a. transverse (betatron) motion s.y. lee, - iu bphysics.indiana.edu/~shylee/ocpa12/ocpa12_g2.pdfa....
TRANSCRIPT
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
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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.