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Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 1
Betatron Motion with Coupling of Horizontal and Vertical Degrees of
Freedom – Part II
USPAS, Fort Collins, CO, June 13-24, 2016
S.A. Bogacz, G.A. Krafft, S. DeSilva and R. Gamage
Jefferson Lab and Old Dominion University
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 2
Outline
Practical Examples:
Vertex-to-plane adapter for electron cooling (Fermilab)
Spin Rotator for Figure-8 Collider ring
Ionization cooling channel for Neutrino Factory and Muon Collider
Generalized vertex-to-plane transformer insert
V. Lebedev, A. Bogacz, ‘Betatron Motion with Coupling of Horizontal and Vertical Degrees of Freedom’, 2000, http://dx.doi.org/10.1088/1748-0221/5/10/P10010
USPAS, Fort Collins, CO, June 13-24, 2016
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 3 USPAS, Fort Collins, CO, June 13-24, 2016
Axisymmetric rotational distribution - Twiss functions
v Fermilab electron cooling The electron beam distribution
is axially symmetric, and uncoupled at the cathode:
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
00
00
00
00
0000
0000
1
βα
αγ
βα
αγ
εTBΞ
where 0/ PmkTr ccT =ε is the thermal emittance of the beam
Electrostatic accelerator Solenoid
Gaussian distribution
Rotational distribution
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 4 USPAS, Fort Collins, CO, June 13-24, 2016
Axisymmetric rotational distribution - Twiss functions
♦ At the exit of the solenoid the electron beam distribution is still axially symmetric
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
Φ−
Φ+Φ
Φ
Φ−Φ+
==
000
002
00
000
0002
0
00
00
1
βαβ
αβγβ
ββα
βαβγ
εTB
Tin ΦΞΦΞ
where
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
Φ−
Φ=
10001000100001
Φ
♠ cPeB 02/=Φ is the rotational focusing strength of the solenoid edge
♠ B is the solenoid magnetic field.
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 5 USPAS, Fort Collins, CO, June 13-24, 2016
Axisymmetric rotational distribution - Twiss functions
♦ The eigen-vectors of the rotational distribution:
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
+−
+−
=
βα
ββα
β
22
22
ˆ
ii
i
i
1v ,
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
+−
+−
=
βα
ββα
β
22
22
ˆ 2
i
ii
i
v
♠ It corresponds to u = 1/2, 2/21 πνν ==
♦ Then, the matrix V̂ is
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−
−
−
=
ββα
βα
β
βββα
βββα
ββ
21
21
0021
21
00
V̂
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 6 USPAS, Fort Collins, CO, June 13-24, 2016
Axisymmetric rotational distribution - Twiss functions
• 4D-emmitance conservation:
2
21 Tεεε =
• Rotational emittance estimate
( ) ( )Φ=Φ=Φ== 02 βεθε Trot rrrr
♦ Comparing left and right hand sides of the equation
TTin UVVUΞ ˆ
/10000/10000/10000/1
ˆˆ
2
2
1
1
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
ε
ε
ε
ε
♠ One obtains
.21
,21
,12
,12
0
1
020
22
01
020
21
20
2
0
20
2
0
0
0
β
ε
ββ
εε
εβββ
εε
β
αα
β
ββ
β
β
Φ⎯⎯ →⎯
Φ+Φ+=
Φ⎯⎯ →⎯Φ−Φ+
=
Φ+=
Φ+=
>>Φ
>>Φ
TT
TT
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 7
Spin rotators for Figure-8 Collider Ring
-10000
-8000
-6000
-4000
-2000
0
2000
4000
6000
8000
10000
-20000 -15000 -10000 -5000 0 5000 10000 15000 20000x [cm]
z [cm]
Figure-8 Collider Ring - Footprint
total ring circumference ~1000 m
60 deg. crossing
USPAS, Fort Collins, CO, June 13-24, 2016
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 8
Spin Rotator - Ingredients…
Arc end 4.4 0 8.8 0
Spin rotator ~ 46 m
BL = 11.9 Tesla m BL = 28.7 Tesla m
320 230
15
0
0.15
-0
.15
BETA
_X&Y
[m]
DIS
P_X&
Y[m
]
BETA_X BETA_Y DISP_X DISP_Y
USPAS, Fort Collins, CO, June 13-24, 2016
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 9
Locally decoupled solenoid pair
solenoid 4.16 m solenoid 4.16 m decoupling quad insert
BL = 28.7 Tesla m
M = C
- C 0 0
17.9032 0
15
0
5 0
BETA
_X&Y
[m]
BETA_1X BETA_2Y BETA_1Y BETA_2X
USPAS, Fort Collins, CO, June 13-24, 2016
Hisham Sayed, PhD Thesis ODU, 2011
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 10
solenoid 4.16 m solenoid 4.16 m decoupling quad insert
BL = 28.7 Tesla m
M = C
- C 0 0
17.9032 0
15
0
1 -1
BETA
_X&Y
[m]
DIS
P_X&
Y[m
]
BETA_X BETA_Y DISP_X DISP_Y
Locally decoupled solenoid pair
USPAS, Fort Collins, CO, June 13-24, 2016
Hisham Sayed, PhD Thesis ODU, 2011
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 11
Universal Spin Rotator - Optics
4.4 0 8.8 0
Spin rotator ~ 46 m
BL = 11.9 Tesla m BL = 28.7 Tesla m
3 7 42 8 8
M o n S e p 0 6 1 7 : 4 4 : 4 5 2 0 1 0 O p t i M - M A I N : - C : \ W o r k i n g \ E L I C \ M E I C \ O p t i c s \ 5 G e V E l e c t e . R i n g \ h a l f _ r i n g _ i n _ s t r a i g h t _ 1 2
30
0
1-
1
BE
TA
_X
&Y
[m
]
DI
SP
_X
&Y
[m
]
B E T A _ XB E T A _ YD I S P _ XD I S P _ Y
374 288
30
0
1 -1
BETA
_X&Y
[m]
DIS
P_X&
Y[m
]
BETA_X BETA_Y DISP_X DISP_Y
5 GeV
USPAS, Fort Collins, CO, June 13-24, 2016
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 12 USPAS, Fort Collins, CO, June 13-24, 2016
Ionization Cooling in an Axially Symmetric Channel
v A single-particle phase-space trajectory along the beam orbit can be expressed as:
( ) ,)(ˆ)(ˆRe)(ˆ ))((22
))((11
2211 sisi esess µψµψ εε +−+− += vvx
♦ One can rewrite the above equations in the following compact form
)()(ˆ)(ˆ sss aVx =
where
⎥⎦⎤
⎢⎣⎡ ʺ−ʹʺ−ʹ= )(ˆ),(ˆ),(ˆ),(ˆ)(ˆ 2211 sssss vvvvV
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+
+
+
+
=
))(sin())(cos())(sin())(cos(
)(
222
222
111
111
ssss
s
µψε
µψε
µψε
µψε
a
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 13 USPAS, Fort Collins, CO, June 13-24, 2016
Ionization Cooling in an Axially Symmetric Channel
♦ In the case of axially symmetric focusing the eigen-vectors reduce to:
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
+−
+−
=
βα
ββα
β
22
22
ˆ
ii
i
i
1v ,
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
+−
+−
=
βα
ββα
β
22
22
ˆ 2
i
ii
i
v
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−
−
−
=
ββα
βα
β
βββα
βββα
ββ
21
21
0021
21
00
V̂
♠ here we used that u = 1/2, ν1 = ν2 =π/2
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 14 USPAS, Fort Collins, CO, June 13-24, 2016
Ionization Cooling in an Axially Symmetric Channel
v Cooling Description ♦ Ionization cooling due to energy loss in a thin absorber can be
described as:
δθθθ ⊥⊥⊥ −≡Δ
−=Δpp ,
♠ here the longitudinal energy restoration by immediate re-acceleration is assumed
♦ Using canonical variables the above cooling equation can be written as:
inout xRRx ˆ
1000010000100001
ˆ 1−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−=
δ
δ
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 15 USPAS, Fort Collins, CO, June 13-24, 2016
♦ Canonical variables
.2
,2
xRyp
yRxp
y
x
+ʹ=
−ʹ= PceBR s /= - longitudinal magnetic field
♦ Relation between geometrical and canonical variables Rxx =ˆ ,
where
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
≡
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
≡
1002010002100001
,,ˆ
R
Ry
x
pypx
y
x
y
x Rxx
θ
θ
,
A ‘cap’ denotes transfer matrices and vectors related to the canonical variables.
Hamiltonian formulation - equations of motion
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 16 USPAS, Fort Collins, CO, June 13-24, 2016
Ionization Cooling in an Axially Symmetric Channel
♦ Employing amplitude vector representation: )()(ˆ)(ˆ sss aVx = , one can rewrite the cooling equation as:
inout aVRRaV ˆ
1000010000100001
ˆ 1−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−=
δ
δ
and finally
inout aVRRVa ˆ
1000010000100001
ˆ 11 −−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−=
δ
δ
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 17 USPAS, Fort Collins, CO, June 13-24, 2016
Ionization Cooling in an Axially Symmetric Channel
♦ Carrying out the above calculation explicitly one obtains:
inout
RR
RR
RR
RR
aa
⎪⎪⎪⎪
⎭
⎪⎪⎪⎪
⎬
⎫
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
+−
−
+−−
+−−
+−
−
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
21
21
21
21
21
21
21
21
1000010000100001
βαα
β
αββ
α
αββ
α
βαα
β
δ
♦ 2D emittances after cooling are given by the following formula:
( )[ ] ( )[ ]
( )[ ] ( )[ ] )(sin1cos211
)(sin1cos2112
21222
2
2211
221
43
21
δδφβφαεεδβεε
δδφβφαεεδβεε
ORRaa
ORRaa
outout
outout
+−−++−=+≡ʹ
++−+−−=+≡ʹ
where 2121 ψψµµφ +++=
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 18 USPAS, Fort Collins, CO, June 13-24, 2016
Ionization Cooling in an Axially Symmetric Channel
v Two alternative descriptions of cooling
♦ After passing through a thin absorber one computes ♠ a new 4D phase space ♠ new partial emittances ♠ new beta-functions
♦ Or one can compute everything relative to the unperturbed beta-functions
♠ Seems like more convenient approach, although partial emittances (actions) depend on betatron phases
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 19 USPAS, Fort Collins, CO, June 13-24, 2016
Ionization Cooling in an Axially Symmetric Channel
v If cooling effect of one absorber is sufficiently small one can perform averaging over betatron phases. That yields
( )( )δβεε
δβεε
RR
+−≈Δ
−−≈Δ
11
22
11 ⇒
dsdp
pR
dsd
dsdp
pR
dsd
0
2
2
0
1
1
11
11
βεε
βεε
+−≈
−−≈
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 20 USPAS, Fort Collins, CO, June 13-24, 2016
Ionization Cooling in an Axially Symmetric Channel
♦ Canonical momentum of a single particle
( )2
ˆ
0000000001001000
ˆˆ
0000000001001000
ˆ 21 εε −=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
=−= aVaVxxMTT
xy ypxp
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 21 USPAS, Fort Collins, CO, June 13-24, 2016
Ionization Cooling in an Axially Symmetric Channel
♦ Second order moments of the Gaussian distribution
(Note that for a single particle - 2/εε =rms and we use rms. emittances below)
( )
( )
( )
0
,2
,441
,
,
2121
21
222
21
2122
==
−=⇒−
=−=
++
==
+−==
+==
yx
xy
yx
yx
ppxy
Mypxp
pp
ypxp
yx
εεεε
εεβα
εεα
εεβ
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 22 USPAS, Fort Collins, CO, June 13-24, 2016
Ionization Cooling in an Axially Symmetric Channel
v Beta-function for Particle Motion with Axial-symmetric Solenoidal Focusing
♦ Equation for the square root of the beta-function is similar to the
equation for Floque-function in the case of uncoupled motion:
( )0
41
4 3
2
2
2
=−+β
ββ R
dsd .
♠ The standard recipe determines the alpha-function:
dsdβ
α21
−=
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 23 USPAS, Fort Collins, CO, June 13-24, 2016
Beta functions in axially symmetric FOFO cell
c1 L[cm]=130 B[kG]=38.1 Aperture[cm]=10 c2 L[cm]=80 B[kG]=-34.3 Aperture[cm]=10
7.20
150
50
BETA_X&Y[m]
DISP_X&Y[m]
BETA_X BETA_Y
7.20
0.5
0PHASE_X&Y
Q_X Q_Y
abs abs
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 24
20
Fri Mar 10 09:55:38 2006 OptiM - MAIN: - D:\Cooling Ring\SolRing\snake_new.opt
0.5
0PH
ASE
_X&Y
Q_X Q_Y
Periodic Cell - Optics
20
Sat Mar 04 23:06:09 2006 OptiM - MAIN: - D:\Cooling Ring\SolRing\snake_new.opt
70
2-2
BE
TA
_X
&Y
[m]
DIS
P_X
&Y
[m]
BETA_X BETA_Y DISP_X DISP_Y
‘inward half-cell’ ‘outward half-cell’
betatron phase adv/cell (h/v) = π/2π
USPAS, Fort Collins, CO, June 13-24, 2016
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 25
Periodic Cell – Magnets ‘inward half-cell’
solenoids: L[cm] B[kG] 22 105 22 105
‘outward half-cell’
dipoles: $L=20; => 20 cm $B= 25; => 25 kGauss $Ang=$L*$B/$Hr; => 0.4996 rad $ang=$Ang*180/$PI; => 28.628 deg
quadrupoles: L[cm] G[kG/cm] 8 1.79754 8 -0.3325 8 1.79754
20
Sat Mar 04 23:06:09 2006 OptiM - MAIN: - D:\Cooling Ring\SolRing\snake_new.opt
70
2-2
BE
TA
_X
&Y
[m]
DIS
P_X
&Y
[m]
BETA_X BETA_Y DISP_X DISP_Y
betatron phase adv/cell (h/v) = π/2π
USPAS, Fort Collins, CO, June 13-24, 2016
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 26
‘Snake’ cooling channel – Dispersion suppression
60
Fri Mar 10 10:15:57 2006 OptiM - MAIN: - D:\Cooling Ring\SolRing\end_snake_new.opt
70
2-2
BE
TA
_X
&Y
[m]
DIS
P_X
&Y
[m]
BETA_X BETA_Y DISP_X DISP_Y 60
Fri Mar 10 10:16:18 2006 OptiM - MAIN: - D:\Cooling Ring\SolRing\end_snake_new.opt
70
2-2
BE
TA
_X
&Y
[m]
DIS
P_X
&Y
[m]
BETA_X BETA_Y DISP_X DISP_Y
periodic cells entrance cell exit cell
USPAS, Fort Collins, CO, June 13-24, 2016
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 27
Muon Cooling Channel - Optics
100
Wed Jun 21 11:47:31 2006 OptiM - MAIN: - D:\Cooling Ring\Snake Channel\snake_100_1.opt
50
3-3
BE
TA
_X
&Y
[m]
DIS
P_X
&Y
[m]
BETA_X BETA_Y DISP_X DISP_Y
n periodic PIC/REMEX cells (n=2) beam extension beam extension
RF cavity skew quad
disp. anti-suppr. disp. suppr.
snake - footprint
0
50
100
150
200
0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000
z [ c m ]
x [ c m ]
USPAS, Fort Collins, CO, June 13-24, 2016
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 28 USPAS, Fort Collins, CO, June 13-24, 2016
Vertex-to-plane transformer insert
SolenoidSkew-quadrupole system
Uncoupledaxial-symmetricdistribution
Rotational distribution
Flat distribution
xPx
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 29 USPAS, Fort Collins, CO, June 13-24, 2016
Vertex-to-plane transformer insert
♦ Eigen-vectors of the decoupled motion in the coordinate system rotated by 450
⎥⎦
⎤⎢⎣
⎡≡
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
+−
+−
⎯⎯ →⎯
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
+−
+−
⎯⎯ →⎯
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡+
− ==
2
222
1
1
1
1
1
1
.deg45
1
1
1
22
22
21
00
11
FF
βα
ββα
β
βαββαβ
βαβ
ααββ
i
i
i
ii
rotation
♠ Rotational eigen-vectors
⎥⎦
⎤⎢⎣
⎡
2
2
FFi
⎥⎦
⎤⎢⎣
⎡
2
2
FFi
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 30 USPAS, Fort Collins, CO, June 13-24, 2016
Vertex-to-plane transformer insert
♦ Focusing system with 450 difference between the horizontal and vertical betatron phase advances will transform the initial vertex distribution into the flat one
♦ The resulting 2D emittances are as follows
.1
,1 0
20
22
020
21ββ
εε
ββ
εε
Φ+Φ+=
Φ−Φ+= TT
♦ Lattice implementation – Twiss functions, beam sizes etc.
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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 31 USPAS, Fort Collins, CO, June 13-24, 2016
Vertex-to-plane transformer insert
Ekin = 10 MeV, Tc = 0.2 eV, Rc = 0.5 cm, Bsol = 1 kG,
⇒ ε1 = 7.14⋅10-3 cm, ε2 = 3.24⋅10-8 cm
2.62647 0
1 0
5 0
BET
A_X
&Y[
m]
BETA_1X BETA_2Y BETA_1Y BETA_2X
2.62647 0
1 0
1 -1
Bet
atro
n si
ze X
&Y[
cm]
Alp
haXY
[-1, +
1]
Ax Ay AlphaXY 2.62647 0
0.5
0
0.5
0
PHA
SE/(2
*PI)
Q_1 Q_2 Teta1 Teta2
2.62647 0
1 0
5 0
BET
A_X
&Y[
m]
BETA_1X BETA_2Y BETA_1Y BETA_2X
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 32 USPAS, Fort Collins, CO, June 13-24, 2016
Summary
v Relationships between the eigen-vectors, beam emittances and the beam ellipsoid in 4D phase space ♦ From the beam ellipsoid to the eigen-vectors (equivalence of both
pictures) v New parametrization of eigen-vectors in terms of generalized Twiss
functions ♦ Complete Weyl-like representation
♠ 10 independent parameters to fully describe the motion ♠ transport line ambiguities resolved
♦ Developed software based on this representation allows effective analysis of coupled betatron motion for both circular accelerators and transfer lines (OptiM).
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 33
Alex Bogacz
Ring Design with Imaginary γt
USPAS, Fort Collins, CO, June 13-24, 2016
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 34
Perturbed FODO Cell
ΔGF1 ΔGF2
17.528 0
40
0
5 -5
BET
A_X
&Y[
m]
DIS
P_X&
Y[m
]
BETA_X BETA_Y DISP_X DISP_Y
Arc Cell - Perturbed 900 FODO
2 x 900 FODO
ΔGF1 ΔGF2
M56 =Dρds∫
M56 – large, positive M56 – very small, still positive
17.528 0
40
0
5 -5
BET
A_X
&Y[
m]
DIS
P_X&
Y[m
] BETA_X BETA_Y DISP_X DISP_Y
γ =56
tSM
USPAS, Fort Collins, CO, June 13-24, 2016
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 35
‘Missing Dipole’ Arc Cell - Negative M56 Cell
M56 =Dρds∫
26.2911 0
40
0
5 -5
BET
A_X
&Y[
m]
DIS
P_X&
Y[m
]
BETA_X BETA_Y DISP_X DISP_Y
ΔGF ΔGF
M56 – small negative
26.2911 0
40
0
5 -5
BET
A_X
&Y[
m]
DIS
P_X&
Y[m
]
BETA_X BETA_Y DISP_X DISP_Y
ΔGF ΔGF
M56 – always positive
900 FODO – Empty - 900 FODO
USPAS, Fort Collins, CO, June 13-24, 2016
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 36
21.808 0
40
0
5 -5
BET
A_X
&Y[
m]
DIS
P_X&
Y[m
]
BETA_X BETA_Y DISP_X DISP_Y
‘New’ Arc Cell - Negative M56 Cell
26.2911 0
40
0
5 -5
BET
A_X
&Y[
m]
DIS
P_X&
Y[m
]
BETA_X BETA_Y DISP_X DISP_Y
900 FODO – Empty - 900 FODO
M56 – always positive
M56 – negative
USPAS, Fort Collins, CO, June 13-24, 2016
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 37
‘Missing Dipole’ vs Perturbed 900 FODO
2 x 900 FODO
M56 =Dρds∫
M56 – small, but negative
M56 – very small, still positive
17.528 0
40
0
5 -5
BET
A_X
&Y[
m]
DIS
P_X&
Y[m
]
BETA_X BETA_Y DISP_X DISP_Y 21.808 0
40
0
5 -5
BET
A_X
&Y[
m]
DIS
P_X&
Y[m
]
BETA_X BETA_Y DISP_X DISP_Y
900 FODO – Empty - 900 FODO
USPAS, Fort Collins, CO, June 13-24, 2016
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 38
85.1122 0
40
0
5 -5
BET
A_X
&Y[
m]
DIS
P_X&
Y[m
]
BETA_X BETA_Y DISP_X DISP_Y
Arc Optics
Perturbed 1200 FODO
77.8675 0
40
0
5 -5
BET
A_X
&Y[
m]
DIS
P_X&
Y[m
]
BETA_X BETA_Y DISP_X DISP_Y
Perturbed 900 FODO
Negative M56 Lattice
16 bends
18 bends
2 x cell
2 x cell
USPAS, Fort Collins, CO, June 13-24, 2016
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 39
P. McIntyre Texas A&M
Dual-dipole + Quad Cryomodule
Dual-dipole Quad
Arc Cell - Super-ferric Magnets
Bend: Lb = 120 cm (magnetic length) Lead ends: 2×22 cm B = 3.13 Tesla bend ang. = 8.12 deg. Sagitta = 2.1 cm
Sextupole: Ls = 10 cm S = 780 Tesla/m2
Quadrupole: Lq = 40 cm G = 10-37 Tesla/m Correctors (H/V): 20 cm BPM can: 20 cm
Magnet aperture radius (inj. at 285 MeV) 6σrms + 10 mm = 41 mm
39
Bend Sextupole Correctors BPM
Quad
17.2 4.4
30
0
3 -3
BET
A_X
&Y[
m]
DIS
P_X&
Y[m
]
BETA_X BETA_Y DISP_X DISP_Y Bend
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 40
Arc-Straight Optics
128.463 0
40
0
5 -5
BET
A_X
&Y[
m]
DIS
P_X&
Y[m
]
BETA_X BETA_Y DISP_X DISP_Y
Arc (85 meters) Straight (43 meters)
USPAS, Fort Collins, CO, June 13-24, 2016
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 41
injection extraction RF cavity
Crossing angle: 75 deg.
Figure-8 Booster for JLEIC
γ =56
tSM
M56 =Dρds∫
Ekin = 285 MeV – 7.062 GeV Ring circumference: 273 m (~2200/8)
41 USPAS, Fort Collins, CO, June 13-24, 2016
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 42
275.087 0
40
0
5 -5
BET
A_X
&Y[
m]
DIS
P_X&
Y[m
]
BETA_X BETA_Y DISP_X DISP_Y
Inj. Arc Straight Straight Arc
= −56 98M cm
γ = =56
16.8tSM
i
= 275S m
Booster Lattice (8 GeV, γt = 16.8 i )
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 43
275.087 0
40
0
5 -5
BET
A_X
&Y[
m]
DIS
P_X&
Y[m
]
BETA_X BETA_Y DISP_X DISP_Y
Arc Quadrupoles: Lq = 40 cm G = 12-65 Tesla/m
Arc Bends: Lb = 120 cm B = 3.13 Tesla bend ang. = 8.12 deg. Sagitta = 2.1 cm
Straight Quads: Lq = 40 cm GF = 12.6 Tesla/m GD = -24.5 Tesla/m
Booster Lattice (8 GeV, γt = 16.8 i )
Lattice configured with super-ferric magnets
43
Proton beam energy (total) GeV 1.2 - 8
Circumference m 275
Straights’ crossing angle deg 79.8
Arc length m 103 / 85
Straight section length m 43
Maximum hor. / ver. β functions m 22 / 22
Maximum hor. dispersion m 4.3
Hor. / ver. betatron tunes νx,y 7.87 / 5.85
Hor. / ver. natural chromaticities ξx,y -6.8 / -4.6
Momentum compaction factor α -3.6 10-3
Hor. / ver. normalized emittance εx,y µm rad 1 / 1
Maximum hor. / ver. rms beam size at inj. σx,y mm 5.1 / 5.1
USPAS, Fort Collins, CO, June 13-24, 2016
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 44
8 GeV Booster design based on imaginary γt lattice Negative M56 lattice based on lightly perturbed 900 FODO
Footprint fits into the same circumference of 275 meters (2200/8).
Lattice configured with super-ferric 3 Tesla magnets (Texas A&M design)
stronger bends (3.1 Tesla vs 2.7 Tesla)
substantially lower magnet aperture (4.1 cm vs 5.2 cm)
fewer number of bends (32 vs 36)
same number of quads (10 extra arc quads and 10 fewer straight quads)
Much lower betas (22 m vs 38 m), better tunability, less sensitivity to errors
Lower dispersion tunes and natural chromaticity
Superior control of chromaticity with 2 families of sextupoles in each plane
Lower quadrupole strengths (overall)
Summary
USPAS, Fort Collins, CO, June 13-24, 2016
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