq=0.5m–2, L1=3.5m

Periodic structure: Many accelerators are designed with the periodic condition: K(s+L)=K(s). The transfer matrix is periodic:

M(s1+L|s1)=MnMn-1Mn-2…M2M1=M(s1)M(s2+L|s2)=M1MnMn-1Mn-2…M2=M(s2)=M1M(s1)M1

-1

M(s4+L|s4)=M3M2M1MnMn-1Mn-2…M4=M(s4)

Each M(s) matrix is a product of identical number of matrices. They are related by similarity transformation. The eigen-values of the periodic matrix M(si) are identical.

M1M2………….Mn M1M2………….Mn

,cossincos

sin

sinsincos

)(M

JsiIs

The most general representation of the matrix M(s) with unit modulus is given by the Courant-Snyder parameterization.

4

51

2

3

As particles move through periods of an accelerator, the transfer matrix becomes

sincos je j

22 1or , , ,10

01

IJJI

Example: FODO cell

0sin

)2

sin1(2

sin

)2

1(21

11

Lf

LL

fL

fL

22sin ,

21cos 1

2

21

A FODO cell is a basic block in beam transport, where the transfer matrices for dipoles (B) can be approximated by drift spaces, and QF and QD are the focusing and defocusing quadrupoles.

sincossin

sin

sincos)(M

s

Tr(M)cos 21

sincos

sin

sinsincos

1

01

10

1

1L

01

10

1

1

01 4

3L

114L

ff

Example: FODO cell

Questions:1) Will Φ of these above matrix identical?2) Will α and β of these matrices identical? 3) What is the meaning of these parameters?

sincos

sin

sinsincos

1

01

10

1

1L

01

10

1

1

01 2

L

112L

ff

sincos

sin

sinsincos

1L

01

10

1

1L

01

10

1

11

ff

Lq=1m, Kq=1m–2, L1=3.5m

fL

fL

22sin ,

21cos 1

2

21

Thin lens – use with care

In this FOFDO cell with Lq=1.0 m, L_dipole=2.0 m, drift length of 0.25 m, and thus L1=3.5 m. Thin lens approximation is good except when the focusing strength is high. The percentage error at high focusing gradient can be larger than 11%.

0

20

40

60

80

100

120

140

160

180

0 0.1 0.2 0.3 0.4 0.5 0.6

Φ (deg)

Thin lens

K (m–2)

sin

)2

sin1(2

sin

)2

1(21

11

max

LfLL

x0=10 mm, x0’=0, y0=0, y0’=5 mrad

Particle Trajectories

RED: x(s); GREEN: y(s); BLUE: dispersion*10z-axis (m)

200180160140120100806040200

x(m

m),

y(m

m),

10*d

isp(

m)

25

20

15

10

5

0

-5

-10

-15

-20

-25

-30

Φy≈105º,Φy≈103ºSector dipole

What is β-function?Consider an accelerator made of 18 FODO cells (C=360 m) with tune 4.793. The phase advance of each FODO cell is 96º.

The trajectories are bounded by ±β(s). Thus β-function is also called the envelope function. One complete betatron oscillation takes less than 4 cells.

Trajectories of a particle with x0=5.82 mm and x0’=0 at the center a focusing quadrupole are shown below (1/3 of the ring), where the first 4 revolutions are marked with magenta, blue, green and red colors. The remaining 16 revolutions are traced out in thin black lines.

What is the meaning of the phase advance per cell?Consider an accelerator made of 18 FODO cells (C=360 m) with tune 3.42. The phase advance of each FODO cell is 68º.Trajectories of a particle with x0=5.65 mm and x0’=0 at the center a focusing quadrupole are shown below (1/3 of the ring), where the first 4 revolutions are marked with magenta, blue, green and red colors. The remaining 16 revolutions are traced out in thin black lines.

The trajectories are bounded by ±β(s). Thus β-function is also called the envelope function. One complete betatron oscillation takes about 5.3 cells.

Mij is the ij-th component of the matrix M(s2,s1)

Fermilab Booster

AGS

D 0

sin

)2

sin1(2 1

L

fL22

sin 1

Examples of lattice design

Φ≈120º

Φ≈53º

AGS Booster is a missing dipole FODO cell. The straight sections can be used for injection, extraction, and rf cavities. How does the dispersion function change as you moves the missing dipole sections?

AGS Booster:C= 201.78 m Qx=4.641 Qy=4.576Qx'=−7.205 Qy'=−2.864αc= 0.046 γtr=4.67 βxmax=13.47 m βxmax=13.33 mDx(max)=2.89 m

Stability of accelerator cells: FODO cell example

xxx

xx

xx

xxx

ff sincossin

sin

sincos1L

01

10

1

1L

01

10

1

21

11

zzz

zz

zz

zzz

ff sincossin

sin

sincos1L

01

10

1

1L

01

10

1

21

11

211221

2

12

211221

2

12

22212

1cos

22212

1cos

XXXXff

LfL

fL

XXXXff

LfL

fL

z

x

.1|cos| ,1|cos| zx

Stability condition: (necktie diagram)

Phase space points of a particle moving in AGS of each revolution form ellipses. The phase space area enclosed is 2πJ. The actions in the plot are πJx= πJz=0.5π mm-mrad. The ellipses are located at the end of the first dipole (left plots) and the end of the fourth dipole (right plot) of the AGS lattice. The scale for the ordinate x or z is in mm, and that for the coordinate x′ or z′ is in mrad. Left plots: βx=17.0 m, αx=2.02, βz=14.7 m, and αz=−1.84. Right plots: βx=21.7 m, αx=−0.33, βz=10.9 m, and αz=0.29.

sincossin

sin

sincos)(M

s

Let M be the one-turn map:

0

0Myy

yy n

n

Let λ1 and λ2 be the eigenvalues of the Matrix M and υ1 and υ2be corresponding eigenfunctions. Thus we find:

The stability of particle motion is given by | λ1| ≤1 and |λ2 |≤1. This is realized by the condition: |Trace(M)| ≤ 2.

Stability of betatron motion

Floquet transformation: consider Hill’s equation:

1. Floquet theorem: If the focusing function obeys K(s)=K(s+L), we can choose the solution with the properties: w(s)=w(s+L), and ψ(s+L)=ψ(s)+Φ, where Φ is the phase advance in one period.

)()( sjesawy Substituting the ansatz:

2

1w

01)( 3 w

wsKwwe find

For any 2nd order differential equation

)('

)(),(

)(')(

0

00 sy

syssM

sysy

we represent the solution in transfer matrix:

2. Transfer matrix

sincossin

sin

sincos)(

2

wwwww

sM

,1 ,21 ,)(

22

ws dssssws

s0

1)( ,)()(

sincos

sin

cos)(sin

sincos),(

21

21

1

1212

2

1

1

2

2

1

21

2211

1

2

ww

wwwwssM

ww

ww

ww

wwwwww

ww

)()( sjesawy Using we find

Now, we apply the Floquet theorem:

The transfer matrix in one period can be identified with the Courant-Snyder parametrization:

sincossin

sin

sincos

01)( 3 w

wsKw

sincos

sin

cos)(sin

sincos),(

21

21

1

1212

2

1

1

2

2

1

21

2211

1

2

ww

wwwwssM

ww

ww

ww

wwwwww

ww

The transfer matrix from s1 to s2:

2211 ww

Betatron tune: The betatron phase change in one revolution is the betatron tune νy, or Qy. It is the number of betatron oscillations in one revolution, is

)()( ssw The amplitude function is we obtain

≡ 12

Floquet transformation:

Define the normalized coordinate dsy s

11 ,

yB

B

yyydsd

dsd

dd

22/32

222/32

2

2

4'''

21''

BB

dd

2/3222

2

022

2

dd

If ∆B=0, we find

A simple harmonic oscillator

BBzsKz

BBxsKx

BBKyy x

zz

x

)( ,)( "

Since the solution of Hill’s equation is

we find

The generating function for (y,y’) to (ψ,J) transformation is

The invariant of betatron motion: action-angle transformation

Hill’s equation can be derived from the Hamiltonian

Define:(y,Py) form a normalized phase space coordinateswith y2+Py

2=2βJ, here J is called action.

Recover betatron phase equation

The action J is invariant!

y

Py

Courant-Snyder Invariant

Jyyyyyyy 2)(12 2222

We define the second moments σ-matrix of a beam of particles with distribution function ρ as:

The transformation of the σ-matrix is:

Define the rms Emittance for a beam of particles

If the beam distribution function is a function of the Courant-Snyder invariant, we find

An invariant distribution is a function of

A beam with an rms emittance 10 π-mm-mrad, what will be the rms betatron amplitude of a beam at the location with β=10 m? Ans: 10 mm!

,

, 10m 10 mm mrad 10mm

The rms emittance is invariant in linear transport:

we find

However, the new Hamiltonian Ḧ depends on the variable s. Because β(s) is not a constant, the phase advance is modulated along the accelerator orbital trajectory. Sometimes it is useful to obtain a global Fourier expansion of particle motion by using the generating function

We have carried out coordinate (Floquet) transformation to change the phase space coordinates (y,y’) into action angle variable (J,ψ):

In terms of the action angle coordinates, the phase space coordinates are given by

and

We change the time coordinate from s to θ, the new Hamiltonian is re-scaled to

2 cos2 sin2 cos2 sin)

In summary, the following Floquet transformation transforms the betatron Hamiltonian to a simple harmonic oscillator for both horizontal and vertical motion.

Some design examples:Large colliders are normally made of arcs and insertion regions (IR), where arcs are made of FODO cells for beam transport, and IRs are used for physics experiments. The IR matches all optical functions for special properties relevant to physics experiments.

THE RHIC LATTICES.Y. Lee, J. Claus, E.D. Courant, H. Hahn and G. Parzen, PAC, 1626 (1985)

∗ ∗

1) S.X.Fang and S.Y.Chen, Proc. 14-th Int. Conf. on High Energy Acc. (1989) 51.2) C.Zhang of BEPC Group, Proc. 15-th Int. Conf. on High Energy Acc. (1992).

See Lecture Note of Q. Qin in ocpa accelerator school 2002, Singapore.BEPC operational since May 1989.

• Synchrotron radiation facilities are designed to minimize emittance and retain a long straight section for IDs.

• Low energy proton synchrotrons can use the dipole for horizontal focusing, and edge angle for vertical focusing.

• High power accelerators are designed by taking into account the effects of space charge and transition energy crossing into consideration.

CIS: Circumference =17.364 m, Inj KE= 7 MeV, extraction: 240 MeVDipole length = 2 m, 90 degree bend, edge angle = 12 deg.

ALPHA (C=20m). An electron storage ring for Radiation effect experiments.

Nader Al Harbi & S.Y. Lee, RSI, 74, 2540 (2003).epBB /22

Ldip=3.0 m, ρ=1.91 m, Edge_angle=8.5°Circum=28.5 m, Qx=1.68, Qz=0.71, KE_tr=356 MeV

A2: Effects of Linear Magnetic field Error

BBzsKz

BBxsKx x

zz

x

)( ,)(

, , , , quad; skew : quad, :, , dipole; (vertical) skew : dipole, :

,

1010101011

000000

0

xaBBzaBBzbBBxbBBabaBBbBBab

jzxjabBBjB

xzxz

xz

n

nnnxz

00 )( ,)( azsKzbxsKx zx Dipole field error:

zbzsKzxbxsKx zx 11 )( ,)( Quadrupole field error:

A.2.1 Effect of dipole field error: We consider a single localized dipole error with the kick angle given by θ=∆Bℓ/Bρ. Because of the dipole field error, the reference orbit is perturbed! The idea is to find a new closed orbit that include the dipole field error. )()( 0ssysKy y

The closed orbit condition is:

000

00

00

0000 sincos

sin

sinsincos

M

)cos(sinsin2

,cossin2 00

00

yy

Where Φ0=2πν, ν is the betatron tune, the parameters α0, β0, and γ0 are values of the Courant-Snyder parameters at the kicker location. The solution is

0

00

0

0 Myy

yy

We have solved the closed orbit at one point s0. The closed orbit of the accelerator can be obtained by making mapping matrix:

0

00

co

),()()(

yy

ssMsysy

),()( 0co ssGsy

|])()(|cos[sin2

)()(),( 0

00 ss

ssssG

The closed orbit as a function of the longitudinal distance for the AGS booster resulting from a horizontal kicker with kick-angle θ = 6.82 mr at the location marked by a straight line. Since the betatron tune of 4.82 is close to the integer 5, the closed orbit is dominated by the fifth error harmonic.

Define ′Q: If you plot ξ vs η, what does the closed orbit graph look like? ∆ ∆ ξ

η

A kicker does not change position. It changes only the angle.

BsBysKy y)()(

ds

An accelerator with circumference 360 m is made of 18 FODO cells. The betatron tunes of the synchrotron are νx=4.793 and νz=4.783 respectively. If one of the 36 dipoles has an error of -2 mrand another has error of -1 mr, The closed orbit deviation from the designed orbit is

Floquet transformation

TLS orbit vs dipole field error: Lecture note by C.C. Kuo (2002 OCPA Singapore)

xco

yco

Things are much more complicated!An accelerator with C=360 m is made of 18 FODO cells. The betatron tunes of the synchrotron are νx=4.793 and νz=4.783 respectively. If all 36 dipoles has a random error of 0.1% in amplitude, the correction of closed orbit needs patience and careful analysis. But a set of orbit correctors can be used to minimize the closed orbit.

An accelerator with circumference 360 m is made of 18 FODO cells. The betatron tunes of the synchrotron are νx=4.793 and νz=4.783 respectively. If one of the 36 dipoles has an error of 1%, estimate the maximum closed orbit deviation from the designed orbit.

xF 35.0xD 5.17f 6.73DxF 4.33DxD 1.99

,)()(2/3

k

jkkef

BB

dse

BBde

BBf jkjk

k

)()(

21)()(

21 2/12/3

0

00022

2

co)cos(||)( )()(

0 kkfse

kfssy kk

kjk

k

k

Using the data, we estimate the maximum closed orbit to be about 37 mm.

Fourier expansion:

Floquet transformation BsBysKy y)()(

ds

12 β 1% 10degree 0.0011

),()( 0co ssGsy

Note that the closed orbit is described by Green’s function. When the betatron tune is an integer, the closed orbit diverges. Each time, when the particle arrives the same location will receive a coherent kick and the particle becomes unstable.

|])()(|cos[sin2

)()(),( 0

00 ss

ssssG

Left, a schematic plot of the closed-orbit perturbation due to an error dipole kick when the betatron tune is an integer. Here py=βyΔy′=βyθ, where θ is the dipole kick angle and βy is the betatron amplitude function value at the dipole.

Right, a schematic plot of the particle trajectory resulting from a dipole kick when the betatron tune is a half-integer; here the angular kicks from two consecutive orbital revolutions cancel each other.

For the distributed dipole field error, the closed orbit becomes

BsBsssds

ssy

Cs

s

)(|])()(|cos[)(sin2

)()( 0

000co

Making coordinate transformation:

ddssssss

dsss

s

);()( ),()( ,)(

1)( 00

0

|]|[cos)()(sin2

)()( 2/3

co

BBd

ssy

Cs

s

The closed orbit becomes

,)()(2/3

k

jkkef

BB

dse

BBde

BBf jkjk

k

)()(

21)()(

21 2/12/3

0

00022

2

co)cos(||)( )()(

0 kkfse

kfssy kk

kjk

k

k

iiii

Cs

s

ssss

BsBsssds

ssy

|])()(|cos[)(sin2

)(

)(|])()(|cos[)(sin2

)()( 0

000co

rmsi

ii Ns

ss

sy

sin22)(

)(sin22

)()( 22/12

co

The sensitivity factor of an accelerator is defined the ratio of rms closed orbit distortion and the rms dipole error.

55ysensitivitrmsco,

rmsco,

x

Extended Matrix Method for the Closed Orbit

The closed-orbit condition for a thin dipole is equivalent to

C0 is the orbit length of the unperturbed orbit, and higher order terms associated with betatron motion are neglected. Since a dipole field error gives rise to a closed-orbit distortion, the circumference of the closed orbit may be changed as well. We consider the closed-orbit change due to a single dipole kick at s = s0 with kick angle θ0, the change in circumference as

The path length of the Frenet-Serret coordinate system is

Applications of dipole field error:

1. closed orbit bump:),()( 0co ssGsy

|])()(|cos[sin2

)()(),( 0

00 ss

ssssG

where θi = (ΔBs)i/Bρ and (ΔBs)i are the kick-angle and the integrated dipole field strength of the i-th kicker. The closed orbit is zero outside these four dipoles yco(s4)=0, y′co(s4) = 0.

iii

i ss

sx

)(cossin2

)()(

2

1co

Pulse from linac Extracted beam

Lambertsonseptum

Kicker 1 Kicker2

2

112

0coscos 212211 21

Electrostatic kicker:

Kicker Strength

-5

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16 18

x co

(mm

)

s (m)

θk = 24 mradθk = 18 mradθk = 12 mrad

θk = 6 mrad

field electric gapElight of speed c

kicker theoflength LMeV 60at ][2.0

where,

TmBBc

LEk

For one turn injection and extraction, the integrated field strength is 0.60 MV at 25 MeV electron beam energy. Choosing a length of L=0.5 m, the applied voltage on two plate is 60 kV.

Using chicane-dipoles, one can provide local orbit bumps!

2. Injection and extraction kicker

θk =∫Bkds/Bρ is the kicker strength (angle), Bk is the kicker dipole field, βx(sk) is the betatron amplitude function evaluated at the kicker location, βx(s) is the amplitude function at location s, and x(s) is the phase advance from sk of the kicker to location s. The quantity in curly brackets is called the kicker lever arm.

A schematic drawing of the central orbit xc, bumped orbit xb, and kicked orbit xk in a Lambertson septum magnet. The blocks marked with X are conductor-coils, The ellipses marked beam ellipses with closed orbits xc, xb, and xk. The arrows indicated a possible magnetic field direction for directing the kicked beams downward or upward in the extraction channel.

Beam Injection

1. Strip, charge-exchange injection scheme2. Betatron phase-space painting, cooling,

radiation damping3. Momentum phase-space stacking

a. Fast single turn extraction and box-car injectionb. Slow extraction

Beam Extraction

3. rf kicker

The discrete nature of the localized kicker generates error harmonics n + νm for all n ∈ (−∞,∞). If the betatron tune is 8.8, large betatron oscillations can be generated by an rf dipole at any of the following modulation tunes: νm = 0.2, 0.8, 1.2, 1.8, . . . . The localized repetitive kicks generate sidebands around the revolution lines.

The coherent betatron motion of the beam in the presence of an rf dipole at νm ≈ ν (modulo 1) with initial condition y = y′ = 0 is

The lower curve shows the measured vertical betatron oscillations at one BPM in the IUCF Cooler resulting from an rf dipole kicker at the betatron frequency. The rf dipole was turned on for 512 revolutions, and the beam was imparted by a one-turn kicker after another 512 revolutions. The betatron amplitude grew linearly during the rf knockout-on time.

The upper curve shows the fractional part of the betatron tune obtained by counting the phase advance in the phase-space map using data of two BPMs.

The beam profile measured from an ionization profile monitor (IPM) at the AGS during the adiabatic turn-on/off of an rf dipole. The beam profile appeared to be much larger during the time that the rf dipole was on because the profile was an integration of many coherent synchrotron oscillations. After the rf dipole was adiabatically turned off, the beam profile restored back to its original shape (Graph courtesy of M. Bai at BNL).

4. Orbit response matrix (ORM) and accelerator modelingClosed orbit vs dipole field change: We consider a set of small dipole perturbation given by θj , j=1, … Nb, where Nb is the number of dipole kickers. The measured closed orbit yi at the ith beam position monitors from a dipole perturbation is

The ORM method minimizes the difference between the measured and model matrices Rexp and Rmodel:

A.2.2 Effect of quadrupole field error:

0)( )( ,)( 0

ykKyBBzsKz

BBxsKx x

zz

x Let the transfer matrix of the unperturbed betatron be

00000 2 ,sincos)( JIsM

)(

)(

)()(

)(s

ss

ssJ

The perturbation of a thin quadrupole is

10

)(

1)(

111 dssk

sm

Substituting x = xco + xβ and z = zβ into the sextupole field, we obtain

The transfer matrix of the one-turn map is M(s1)=M0(s1)m(s1)

The transfer matrix of the one-turn map is M(s1)=M0(s1)m(s1)

010

01

1101001

11010101 sincos

sin

)(]sin[cossin)(sinsincos

)(M

dssk

dssks

011121

021 sin)(cosTr(M)cos dssk

111111111 )(41 ,)(

41 ,)(

21 dsskdsskdssk

Betatron tune shift

Φ=Φ0+ΔΦ

The betatron tune of the accelerator is shifted by the quadrupolefield error.

The betatron amplitude function is also changed by the quadrupolefield error. The betatron amplitude function can be obtained by a one-turn map: ),()(),()(M 211122 ssMsmsCsMs

)(2cos)(sin2

1120111

02

2

dssk

)(2cos)(sin2

1)()(

1201110

skdsss Cs

s

For distributed quadrupole field errors, the β-function perturbation is

Betabeat: betatron amplitude function perturbation

02022122 cossinsin)(M s

)2(sin)(sin)()(M 1202102111122 dssks

The M12 element is β2sinΦ. The difference (M-M0)12 is

The betatron amplitude function diverges at an integer or half-integerthe betatron tune! We expand the perturbation as

p

jppeJsk )(2

0 dseskJ jpp

)(21

Half integer stopband

where ϕ=(1/ν0)ds/β. We find Δβ/β satisfies

)(2)()(4

)()( 22

0202

2

skss

ss

dd

jp

p

p ep

Jss

)2/(2)(

)(22

0

0

)(2cos)(sin2

1)()(

1201110

skdsss Cs

s

)(2cos)()(sin2)(

)(1011

22

10

0

kdss

01)( 31

11 w

wsKw

,1 ,21 ,)(

22

ws

01)]()([ 32

22 w

wsksKw

∆w=w2−w1

wskww

wsKw )(3)( 4

Practice of Floquet transformation

Floquet transformation: dsw s

11 ,

21)(

2222

2222

2

4)(34

'"21)()(

skKsk

dd

)(2)()(4

)()( 222

2

2

skss

ss

dd

Left, schematic plot of a particle trajectory at a half-integer betatron tune resulting from an error quadrupole kick py = βyΔy′ = −βyy/f, where f is the focal length, y is the displacement from the quadrupole center, and βy is the betatron amplitude function at the quadrupole. The quadrupole kick is proportional to the displacement y. At a half-integer betatron tune, the betatron coordinate changes sign in each consecutive revolution and the kick angles coherently add in each revolution to produce unstable particle motion. Right: cancellation of two kicks by zero tune shift π-doublets, that produce only local perturbation to betatron motion.

Floquet transformation:

The equilibrium radius is R=1, i.e. the envelope is Y=βε.

If the injected beam is mismatched, the solution of the envelope is1 cos 2The beam envelope oscillates at 2ν, twice the betatron tune!

What happens if a beam has beam profile mismatch to the Courant-Snyder invariant? Envelope oscillation of a mis-matched beam

Example of one quadrupole error in FODO cell latticeConsider a simple accelerator lattice made of 18 FODO cells with half cell length 10-m, and dipole length 8 m bending angle 10o. The betatrontunes are set at νx=4.79302 and νz=4.78298 by quadrupoles. Now, consider an 1% decrease in focusing quadrupole strength at the end of the 10th cell. The betatron amplitude function perturbation is dominated by harmonics nearest [2νx] and [2νz].

Perturbation of betatron amplitude functions vs (either x or z) resulting from 1% decrease in gradient strength of the 10th focusing quadrupole. Since βx/βz∼6.37 at the focusing quadrupole location, the resulting error Δβx/βx is about 6.37Δβz/βz. A single kick at the error quadrupole location can be identified in the top 2 plots. The bottom plot shows the effect of quadrupole error on dispersion function shown as ΔDx/√βx vs =x. A single kick at the error quadrupole location is visible to the dispersion closed orbit.

Betabeat Can you spot the locations of quadrupole error? What can you say about the betabeat?

+2% and -2% errors with phase advance about 2pi Qx = 4.406299 Qy = 4.391029 18FODO cells

Qx=4.406

What happens if the −2% is changed to +2%?

What happens if two quadrupoles are separated by π phase advance?

Why stopband of two quadrupoles ±2% cancel each other?

What happens if the stopband is exactly cancelled for the resonance harmonic in the horizontal plane?

What happens to the vertical betatron amplitude function?

Note that the betatron amplitude function diverges when the betatron tune is integer or half-integer!

)(2)()(4

)()( 22

02

2

skss

ss

dd

,)(20

p

jppeJsk

dseskJ jpp

)( 21

jp

p

p ep

Jss

)2/(2)(

)(22

0

0

Effect of Linear resonances

111 )(41 dssk

1. Betatron amplitude function measurement

2. Tune jump

3. Half-integer stopband corrections

Applications of quadrupole error

The horizontal and vertical tunes, determined by the FFT spectrum of the betatron oscillations, vsquadrupole field strength. The slope can be used to determine the average betatron amplitude function in a quadrupole.

The fractional parts of betatron tunes were qx=4−νx and qz=5−νz,. The experimental result of fractional horizontal tune appeared to “increase” with the strength of the horizontal defocusing quadrupole.

Q: Is the quadrupole focusing or defocusing? At this location, what can you say about the betatron amplitude functions? Is this machine a FODO cell like?

111 )(41 dssk

See Lecture Note of Q. Qin, ocpa accelerator school 2002. Data: X. Wen (IHEP).

Transverse beam position monitors (BPMs) or pickup electrodes (PUEs) are usually composed of two or four conductor plates or various button-like geometries.

Betatron oscillations in the presence of a dipole kick

Courant-Snyder Invariant

A circulating particle passes through the pickup electrode (PUE) at fixed time intervals T0, where T0 = 2πR/βc is the revolution period, R is the average radius, and βc is the speed. The current of the orbiting charged particle observed at the PUE is

Transverse spectra of a particle

A schematic drawing of current pulses in the time domain (upper plot), in the frequency domain (middle plot), and observed in a spectrum analyzer (bottom plot). The rf current is twice the DC current because negative frequency components are added to their corresponding positive frequency components.

If we apply a transverse impulse (kick) to the beam bunch, or if there is a betatron oscillation about the closed orbit. The BPM measures the transverse coordinates of the centroid of the beam charge distribution (dipole moment), given by

In the frequency domain, the betatron oscillations obtained from the BPM signals contain sidebands around the revolution frequency lines, i.e. f = (n ± Qy)f0 with integer n. Naturally, the betatron tune can be measured by measuring the frequency of betatron sidebands.

The spectrum around the rf frequency of 476.312 MHz for the SPEAR3. The harmonic number is 372. Thus the upper betatron sideband corresponds to n = 366, and the lower betatron sideband corresponds to n = −378. The corresponding vertical betatron tune is about 6.18. The betatron sidebands are classified into fast wave, backward wave, and slow wave. The betatron sidebands are invisible during production runs. The sidebands appear only when the vertical chromaticity is set to 0. (Graph: courtesy of Xiaobiao Huang).

372−378+6.16 +366+6.16

For beam distribution with a finite time span:

The frequency spectrum has a cut off depending on the bunch length, e.g.

Form factor:

Form factor:

The form factors for Fourier spectra of a beam with Gaussian, rectangular, elliptical uniform, and triangular distributions with rms bunch length σt = 1 ns. The form factor serves as the envelope of the revolution comb lines. All distributions with same rms bunch length has the same 6dB roll-off frequency, froll−off=0.187/σt . The coherent signal of a rectangular bunch can extend beyond the rolloff frequency.

Beam power (10 dB /division) vs frequency. Top: The epctra (0-3 GHz) of a TLS production beam. There are 154 bunches in 200 buckets with frf = 499.6438 MHz. The rms bunch length is about 24ps. Bottom: The spectrum around the frequency frf with span 20 kHz. The power of the synchrotron mode is about 82 dB below that of the revolution harmonic (Graph courtesy of Yi-Chih Liu).

Turn-by-turn data can also be used for accelerator modeling

Consider the betatron motion:We can organize the turn-by-turn betatron coordinates in a data matrix, X. If we carry out singular value decomposition, we find

)(

),...,,(22

21

21

sUUUUUU

x

n

Note that The eigenvalues of betatron modes increase with the number M of BPMs and the number of turns N measured.

What happens if BPM data are noisy? How about the beam energy deviates from the designed value?

+N(s,t) + D(s)[Δp/p](t) + other sources

Spatial wave function

Temporal wave function

A.3 Off-momentum closed orbit and dispersion function Including dipole field errors and quadrupole misalignment, we found the closed orbit for a reference particle with momentum p0. By using closed-orbit correctors, we can achieve an optimized closed orbit that essentially passes through the center of all accelerator components. This closed orbit is called the “golden orbit,” and a particle with momentum p0 is called a synchronous particle. A beam is made of particles with momenta distributed around the synchronous momentum p0. • What happens to particles with momenta different from p0? • What is the effect of off-momentum on the closed orbit?

For a particle with momentum p, the momentum deviation is ∆p=p-p0and the fractional momentum deviation is δ=∆p/p0, which is typically small of the order of 10–6 to 10–3. Since δ is small, we study the motion of off-momentum particles in perturbation expansion of δ.

1 ,0

0

ppppp

)()(2 )( ,)1(

)()(1

)1()1()1(1

211

11

,

222

2

2

2

2

00

OsKsKxsKsKx

xKx

xxKxxx

ppppp

.)1( ,)1( 20202

xpp

BBzx

pp

BBxx xz

Expanding the betatron equation of motion, we obtain

xBB

BBsKxsKx z

1

12 ,)( ,

)1(1)(

)1(1

For a planar accelerator, the horizontal betatron equation of motion for particles with nonzero δ is inhomogeneous. The solution of the inhomogeneous equation is a linear combination of the particular solution and the solution of the homogeneous equation, i.e.

Dxx ,0)( xKsKx xx

)(1)(

ODKsKD xx

)(1)( 2 sKsK x

The solution of the homogeneous equation is the betatron oscillation. The solution of the inhomogeneous equation is called the dispersion function, or the off-momentum closed orbit.

Dxx

,)1(

)(1

)()(2 )( ,)1(

)()(1

2

222

xsKx

OsKsKxsKsKx

For a pure dipole:

Dxxxx co

,1)(12

DsKD

Kx=1/ρ2

For a pure dipole:

,1)(12

co

DsKD

Dxxxx

1 0 0

1 0 1

1 0 0

sin cos sin)cos1( sin cos 2

21

1

M

For pure quadrupoles:

1 0 00 0 1/0 0 1

1 00 cos

0 sin

0

sin

cos

),(

1

0 fK

K

KK

K

ssMK

1 0 00 0 1/0 0 1

1 00 ||cosh

0 ||sinh

0

||sinh||

||cosh

),(||

1

0 fK

K

KK

K

ssMK

Using the Courant-Snyder parameterization for the transfer matrix, we obtain

Closed orbit condition:

Example: FODO cell

Result: (1) The dispersion is proportional to the length of the cell L, the bending angle θ, and inversely proportional to the square of the phase advance. (2) The dispersion at other locations can be obtained by using the transfer matrix M(s2,s1).

The AGS (33 GeV proton synchrotron built in 1960) is simply made of 60 (5×12) FODO cells. The CPS (28 GeV) is simply made of 50 FODO cells.

The betatron amplitude functions for one superperiod of the AGS lattice, which made of 20 combined-function magnets. The upper plot shows βx (solid line) and βz (dashed line). The middle plot shows the dispersion function Dx. The lower plot shows schematically the placement of combined-function magnets. Note that the superperiodcan be well approximated by five regular FODO cells. The phase advance of each FODO cell is about 52.8o.

sin

)2

sin1(2

sin

)2

1(21

11

max

LfLL

Ldip = 2, 4, 6, 8 mL=10m, LQ=1m, P=18

What is the effect of bending radius on dispersion function?

Damping re-partition number

We recall that we define the normalized betatron phase-space coordinates: y2+Py

2=y2+(αy+βy’)2=2βJ.We define the normalized dispersion function coordinates:

The H-function of the dispersion invariant is defined as:

Xd and Pd of AGS lattice. Note that Pd≈0 and Xd≈constant for FODO cells.

Example: APS lattice is made of 40 Double-bend Achromats (DBA) with a total length of 1104m. The momentum compaction factor for all DBA lattice is αc=ρθ2/(6R).

Because of its simplicity and flexibility, DBA lattice is commonly used as basic cells of synchrotron light source design.

Xd and Pd of DBA lattice in the dispersion matching section is a circle. The entire dispersion free straight section is a single point Pd=Xd=0.

Example: The Cooler is designed to have 3 dispersive and 3 dispersion free straight sections. In the dispersive straight section Pd vs Xd is a circle. In the entire dispersion free straight section, we have Pd=Xd=0!

An accelerator with circumference 360 m is made of 18 FODO cells. The betatron tunes of the synchrotron are νx=4.793 and νz=4.783 respectively. If one of the focusing quadrupoles has 1% error, estimate the maximum change in ΔDx.

xF 35.0xD 5.17f 6.73DxF 4.33DxD 1.991 ∆ ∆ ⋯

Floquet transformation: η=ΔD/, ϕ=(1/ν)ds/

/ 12 /

≅ cosFor the above problem, we find |η|~0.03 m, and the maximum ΔD~0.18 m.

/

12 35 ∆ 1 4.33 0.0061

Effect of half-integer stopband on dispersion function/

Qx=4.406

What happens if the −2% is changed to +2%?

We know the effect of half integer stopbands on bettatron amplitude functions!

What is the perturbation on the dispersion function?

Path length, momentum compaction and phase-slip factors:

We recall the Frenet-Serret coordinate system. The path length of the reference orbit in one complete revolution is

]1[])1[()()()()1( 2/12222222xxx dszxdsdzdxdsd

dsDx

CdsxdsdC

0

Here αc is called the momentum compaction factor, which is a measure of the compactness of the orbit length for particles with different momenta. We examine its effect on the ORM modeling.

c dsDC

iiiD

CdsD

CCdCd

11 c

Orbit response matrix method and accelerator modelingClosed orbit vs dipole field change:

The closed orbit of the accelerator resulting from a dipole error at s0 is

),()( 0co ssGsy

|])()(|cos[sin2

)()(),( 0

00 ss

ssssG

What is Rij?

We consider a set of small dipole perturbation given by θj , j=1, … Nb, where Nb is the number of dipole kickers. The measured closed orbit yi at the ith beam position monitors from a dipole perturbation is

Effect of dispersion function on ORMA dipole-kick θj at position sj changes the closed orbit G(si, sj)θj, and change the circumference ΔC=D(sj)θj. 1. Constant momentum: The change of the revolution period is ∆T=∆C/βc=D(sj)θj/βc at a constant velocity. Similarly the rf frequency must be adjusted according to ∆f/f=−∆T/T in order to maintain a constant momentum, The beam motion at this new rf frequency is on-momentum, i.e. δ = 0 and the closed orbit should be

2. Constant path length: If the rf frequency is maintained constant in the ORM measurement, the path length is constant. An equivalent off-momentum “δ”=∆C/(αcC0) is needed to compensate path length change by the dipole bump. Thus the corresponding closed orbit is

,

, +

Another important of the orbit length change is that the particles in synchrotron must synchronize with the rf accelerating voltage. Note that the orbiting time for particle is T=C/v. Thus

Here η is the phase slip-factor.

Pulse from linac Extracted beam

Lambertson septum

Kicker 1 Kicker21. Beam in and out in

one revolution can debunch linacbunches if Δp/p>0.005.

2. The accelerator can also accumulate particles for a storage ring.

ALPHA as a de-bunch ring

Note that a large compaction factor is necessary for achieving de-bunching for the electron beams in a single path!

022

02

11EE

pp

0

0

0

0

00

0

0

0 )/1()11(

EdtdE

EEEEEEEEE

V=V0sin(ωrft+φ)ωrf=hf0

V=V0sin(ωrft+φ)ωrf=hf0

Phase stability and the synchrotron equation of motion:

0 π/2 π 3π/2 2πφ

V0

−V0

0π/2φ s

η<0 η>0

Acceleration

Deceleration

φ s

Lower Energy

Synchronous Energy

Higher Energy

Illustration of the Phase stability: A beam bunch consists of particles with slightly different momenta. A particle with momentum p has its own off-momentum closed orbit Dδ. Since the energy gain depends sensitively on the synchronization of rf field and particle arrival time, what happens to a particle with a slightly different momentum when the synchronous particle is accelerated?

The key answer is the discovery of the phase stability of synchrotron motion by McMillan and Veksler. If the revolution frequency f is higher for a higher momentum particle, i.e. df/dδ>0, the higher energy particle will arrive at the rf gap earlier, i.e. φ<φs. Therefore if the rf wave synchronous phase is chosen such that 0<φs<π/2, higher energy particles will receive less energy gain from the rf gap.

Similarly, lower energy particles will arrive at the same rf gap later and gain more energy than the synchronous particle. This process provides the phase stability of synchrotron motion. In the case of df/dδ<0, phase stability requires π/2<φs<π.

)sin(sin1 snnn eVEE

Synchrotron equation of motion:

1212

nnn EE

The separatrix orbits for η > 0 (γ>γT) with ϕs=2π/3, 5π/6, π, (top) and for η < 0 (γ<γT) with ϕs= 0, π/6, π/3 (bottom). The phase space area enclosed by the separatrix is called the bucket area. The stationary buckets that have largest phase space areas correspond to ϕs=0 (bottom) and π (top) respectively.

1. Particle motion in an accelerator can be described by 3D simple harmonic motion. The transverse oscillations is called betatron motion and the longitudinal motion is called the synchrotron motion.

2. The betatron tunes are number of betatron oscillations per revolution, and the synchrotron tune is the number of synchrotron oscillations per revolution. The betatron tunes increase with the size of the accelerator, while the synchrotron tune is about 10–4 to 10–2.

3. The momentum compaction factor plays an important role in the accelerator. Typically, the momentum compaction factor for FODO cell lattice is αc~1/νx

2. Thus the transition energy is γT~νx. However, the momentum compaction for accelerators can be changed by changing the dispersion function in dipoles.

i

iiDC

dsDCCd

CddsDC

11 , c

Summary:

C=86.8 m

K=45 – 270 keVI=0-4 A

How to measure D(s)? xco(s)=D(s). Lecture note by Q. Qin in OCPA accelerator school 2002 in Singapore. Data: X. Wen, IHEP

.)1( ,)1( 20202

xpp

BBzx

pp

BBxx xz

1/ 0pp

Dxx 0)( ,0)( xKsKzxKsKx zzxx

)(1)(

ODKsKD xx

)()]([)( ),()(

,)()](2[)( ),(1)( 22

sKsKsKsKsK

sKsKsKsKsK

zzz

xxx

A.4 Chromatic aberrationInhomogeneous equation

Note that the betatron motion for off momentum particle is perturbed by a chromatic term. The betatron tunes must avoid half-integer resonances. But, the quadrupole error is proportional to the designed quadrupole field. They are called systematic chromatic aberration. It is an important topic in accelerator physics.

xBB

BBsKxsKx z

1

12 ,)( ,

)1(1)(

)1(1

1. Tune shift, or tune spread, due to chromatic aberration:

ddCCdssKs

ddCCdssKs

zzzzzz

xxxxxx

/ ,)()(

/ ,)()(

41

41

The chromaticity induced by quadrupole field error is called natural chromaticity. For a simple FODO cell, we find

yyy

ixixxx

ffNC

fdssKs

2/)2/tan(

/)()(

minmax41FODO

nat,

41

41

sin

))2/sin(1(2 ,sin

))2/sin(1(2 1min

1max

LL fL22

sin 1

We define the specific chromaticity as zzzxxx CC / ,/

The specific chromaticity is about −1 for FODO cells, and can be as high as −4 for high luminosity colliders and high brightness electron storage rings.

Chromaticity measurement:The chromaticity can be measured by measuring the betatron tunes vs the rffrequency f, i.e. /

M. Yoon and T. Lee, RSI 68, 2651 (1997)

The Natural chromaticity can be obtained by measuring the tune variation vs the bending-magnet current at a constant rf frequency. Change of the bending-magnet current is equivalent to the change of the beam energy. Since the orbit is not changed, the effect of the sextupole magnets on the beam motion can be neglected. The Figure shows the horizontal and vertical tune vs the bending-magnet current in the PLS storage ring.

The data give Cx=−18.96, Cz=−13.42; vs theory: Cx=−23.36, Cz=−16.19.

M. Yoon and T. Lee, RSI 68, 2651 (1997)

/ / /

Fermilab Booster (X. Huang, Ph.D. thesis 2005): The measured horizontal chromaticity Cx when SEXTS is on (triangles) or off (stars), and the measured vertical chromaticity Czwhen SEXTS is on (dash, circles) or off (squares). The error bar is estimated to be 0.5. The natural chromaticitiesare Cnat,z=−7.1 and Cnat,x=−9.2 for the entire cycle.

BNL AGS (E. Blesser 1987): Chromaticities measured at the AGS.

yyyC

2/

)2/tan(FODOnat,

Examples:

∗ ∗CIR=

Contribution from 6-IRs

Contribution outside IRs

0

10

20

30

40

50

60

70

80

1 2 3 4 5 6 7 8 9 10

*(m)

The total chromaticity is composed of contributions from low -quads and the rest of accelerators that is made of FODO cells. Contribution of low triplets in an IR to the natural chromaticity is

CIR= The decomposition to fit the data is Δs≈35 m in RHIC.

β and D vs Δp/pν vs Δp/p

Other important chromatic aberrations are the off-momentum betabeat, and perturbation to the dispersion function.

2. Chromaticity correction:The chromaticity can cause tune spread to a beam with momentum spread ∆ν=Cδ. For a beam with C=−100, δ=0.005, ∆ν=0.5. The beam is not stable for most of the machine operation. Furthermore, there exists collective (head-tail) instabilities that requires positive chromaticity for stability! To correct chromaticity, we need to find magnetic field that provide stronger focusing for off-(higher)-momentum particles. We first try sextupole with

3203

1220 Re , jzxbBAjzxbBBjB sxz

BBzsKz

BBxsKx x

zz

x

)( ,)(

Dxx

zxbBDzbBxzbBBzxDDxbBzxbBB

x

z

222)2()(

202020

222220

2220

zz

0))(( ,0))(( 22 zDKsKzxDKsKx zx

Let K2=−2B0b2/Bρ= − B2/Bρ, we obtain:

With sextupoles, the chromaticities becomes

dssDsKsKsC

dssDsKsKsC

zzz

xxx

)]()()()[(

)]()()()[(

241

241

))2/sin(1(2)2/sin( ,

))2/sin(1(2)2/sin(

212SDD2D

212SF2F

f

KSf

KS

For FODO cells, the integrated sextupole strength is

For high energy colliders and high brightness synchrotron light sources, the sextupolestrength can be much higher. Even more important is the effect of the systematic half-integer stopbands.

dsesKsJ sjpyypy

y )(, )()(

21

)()]([)( ),()(

,)()](2[)( ),(1)( 22

sKsKsKsKsK

sKsKsKsKsK

zzz

xxx

3. Betabeat – systematic stopbands

)(2)()(4

)()( 22

02

2

sKss

ss

dd

)(2sin)()(sin2)(

)(1011

22

10

0

Kdss

,)(20

p

jppeJsK dsesKJ jp

p

)(

21

Half integer stopband

jp

p

p ep

Jss

)2/(2)(

)(22

0

0

What symmetry can do to stopbands?

Systematic chromatic half-integer stopband width

The effect of systematic chromatic gradient error on betatron amplitude modulation can be analyzed by using the chromatic half-integer stopbandintegrals

We consider a lattice made of P superperiods, where L is the length of a superperiod with K(s + L) = K(s), β(s + L) = β(s). Let C = PL be the circumference of the accelerator. The integral becomes

1. Particle motion in an accelerator can be described by 3D simple harmonic motion. The transverse degree of freedom is called betatron motion and the longitudinal degree of freedom is called the synchrotron motion.

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Summary:

2. The betatron tunes are number of betatron oscillations per revolution, and the synchrotron tune is the number of synchrotron oscillations per period.

3. The momentum compaction factor plays an important role in the accelerator. Typically, the momentum compaction factor for FODO cell lattice is αc~1/νx

2. Thus the transition energy is γT~νx. However, the momentum compaction for accelerators can be changed by changing the dispersion function in dipoles.

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4. Phase stability and the synchrotron equation of motion:

V=V0sin(ωrft+φ)ωrf=hf0

The betatron tunes increase with the size of the accelerator, while the synchrotron tune is about 10–4 to 10–2.