Betatron Motion with Coupling of Horizontal and Vertical Degrees...
Transcript of Betatron Motion with Coupling of Horizontal and Vertical Degrees...
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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 10-21, 2013
Alex Bogacz, Geoff Krafft and Timofey Zolkin
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
Lecture 9 Coupled Betatron Motion II 2
Outline
Practical Examples:
Spin Rotator for Figure-8 Collider ring
Vertex-to-plane adapter for electron cooling (Fermilab)
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 10-21, 2013
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
Lecture 9 Coupled Betatron Motion II 3 USPAS, Fort Collins, CO, June 10-21, 2013
Axisymmetric rotational distribution Twiss functions
Fermilab electron cooling
The electron beam distribution
is axially symmetric, and
uncoupled at the cathode:
00
00
00
00
00
00
00
00
1
T
BΞ
where 0/ PmkTr ccT is the
thermal emittance of the beam
Electrostatic accelerator
Solenoid
Gaussian distribution
Rotational distribution
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Lecture 9 Coupled Betatron Motion II 4 USPAS, Fort Collins, CO, June 10-21, 2013
Axisymmetric rotational distribution Twiss functions
At the exit of the solenoid the electron beam distribution is still
axially symmetric
000
00
2
00
000
000
2
0
0
0
0
0
1
T
B
T
in ΦΞΦΞ
where
100
0100
010
0001
Φ
cPeB 02/ is the rotational focusing strength of the
solenoid edge
B is the solenoid magnetic field.
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Lecture 9 Coupled Betatron Motion II 5 USPAS, Fort Collins, CO, June 10-21, 2013
Axisymmetric rotational distribution Twiss functions
The eigen-vectors of the rotational distribution:
2
2
2
2
ˆ
ii
i
i
1v ,
2
2
2
2
ˆ2
i
ii
i
v
It corresponds to u = 1/2, 2/21
Then, the matrix V̂ is
2
1
2
1
00
2
1
2
1
00
V̂
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Thomas Jefferson National Accelerator Facility
Lecture 9 Coupled Betatron Motion II 6 USPAS, Fort Collins, CO, June 10-21, 2013
Axisymmetric rotational distribution Twiss functions
4D-emmitance conservation:
2
21 T
Rotational emittance estimate
0
2
Trot rrrr
Comparing left and right hand sides of the equation
TT
in UVVUΞ ˆ
/1000
0/100
00/10
000/1
ˆˆ
2
2
1
1
One obtains
.21
,21
,12
,12
0
1
0
2
0
22
0
1
0
2
0
21
2
0
2
0
2
0
2
0
0
0
TT
T
T
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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 20000
x [cm]
z [cm]
Figure-8 Collider Ring - Footprint
total ring circumference ~1000 m
60 deg. crossing
USPAS, Fort Collins, CO, June 10-21, 2013
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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
0.1
5
-0.1
5
BE
TA
_X
&Y
[m]
DIS
P_X
&Y
[m]
BETA_X BETA_Y DISP_X DISP_Y
USPAS, Fort Collins, CO, June 10-21, 2013
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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
BE
TA
_X
&Y
[m]
BETA_1X BETA_2Y BETA_1Y BETA_2X
USPAS, Fort Collins, CO, June 10-21, 2013
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 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
1
5
0
1
-1
BE
TA
_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 10-21, 2013
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
BE
TA
_X
&Y
[m]
DIS
P_X
&Y
[m]
BETA_X BETA_Y DISP_X DISP_Y
5 GeV
USPAS, Fort Collins, CO, June 10-21, 2013
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Thomas Jefferson National Accelerator Facility
Lecture 9 Coupled Betatron Motion II 12 USPAS, Fort Collins, CO, June 10-21, 2013
Ionization Cooling in an Axially Symmetric Channel
A single-particle phase-space trajectory along the
beam orbit can be expressed as:
,)(ˆ)(ˆRe)(ˆ))((
22
))((
112211 sisi
esess vvx
One can rewrite the above equations in the following compact form
)()(ˆ)(ˆ sss aVx
where
)(ˆ),(ˆ),(ˆ),(ˆ)(ˆ 2211 sssss vvvvV
))(sin(
))(cos(
))(sin(
))(cos(
)(
222
222
111
111
s
s
s
s
sa
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Lecture 9 Coupled Betatron Motion II 13 USPAS, Fort Collins, CO, June 10-21, 2013
Ionization Cooling in an Axially Symmetric Channel
In the case of axially symmetric focusing the eigen-vectors reduce to:
2
2
2
2
ˆ
ii
i
i
1v ,
2
2
2
2
ˆ2
i
ii
i
v
2
1
2
1
00
2
1
2
1
00
V̂
here we used that u = 1/2, 1 = 2 = /2
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
Lecture 9 Coupled Betatron Motion II 14 USPAS, Fort Collins, CO, June 10-21, 2013
Ionization Cooling in an Axially Symmetric Channel
Cooling Description
Ionization cooling due to energy loss in a thin absorber can be
described as:
p
p ,
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
1000
0100
0010
0001
ˆ 1
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Thomas Jefferson National Accelerator Facility
Lecture 9 Coupled Betatron Motion II 15 USPAS, Fort Collins, CO, June 10-21, 2013
Canonical variables
.2
,2
xR
yp
yR
xp
y
x
PceBR s / - longitudinal magnetic field
Relation between geometrical and canonical variables
Rxx̂ ,
where
1002
0100
0210
0001
,,ˆ
R
R
y
x
p
y
p
x
y
x
y
xRxx
,
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 10-21, 2013
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
1000
0100
0010
0001
ˆ 1
and finally
inout aVRRVa ˆ
1000
0100
0010
0001
ˆ 11
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
Lecture 9 Coupled Betatron Motion II 17 USPAS, Fort Collins, CO, June 10-21, 2013
Ionization Cooling in an Axially Symmetric Channel
Carrying out the above calculation explicitly one obtains:
inout
RR
RR
RR
RR
aa
2
1
2
12
1
2
12
1
2
12
1
2
1
1000
0100
0010
0001
2D emittances after cooling are given by the following formula:
)(sin1cos211
)(sin1cos211
2
212
22
2
2
211
22
1
43
21
ORRaa
ORRaa
outout
outout
where 2121
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Thomas Jefferson National Accelerator Facility
Lecture 9 Coupled Betatron Motion II 18 USPAS, Fort Collins, CO, June 10-21, 2013
Ionization Cooling in an Axially Symmetric Channel
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 10-21, 2013
Ionization Cooling in an Axially Symmetric Channel
If cooling effect of one absorber is sufficiently small one
can perform averaging over betatron phases. That
yields
R
R
1
1
22
11
ds
dp
p
R
ds
d
ds
dp
p
R
ds
d
0
2
2
0
1
1
11
11
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
Lecture 9 Coupled Betatron Motion II 20 USPAS, Fort Collins, CO, June 10-21, 2013
Ionization Cooling in an Axially Symmetric Channel
Canonical momentum of a single particle
2ˆ
0000
0000
0100
1000
ˆˆ
0000
0000
0100
1000
ˆ 21aVaVxxMTT
xy ypxp
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Thomas Jefferson National Accelerator Facility
Lecture 9 Coupled Betatron Motion II 21 USPAS, Fort Collins, CO, June 10-21, 2013
Ionization Cooling in an Axially Symmetric Channel
Second order moments of the Gaussian distribution
(Note that for a single particle - 2/rms and we use rms. emittances below)
0
,2
,4
41
,
,
21
21
21
222
21
21
22
yx
xy
yx
yx
ppxy
Mypxp
pp
ypxp
yx
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Thomas Jefferson National Accelerator Facility
Lecture 9 Coupled Betatron Motion II 22 USPAS, Fort Collins, CO, June 10-21, 2013
Ionization Cooling in an Axially Symmetric Channel
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:
04
1
43
2
2
2R
ds
d .
The standard recipe determines the alpha-function:
ds
d
2
1
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Lecture 9 Coupled Betatron Motion II 23 USPAS, Fort Collins, CO, June 10-21, 2013
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
15
0
50
BE
TA
_X
&Y
[m]
DIS
P_X
&Y
[m]
BETA_X BETA_Y
7.20
0.5
0P
HA
SE
_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
0P
HA
SE
_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) = /2
USPAS, Fort Collins, CO, June 10-21, 2013
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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
7
0
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) = /2
USPAS, Fort Collins, CO, June 10-21, 2013
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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 10-21, 2013
Operated by JSA for the U.S. Department of Energy
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 10-21, 2013
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Thomas Jefferson National Accelerator Facility
Lecture 9 Coupled Betatron Motion II 28 USPAS, Fort Collins, CO, June 10-21, 2013
Vertex-to-plane transformer insert
SolenoidSkew-quadrupole system
Uncoupledaxial-symmetricdistribution
Rotational distribution
Flat distribution
x
Px
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Thomas Jefferson National Accelerator Facility
Lecture 9 Coupled Betatron Motion II 29 USPAS, Fort Collins, CO, June 10-21, 2013
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
2
2
2
2
2
1
0
0
1
1
F
F
i
i
i
ii
rotation
Rotational eigen-vectors
2
2
F
F
i
2
2
F
Fi
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Thomas Jefferson National Accelerator Facility
Lecture 9 Coupled Betatron Motion II 30 USPAS, Fort Collins, CO, June 10-21, 2013
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
2
0
22
0
2
0
21TT
Lattice implementation – Twiss functions, beam sizes etc.
Operated by JSA for the U.S. Department of Energy
Thomas Jefferson National Accelerator Facility
Lecture 9 Coupled Betatron Motion II 31 USPAS, Fort Collins, CO, June 10-21, 2013
Vertex-to-plane transformer insert
Ekin = 10 MeV, Tc = 0.2 eV, Rc = 0.5 cm, Bsol = 1 kG,
1 = 7.14 10-3 cm, 2 = 3.24 10-8 cm
2.62647 0
1
0
5
0
BE
TA
_X
&Y
[m]
BETA_1X BETA_2Y BETA_1Y BETA_2X
2.62647 0
1
0
1
-1
Beta
tro
n s
ize X
&Y
[cm
]
Alp
haX
Y[-
1,
+1]
Ax Ay AlphaXY 2.62647 0
0.5
0
0.5
0
PH
AS
E/(
2*P
I)
Q_1 Q_2 Teta1 Teta2
2.62647 0
1
0
5
0
BE
TA
_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 10-21, 2013
Summary
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)
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).