HLAB MEETING -- Paper -- T.Gogami 30Apr2013. Experiments with magnets (e,eK + ) reaction.
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Transcript of HLAB MEETING -- Paper -- T.Gogami 30Apr2013. Experiments with magnets (e,eK + ) reaction.
![Page 1: HLAB MEETING -- Paper -- T.Gogami 30Apr2013. Experiments with magnets (e,eK + ) reaction.](https://reader035.fdocuments.us/reader035/viewer/2022070306/5516e4ed550346fe558b46b9/html5/thumbnails/1.jpg)
HLAB MEETING-- Paper --
T.Gogami30Apr2013
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Experiments with magnets(e,e’K+) reaction
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• Dispersive plane• Transfer matrix• R12 , R16
• Emittance• Beam envelope• ・・・
詳細な計算 [参照 ]Transport AppendixK.L.Brown and F.Rothacker
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Paper
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Contents
• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design
![Page 6: HLAB MEETING -- Paper -- T.Gogami 30Apr2013. Experiments with magnets (e,eK + ) reaction.](https://reader035.fdocuments.us/reader035/viewer/2022070306/5516e4ed550346fe558b46b9/html5/thumbnails/6.jpg)
Contents
• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design
![Page 7: HLAB MEETING -- Paper -- T.Gogami 30Apr2013. Experiments with magnets (e,eK + ) reaction.](https://reader035.fdocuments.us/reader035/viewer/2022070306/5516e4ed550346fe558b46b9/html5/thumbnails/7.jpg)
Contents
• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design
![Page 8: HLAB MEETING -- Paper -- T.Gogami 30Apr2013. Experiments with magnets (e,eK + ) reaction.](https://reader035.fdocuments.us/reader035/viewer/2022070306/5516e4ed550346fe558b46b9/html5/thumbnails/8.jpg)
Design requirements
1. Correct beam transport properties2. To reduce the – Weight– Cost– Power
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Dipole, Quadrupole, Sextupole
By(x) = a + bx + cx2 + ・・・・The field of the magnet as a multpole expansion about the central trajectory
Dipole term Quadrupole term Sextupole term
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Dipole elements
R0 = mv/qB0
ObjectImage
Particle of higher momentum
Dipole termQuadrupole termSextupole term
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Contents
• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design
![Page 12: HLAB MEETING -- Paper -- T.Gogami 30Apr2013. Experiments with magnets (e,eK + ) reaction.](https://reader035.fdocuments.us/reader035/viewer/2022070306/5516e4ed550346fe558b46b9/html5/thumbnails/12.jpg)
Field-path integral
Field-path integral B0R0
1 rad
𝑅0=𝑝
𝐵0𝑞
[rad]
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Contents
• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design
![Page 14: HLAB MEETING -- Paper -- T.Gogami 30Apr2013. Experiments with magnets (e,eK + ) reaction.](https://reader035.fdocuments.us/reader035/viewer/2022070306/5516e4ed550346fe558b46b9/html5/thumbnails/14.jpg)
A quadropole element
A) By a separate quadrupole magnetB) By a rotated input or output in a bending magnetC) By a transverse field gradient in a bending magnet
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A quadropole element
A) By a separate quadrupole magnetB) By a rotated input or output in a bending magnetC) By a transverse field gradient in a bending magnet
Extra cost
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Rotated pole edge (1)
Imaging in the dispersive plane( Frequently used to generate first order imaging )
ΔB y=𝜕𝐵 𝑦𝜕 𝑥
𝑥
−𝜃=−𝑠 Δ𝐵 𝑦𝐵0𝑅 0
=−x
𝜕𝜕𝑥
(𝐵 𝑦 𝑠)
𝐵0𝑅 0
B y s=−𝐵0𝑥 tan𝛼
1𝑓 𝑥
=−𝜃𝑥
=−tan𝛼𝑅0
−𝜃=−𝑥tan𝛼𝑅0
Optical focusing power
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Rotated pole edge (2)( Frequently used to generate first order imaging )
Imaging in the non-dispersive plane
ΔB x=𝜕𝐵𝑥𝜕 𝑦
𝑦=𝜕𝐵 𝑦𝜕𝑥
𝑦
𝜑=𝑠 Δ𝐵𝑥𝐵0𝑅 0
=𝑦
𝜕𝜕 𝑥
(𝐵 𝑦 𝑠)
𝐵0𝑅 0
B y s=−𝐵0𝑥 tan𝛼
1𝑓 𝑦
=−𝜑𝑦
=tan𝛼𝑅0
𝜑=−𝑦tan𝛼𝑅0
(Rot B = 0 )
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Rotated pole edge (3)( Frequently used to generate first order imaging )
1𝑓 𝑥
=−tan𝛼𝑅0
Optical focusing power
Dispersive plane
Non-dispersive plane
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Transverse field gradient (1)
𝑑𝑥 ′𝑥
=−𝑑𝑠𝑅02
1𝑓 𝑥
=𝑑𝑠𝑅02
Focusing power
𝑑𝑥 ′𝑥
=−𝜕𝐵 𝑦𝜕 𝑥
𝑑𝑠𝐵0𝑅 0
=𝑛
𝑅02𝑑𝑠
Transverse field gradient is zero (Pure dipole field)
Transverse field gradient is not zero
dB y=𝜕𝐵 𝑦𝜕 𝑥
𝑥
𝑛=−𝑅0𝐵0
𝜕𝐵 𝑦𝜕 𝑥 Field index
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Transverse field gradient (2)
Total focusing power ( Dipole + transverse field gradient )
𝑛=−𝑅0𝐵0
𝜕𝐵 𝑦𝜕 𝑥
Field index
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A) A pure dipole filedFocusing in the dispersive plane
B) A transverse field gradient characterized by n– Focusing in both plane– Sum of the focusing powers is constant
1/fx + 1/fy = (1-n)/(R02)ds – n/R0
2 = ds/R02
C) If n=1/2Dispersive and non-dispersive focusing power: ds/2R0
2
D) If n < 0– Dispersive plane focusing power : strong and positive– Non-dispersive plane focusing power : negative
Transverse field gradient (3)
𝑛=−
𝑅0𝐵0
𝜕𝐵 𝑦𝜕 𝑥
Field index
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Contents
• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design
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Matrix formalism (first order)
x1 = x
x2 = θ = px/pz(CT)
x3 = y
x4 = φ = py/pz(CT)
x5 = l = z – z(CT)
x6 = δ = (pz – pz(CT))/pz(CT)
𝑥𝑖 (𝑠 )=∑𝑗=1
6
𝑅𝑖𝑗 𝑥 𝑗 (0)
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Examples of transport matrices Rij
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Imaging
• R12 = 0– x-image at s with magnification R11
• R34 = 0– y-image at s with magnification R33
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Focal lengths and focal planes
• x-plane
• y-plane
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Dispersion
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Contents
• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design
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Phase ellipse and Beam envelope
√𝜎 22
√𝜎 11x
θ
Phase ellipse
Beam emittance
x
z
√𝜎 11
1/2
Beam Envelope
s = 0 beam size (beam waist)
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Output beam matrix
• Initial beam ellipse• R-matrix
σ (0 )=(𝜎 11(0) 00 𝜎 22(0))
𝜎 (𝑠 )=𝑹 σ (0 ) 𝑹𝑇=(𝜎 11(𝑠) 𝜎 12(𝑠)𝜎12(𝑠) 𝜎 22(𝑠))
𝜎 22 (𝑠) 𝑥2−2𝜎 12 (2 )𝑥𝜗+𝜎 11 (𝑠 )𝜗 2=|𝜎||𝜎|=𝜎 11 (𝑠)𝜎 22 (𝑠)−𝜎 212(𝑠)=𝜎 11(0)𝜎 22(0)
Initial Beam matrix
After a magnet system with an R-matrix (Rij)
Output beam ellipse
• Final beam matrix• Final beam ellipse
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Contents
• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design
![Page 32: HLAB MEETING -- Paper -- T.Gogami 30Apr2013. Experiments with magnets (e,eK + ) reaction.](https://reader035.fdocuments.us/reader035/viewer/2022070306/5516e4ed550346fe558b46b9/html5/thumbnails/32.jpg)
Parameters
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Practical magnet design
A) Bending power
B) Pole gap
C) Coil power
D) Magnet weight : Coil weight
: Steel weight
Key constrains
An advantage B0
R0 Focal length
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“Strong focusing” techniqueLarge pole edge rotation + Large field index
NOVA NV-10 ion implanter
Bend : 70 degreesGap : 5 cmBending radius : 53.8 cmPole gap field : 8 kGParticle : 80 keV antimonyWeight : 2000 lbPole edge rotation : 35 degreesField index : -1.152
x-defocusy-focus
x-focusy-defocus
x : DFDy : FDF
Uniform field bending magnet• Weight : 4000 lb• Pole gap field : 16 kG• Coil power : substantially higher
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SPL with field clamp + ENGE
New magnetic field map Committed to the svn
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Split pole magnet (ENGE)
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Matrix tuning (E05-115)
Before
After
FWHM ~ 4 MeV/c2
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Backup
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Transverse field gradient (2)
Total focusing power ( Dipole + transverse field gradient )
𝑛=−𝑅0𝐵0
𝜕𝐵 𝑦𝜕 𝑥
Field index
Simple harmonic motion
Simple harmonic motion
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