HLAB MEETING-- Paper --

T.Gogami30Apr2013

Experiments with magnets(e,e’K+) reaction

• Dispersive plane• Transfer matrix• R12 , R16

• Emittance• Beam envelope• ・・・

詳細な計算　 [参照 ]Transport AppendixK.L.Brown and F.Rothacker

Paper

Contents

• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design

Contents

• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design

Contents

• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design

Design requirements

1. Correct beam transport properties2. To reduce the – Weight– Cost– Power

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

Dipole elements

R0 = mv/qB0

ObjectImage

Particle of higher momentum

Dipole termQuadrupole termSextupole term

Contents

• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design

Field-path integral

Field-path integral B0R0

1 rad

𝑅0=𝑝

𝐵0𝑞

[rad]

Contents

• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design

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

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

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

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 )

Rotated pole edge (3)( Frequently used to generate first order imaging )

1𝑓 𝑥

=−tan𝛼𝑅0

Optical focusing power

Dispersive plane

Non-dispersive plane

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

Transverse field gradient (2)

Total focusing power ( Dipole + transverse field gradient )

𝑛=−𝑅0𝐵0

𝜕𝐵 𝑦𝜕 𝑥

Field index

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

Contents

• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design

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)

Examples of transport matrices Rij

Imaging

• R12 = 0– x-image at s with magnification R11

• R34 = 0– y-image at s with magnification R33

Focal lengths and focal planes

• x-plane

• y-plane

Dispersion

Contents

• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design

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)

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

Contents

• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design

Parameters

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

“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

SPL with field clamp + ENGE

New magnetic field map Committed to the svn

Split pole magnet (ENGE)

Matrix tuning (E05-115)

Before

After

FWHM ~ 4 MeV/c2

Backup

Transverse field gradient (2)

Total focusing power ( Dipole + transverse field gradient )

𝑛=−𝑅0𝐵0

𝜕𝐵 𝑦𝜕 𝑥

Field index

Simple harmonic motion

Simple harmonic motion