Elliptical Dipole Holger Witte Brookhaven National Laboratory Advanced

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Elliptical Dipole Holger Witte Brookhaven National Laboratory Advanced Accelerator Group 15 December 2011

Elliptical Dipole Holger Witte Brookhaven National Laboratory Advanced Accelerator Group 15 December 2011

Motivation • Bending magnets in muon collider: – exposed to decay particles – a

Motivation • Bending magnets in muon collider: – exposed to decay particles – a few k. W/m – from short lived muons • Distribution is highly anisotropic – large peak at the midplane (Mokhov) • One suggestion: open midplane dipoles – Issue: filed quality 15 December 2011 Nikolai Mokhov, in “Brief Overview of the Collider Ring Magnets Mini-Workshop, Telluride 2011. 2

Task Inside pipe width = 5 cm Inside pipe height = 2 cm Tungsten

Task Inside pipe width = 5 cm Inside pipe height = 2 cm Tungsten liner From: Suggested shield & cos theta dipole dimensions R. B. Palmer, 5/26/11 15 December 2011 3

Methodology developed for Integrable Optics Lattice (FNAL) • Task: generate certain vector potential •

Methodology developed for Integrable Optics Lattice (FNAL) • Task: generate certain vector potential • Singularities • Difficult to approximate with multipole fields • Ideally noncircular aperture – 2 cm horizontal, 4 cm vertical 15 December 2011 4

Vector Potential of Single Line Current • Vector potential at point P due to

Vector Potential of Single Line Current • Vector potential at point P due to current I (in zdirection): y P R r I a • Magnetic field: 15 December 2011 x 5

Methodology • Required: desired Az and coil bore • A~I, therefore: I 1 P

Methodology • Required: desired Az and coil bore • A~I, therefore: I 1 P 2 • P 2: Beam Aperture • Generally: 15 December 2011 A 11=VP @ P 1 for unit current I 1 A 21=VP @ P 2 for unit current I 1 A 12=VP @ P 1 for unit current I 2 6

Methodology: Formalism • Same is true for multiple currents and positions P • Formalism:

Methodology: Formalism • Same is true for multiple currents and positions P • Formalism: A · x = I 1 b • Linear equation system: Ax=b 15 December 2011 I 3 I 4 I I 2 3 P 1 P 2 P 3 P 4 A 11, A 12, . . . are known (can be calculated – unit current Im, calculate Az at Pn) b: also known (this is the vector potential we want) 7

Example: Quadrupole Current 15 December 2011 8

Example: Quadrupole Current 15 December 2011 8

Rectangular Shape Conductor 15 December 2011 Reference Az 9

Rectangular Shape Conductor 15 December 2011 Reference Az 9

From 2 D to 3 D • Power each current strand individually Vector addition

From 2 D to 3 D • Power each current strand individually Vector addition – Very inefficient, clumsy – Not very elegant • Known current distribution • Helical coil: vector addition of two currents, which always intersect at the correct angle 15 December 2011 10

From 2 D to 3 D • Easy if functional relationship is known (i.

From 2 D to 3 D • Easy if functional relationship is known (i. e. cos theta) • Here: In+1 In ds In-1 – (x, y) position known need to determine z • dz=d. I 15 December 2011 11

Quadrupole 15 December 2011 12

Quadrupole 15 December 2011 12

Quadrupole Calculated for two coils 15 December 2011 13

Quadrupole Calculated for two coils 15 December 2011 13

Task Inside pipe width = 5 cm Inside pipe height = 2 cm From:

Task Inside pipe width = 5 cm Inside pipe height = 2 cm From: Suggested shield & cos theta dipole dimensions R. B. Palmer, 5/26/11 15 December 2011 14

Concept: Elliptical Helical Coil Task: Find 2 D current distribution which generates (almost) pure

Concept: Elliptical Helical Coil Task: Find 2 D current distribution which generates (almost) pure dipole field Calculate this for a set of positions on ellipse y (m) A-axis: 9. 1 cm /2 B-axis: 13. 77 cm /2 x (m) 15 December 2011 15

Answer: Current Distribution Normalized current density vs. azimuthal angle 15 December 2011 16

Answer: Current Distribution Normalized current density vs. azimuthal angle 15 December 2011 16

Implementation: Elliptical Helical Coil 40 turns Spacing: 20 mm (= length about 0. 8

Implementation: Elliptical Helical Coil 40 turns Spacing: 20 mm (= length about 0. 8 m + “coil ends”) Single double layer Current in strand: 10 k. A (=400 k. A turns) Average current density: 10 k. A/(20 mmx 1 mm)=500 A/mm 2 15 December 2011 17

Field Harmonics Normalized to Dipole field of 1 T Evaluated for radius of 25

Field Harmonics Normalized to Dipole field of 1 T Evaluated for radius of 25 mm Well behaved: small sextupole component at coil entrance and exit 15 December 2011 18

Field along z B (T) 10 k. A = 1. 1 T z (m)

Field along z B (T) 10 k. A = 1. 1 T z (m) All unwanted field components point symmetric to the origin should disappear (e. g. Bz) for 4 -layer arrangement 15 December 2011 19

Other Geometries? • Well-known: intersecting ellipses produce dipole field • Worse performance J+ J-

Other Geometries? • Well-known: intersecting ellipses produce dipole field • Worse performance J+ J- – Field quality – Peak field on wire • Less flexible • Coil end problem? • Geometry problem – Approximation with blocks • Stresses? 15 December 2011 20

Additional Slides 15 December 2011 21

Additional Slides 15 December 2011 21

Integrable Optics 13 m • Introduce tune shift to prevent instabilities Nonlinear Lens Block

Integrable Optics 13 m • Introduce tune shift to prevent instabilities Nonlinear Lens Block 10 cm 5. 26 F F – Introduces Landau damping • One option for high intensity machines • Key: Non-linear block – Length 3 m 15 December 2011 22

Required Vector Potential • Singularities • Difficult to approximate with multipole fields • Ideally

Required Vector Potential • Singularities • Difficult to approximate with multipole fields • Ideally noncircular aperture – 2 cm horizontal, 4 cm vertical 15 December 2011 23

Integrable Optics - Field 15 December 2011 24

Integrable Optics - Field 15 December 2011 24

Quadrupole Gauging 15 December 2011 25

Quadrupole Gauging 15 December 2011 25

Gauging • Circular coil: constant current in longitudinal direction will cause a uniform vector

Gauging • Circular coil: constant current in longitudinal direction will cause a uniform vector potential A 0 within this circle • Az(x, y)=A 1(x, y)+A 0 • N. b. : • Ergo: changes vector potential but not field • Allows to shift current 15 December 2011 26

Gauging for elliptical coils • For elliptical coils (or other shapes): some modest variation

Gauging for elliptical coils • For elliptical coils (or other shapes): some modest variation of Az • Example: quadrupole • Correction per current strand: 2 k. A • Field: 0. 3 m. T 15 December 2011 27

Methodology • Required: desired vector potential – Defined by application • Required: beam aperture

Methodology • Required: desired vector potential – Defined by application • Required: beam aperture – Defined by application – (Real coil will be slightly larger) 15 December 2011 Az y x Beam Aperture 28

Methodology (cont. ) • Define point P 1 on desired crosssection (known Az) •

Methodology (cont. ) • Define point P 1 on desired crosssection (known Az) • Define current I 1 (for example on coil cross-section) • Az can be calculated from 15 December 2011 I 1 P 1 29