HLAB MEETING -- Paper --

1. HLAB MEETING-- Paper -- T.Gogami 30Apr2013

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

3. Dispersive plane • Transfer matrix • R12 , R16 • Emittance • Beam envelope • ・・・ 詳細な計算　[参照] Transport Appendix K.L.Brown and F.Rothacker

4. Paper

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

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

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

8. Design requirements • Correct beam transport properties • To reduce the • Weight • Cost • Power

9. Dipole, Quadrupole, Sextupole By(x) = a + bx + cx2 + ・・・・ The field of the magnet as a multpole expansion about the central trajectory Sextupole term Dipole term Quadrupole term

10. Dipole elements R0 = mv/qB0 Dipole term Quadrupoleterm Sextupoleterm Particle of higher momentum Image Object

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

12. Field-path integral Field-path integral B0R0  1 rad [rad]

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

14. A quadropole element • By a separate quadrupole magnet • By a rotated input or output in a bending magnet • By a transverse field gradient in a bending magnet

15. A quadropole element • By a separate quadrupole magnet • By a rotated input or output in a bending magnet • By a transverse field gradient in a bending magnet Extra cost

16. Rotated pole edge (1) ( Frequently used to generate first order imaging ) Imaging in the dispersive plane Optical focusing power

17. Rotated pole edge (2) ( Frequently used to generate first order imaging ) Imaging in the non-dispersive plane (Rot B = 0 )

18. Rotated pole edge (3) ( Frequently used to generate first order imaging ) Optical focusing power Dispersive plane Non-dispersive plane

19. Transverse field gradient (1) Transverse field gradient is zero (Pure dipole field) Focusing power Transverse field gradient is not zero Field index

20. Transverse field gradient (2) Total focusing power ( Dipole + transverse field gradient ) Field index

21. Transverse field gradient (3) Field index • A pure dipole filed Focusing in the dispersive plane • 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/R02 = ds/R02 • If n=1/2 Dispersive and non-dispersive focusing power: ds/2R02 • If n < 0 • Dispersive plane focusing power : strong and positive • Non-dispersive plane focusing power : negative

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

23. 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)

24. Examples of transport matrices Rij

25. Imaging • R12 = 0 • x-image at s with magnification R11 • R34 = 0 • y-image at s with magnification R33

26. Focal lengths and focal planes • x-plane • y-plane

27. Dispersion

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

29. Phase ellipse and Beam envelope Phase ellipse Beam Envelope θ x x z 1/2 s = 0  beam size (beam waist) Beam emittance

30. Output beam matrix Initial Beam matrix • Initial beam ellipse • R-matrix After a magnet system with an R-matrix (Rij) Output beam ellipse • Final beam matrix • Final beam ellipse

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

32. Parameters

33. Practical magnet design Key constrains • Bending power • Pole gap • Coil power • Magnet weight : Coil weight : Steel weight An advantage B0 R0 Focal length

34. “Strong focusing” technique Large pole edge rotation + Large field index NOVA NV-10 ion implanter Bend : 70 degrees Gap : 5 cm Bending radius : 53.8 cm Pole gap field : 8 kG Particle : 80 keV antimony Weight : 2000 lb Pole edge rotation: 35 degrees Field index : -1.152 • Uniform field bending magnet • Weight : 4000 lb • Pole gap field : 16 kG • Coil power : substantially higher x : DFD y : FDF x-defocus y-focus x-focus y-defocus

35. SPL with field clamp + ENGE New magnetic field map  Committed to the svn

36. Split pole magnet (ENGE)

37. Matrix tuning (E05-115) Before FWHM ~ 4 MeV/c2 After

38. Backup

39. Transverse field gradient (2) Total focusing power ( Dipole + transverse field gradient ) Simple harmonic motion Simple harmonic motion Field index