Transcript Chromatic Corrections

Chromatic Corrections
Vasiliy Morozov and Yaroslav Derbenev
for the JLab EIC Study Group
Review 09/2010
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Outline
• Symmetry-based IR design concept
• IR linear optics design
• Analysis and compensation of 2nd- and 3rd-order
aberration terms contributing to beam smear at IP
• Chromaticity compensation
• Dynamic aperture tracking
Review 09/2010
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IR Design Challenges
• Low * essential to ELIC’s high-luminosity concept
• Large size of extended beam f * = F2
• Large chromatic spread at IP F ~ Fp/p >> *
requires sextupole compensation
• Non-linear field effects must be accounted for
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Symmetry Concept
•
•
•
•
Dedicated Chromaticity Compensation Block (CCB) symmetric around IP
Model assuming large parallel beam and ignoring angular divergence
Take advantage of symmetry to reduce number of compensation conditions
Conditions for compensation of 2nd-order aberrations at IP
• Chromatic spread (x, y are betatron trajectory components)
2 Dns y 2 ds   ny 2 ds, 2 Dns x 2 ds   nx 2 ds
• Smear due to betatron beam size and 2nd-order dispersion effects
2
n
xy
ds  0,
s

3
n
x
s
 ds  0,
 (n D  n) Dxds  0
s

•
•
Satisfied automatically for
• symmetric x2 and y2
• symmetries of D and ns opposite to symmetry of x
Compensation of
• chromatic tune spread
• chromatic and sextupole beam smear at IP
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IR Design
• Modular approach: IR designed independently to be later integrated
into the ring
• Utilize COSY Infinity
• calculates coefficients M(x|) of expansion of type
x
M ( x |  ) x x y y t  q 

 
, , ,
 , , 
to arbitrary order (+++++) for each of coordinate components
• Design system such that
x( s)   x( s )
y 2 ( s)  y 2 ( s)
D( s)  D( s)
n( s )  n(  s )
ns ( s )  ns (  s )
no ( s )  no (  s )
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Chromaticity Compensation Block
3 symmetry requirements:
x(sf) = -x(si)  M(xf |xi) = -1
y(sf) = y(si)  M(yf |yi) = 1
D(sf) = D(si)  M(xf |q) = 0
xf
matrix:
xf ’
yf

1st-order
yf ’
xi
xi’
yi
yi’
ti
qi
3 parameters: Q1, Q2, Q3
Q1 = -0.174 T @ 5 cm
Q2 = 0.350 T @ 5 cm
Q3 = -0.079 T @ 5 cm
  200 m
  30 mrad
D /10
X-motion
Y-motion
5 cm
Q1 Q2
tf
Q3
Q3
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Q2 Q1
37 m
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Final Focusing Block
Q4
2 conditions: M(xf |xi) = 0, M(yf |yi) = 0
Q5
Q6
4m
2 parameters: Q4  -0.308 T @ 5 cm,
Q5  0.850 T @ 5 cm
To have large y at the final quad’s exit:
Q6  -0.750 T @ 5 cm
1st-order matrix:
xf
xf ’
yf
yf ’
tf
xi
xi’
yi
yi’
ti
qi
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IR Up to IP
1st-order matrix:
xf
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xf ’
yf
yf ’
tf
xi
xi’
yi
yi’
ti
qi
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IR Up to IP
 x  23.2 mm
 y  4.8 mm
Review 09/2010
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Beam Size & 2nd-Order Aberrations
Assume: 5 GeV/c e ,  xN  54 μm,  yN  11μm,  xi  2 km,
 yi  3.5 km, E /E  7.1104
1st-order matrix
Geometric beam size at IP due to emittance
xIP ( xi)
yIP ( yi)
2nd-order aberrations:
xIP ( xi qi ) ~122%
yIP ( yi qi ) ~102%
3rd-order aberrations negligible
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Sextupole Compensation
Make M(xf | xi qi) = 0 and M(yf | yi qi) = 0 by adjusting
s1  0.055 T @ 5 cm and s2  -0.180 T @ 5 cm
s1
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s2
s1
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Sextupole Compensation
Assume: 5 GeV/c e ,  xN  54 μm,  yN  11μm,  xi  2 km,
 yi  3.5 km, E /E  7.1104
1st-order matrix unchanged
Geometric beam size at IP due to emittance
xIP ( xi)
yIP ( yi)
2nd-order aberrations:
xIP ( xi xi) ~ 9.5%
yIP ( xi yi ) ~ 8.0%
yIP ( xi yi) ~ 5.3%
Review 09/2010
xIP ( xi qi )
0%
yIP ( yi qi )
0%
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Sextupole Compensation
Assume: 5 GeV/c e ,  xN  54 μm,  yN  11μm,  xi  2 km,
 yi  3.5 km, E /E  7.1104
3rd-order aberrations:
xIP ( xi3 )
~ 64%
yIP ( xi2 yi )
~ 3.6%
yIP ( yi3 )
~ 21%
xIP ( xi2 qi ) ~ 4.0%
yIP ( xi yi qi ) ~ 6.6%
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Octupole Compensation
Minimize M(xf | xi3) and M(yf | yi3) by introducing 2 pairs of octupoles with
o1  0.092 T @ 5 cm and o2  -0.146 T @ 5 cm
o1
o2
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o2
o1
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Octupole Compensation
Assume: 5 GeV/c e ,  xN  54 μm,  yN  11 μm,  xi  2 km,
 yi  3.5 km, E /E  7.1104
3rd-order aberrations:
xIP ( xi3 )
0%
yIP ( xi2 yi )
0%
yIP ( yi3 )
~ 9.9%
xIP ( xi2 qi ) ~ 4.0%
yIP ( xi yi qi ) ~ 6.6%
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Chromatic Tune Dependence
Up to 5th order in p/p
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Dynamic Aperture & Tracking
• To estimate Dynamic Aperture limitations due to IR
• note symmetry of IR
• note that betatron phase advance in each plane is n
• rather than completing the ring, represent the rest of the ring by linear matrix
M ring
 x sin(2 x )
 cos(2 x )

0
  sin(2 x ) /  x

cos(2 x )


cos(2

)

sin(2

)
y
y
y 

0

 sin(2 y ) /  y
cos(2 y ) 

• Track multiple turns through
M Total  M ring  M IR (5th order)
• Isolate IR effects
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Dynamic Aperture & Tracking
Assume: 5 GeV/c e ,  xN  54 μm,  yN  11 μm,  xi  0.5 km,  yi  3.5 km,
E /E  0,  x  0.500616,  y  0.504374
x
y
9  x
500 turns
2 103
10   y
500 turns
2  10 4
y
x
5 cm
5 cm
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Dynamic Aperture & Tracking
Assume: 5 GeV/c e ,  xN  54 μm,  yN  11 μm,  xi  0.5 km,  yi  3.5 km,
E /E  5 103 ,  x  0.500616,  y  0.504374
x
y
9  x
500 turns
2 103
9  y
500 turns
2  10 4
y
x
5 cm
5 cm
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Summary
• Introduced dedicated Chromaticity Compensation Block
(CCB) symmetric around IP
• Arranged CCB’s magnetic structure and orbital motion to
meet certain symmetry requirements
• Demonstrated compensation of leading-term aberrations at
IP, largest of a few remaining aberrations is under 10% of
beam size
• Demonstrated chromaticity compensation
• Dynamic aperture tracking underway with promising results
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Next Steps
• Further optimization
• Compensation of lowest-order effect of angular divergence
on dynamic aperture
• Larger-scale octupole symmetry (across IP or 2 IP’s) to
improve dynamic aperture
• Integration/matching of IR to the ring
• Chromaticity compensation and tracking using complete ring
• Benchmark numeric results against independent code
• Design ion ring IR (similar but no emittance and polarization
degradation issues)
Review 09/2010
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