Simona Bettoni, Remo Maccaferri Analysis of a proposal for the design of the CLIC damping rings wigglers

Outline  Introduction  The model  2D (Poisson)  3D (Opera Vector Fields-Tosca)  The analysis tools  Field uniformity  Multipoles (axis and trajectory)  Tracking studies  The integrals of motion cancellation  Possible options  The final proposal  The prototype analysis  Method to reduce the integrated multipoles  Conclusions

Damping ring layout

Wigglers/undulators model Large gap & long period Small gap & short period

2D design (R. Maccaferri) BEAM Advantages: Short period Small forces on the heads (curved)

The 3D model

The 3D model (base plane)

The 3D model (extrusions)

The 3D model (conductors) Parameters the script: wire geometry (l_h, l_v, l_trasv) winding “shape” (n_layers, crossing positions) Conductors generated using a Matlab script. Grouping of the conductors.

The analysis tools o Tracking analysis: Single passage: ready/done Multipassage: to be implemented o Field uniformity: ready/done o Multipolar analysis: Around the axis: ready /done Around the reference trajectory: ready x and x’ at the exit of the wiggler

Prototype analysis (CLIC_Wiggler_7.op3) Period (mm)Gap (mm)Number of periodsTotal length (cm) flanges length x z y

Field distribution on the conductors Maximum field and forces (P MAX ~32 MPa) on the straight part  Manufacture: well below the limit of the maximum P for Nb 3 Sn  Simulation: quick to optimize the margin B Mod (Gauss)

The 2D/3D comparison T T T T 2D (Poisson) 3D (Tosca)

Field uniformity (x range = ±2 cm) z (cm)

Multipolar analysis (x range = ±2 cm)

Tracking studies Trajectory x-shift at the entrance = ± 3 cm z x y

Tracking studies: the exit position Subtracting the linear part

Tracking studies: the exit angle

Integrals of motion 1 st integral 2 nd integral CLIC case (even number of poles anti-symmetric) No offset of the oscillation axis Offset of the oscillation axis = 0 for anti-symmetry

Integrals of motion: the starting point 1 st integral 2 nd integral First integral Bz * dySecond integral Bz * dy 5e-5 Gauss*cm-1.94e5 Gauss*cm 2 5e-11 T*m-1.94e-3 T*m 2 = 0 for anti-symmetry (cm)

Lowering the 2 nd integral: what do we have to do? To save time we can do tracking studies in 2D up to a precision of the order of the difference in the trajectory corresponding to the 2D/3D one (~25  m) and only after refine in 3D.

Lowering the 2 nd integral: how can we do? What we can use:  End of the yoke length/height  Height of the yoke  Terminal pole height (|B| > 5 T)  Effectiveness of the conductors → → → Highly saturated

Lowering the 2 nd integral: option 1

The multipoles of the option 1 CLICWiggler7.op3 CLICWiggler8.op3

Lowering the 2 nd integral: option 2 (2D)

Option 1 vs option 2 The “advantage” of the option 2:  Perfect cancellation of the 2 nd integral  Field well confined in the yoke  Possibility to use only one IN and one OUT (prototype) The “disadvantage” of the option 2:  Comments? The “advantage” of the option 1:  Easy to be done The “disadvantage” of the option 1:  No perfect cancellation of the 2 nd integral  Field not completely confined in the yoke  Multipoles get worse 1 st layers (~1/3 A*spire equivalent) All the rest → → → → → start end

Lowering the 2 nd integral: option 2 (3D) If only one IN and one OUT → discrete tuning in the prototype model Fine regulation would be possible in the long model and in the DR (modular)

Tracking studies (optimized configuration) Not optimizedOptimized

Working point: Nb 3 Sn & NbTi I (A)Max|B| (T)By peak (T) *MANUFACTURE AND TEST OF A SMALL CERAMIC-INSULATED Nb3Sn SPLIT SOLENOID, B. Bordini et al., EPAC’08 Proceedings. * Wire diameter (insulated) = 1 mm Wire diameter (bare) = 0.8 mm I (A)Max|B| (T)By peak (T) Nb 3 Sn NbTi Nb 3 Sn NbTi Cu/SC ratio = 1 Non-Cu fraction = 0.53

Possible configurations Possible to increase the peak field of 0.5 T using holmium (Remo), BUT $ Nb 3 Sn2.1 T40 mm20 mm

Working point: comparison

Short prototype status & scheduling

Reduction of the integrated multipoles S. Bettoni, Reduction of the integrated odd multipoles in periodic magnets, PRST-AB, 10, (2007), S. Bettoni et al., Reduction of the Non-Linearities in the DAPHNE Main Rings Wigglers, PAC’07 Proceedings.

Even multipoles → Odd multipoles → In a displaced system of reference: b A k → defined in the reference centered in O A (wiggler axis) b T k → defined in the reference centered in O T (beam trajectory) x’ y’ x y OAOA O T xTxT Left-right symmetry of the magnet Multipoles change sign from a pole to the next one Sum from a pole to the next one The integrated multipoles in periodic magnets

The displacement of the magnetic field axis WITHOUT THE POLE MODIFICATION In each semiperiod the particle trajectory is always on one side with respect the magnetic axis In each semiperiod the particle travels on both sides with respect to the magnetic axis Opportunely choosing the B axis is in principle possible to make zero the integrated octupole in each semiperiod WITH THE POLE MODIFICATION Octupole ↑

Number of poles5 fulls+2 halves Period (cm)64 Magnetic field in the gap (T)~2 Gap (cm)3.7 Particles beamse + e - Beam energy (MeV)511 Excursion of ± 1.3 cm with respect to the axis of the wiggler The application to the DA  NE main rings wigglers

Integrated b 3 T (T/m 2 ) Aligned poles Shifted poles0.07 The results

Conclusions  A novel design for the CLIC damping ring has been analyzed (2D & 3D)  Advantages: o Possibility to have a very small period wiggler o Small forces on the heads  Analysis on the prototype: o Maximum force o Multipolar analysis o Tracking studies o Zeroing the integrals of motion  A method to compensate the integrated multipoles has been presented  Even multipoles cancel from a pole to the next one and odd multipoles canceled by the opportune magnetic axis displacement  How to proceed  Optimization of the complete wiggler model (work in progress): o Best working point definition, if not already (margin) o Modeling of the long wiggler o 2 nd integral optimization for the long model o Same analysis tools applied to the prototype model (forces, multipoles axis/trajectory, tracking) o Minimization of the integrated multipoles

Extra slides

Longitudinal field (By = f(y), several x) Scan varying the entering position in horizontal, variation in vertical:   z = 0.1  m for x-range = ±1 cm   z = 2  m for x-range = ±2 cm

Horizontal transverse field (Bx = f(y), several x) Scan varying the entering position in horizontal, variation in vertical:   z = 0.1  m for x-range = ±1 cm   z = 2  m for x-range = ±2 cm

Controlling the y-shift: cancel the residuals W1W2W3W4 W1W2W3W4 2  m in 10 cm -> 20*2 = 40  m in 2 m

Controlling the x-shift: cancel the residuals (during the operation) Entering at x = 0 cm Entering at x = -  x MAX /2 Entering at x = +  x MAX /2 (opposite I wiggler … positron used for trick) W1W2 … Quadrupoles very close to the beginning of the wiggler or at half distance?

The fit accuracy: an example

Field uniformity (x-range = ±3 cm)

Multipolar analysis (x-range = ±3 cm)

Tracking at x-range = ±3 cm: exit position Subctracting the linear part

Tracking at x-range = ±3 cm: exit angle

Tracking optimized (x-range = ±3 cm)

Holmium option

BINP wire

2nd integral optimization (long model)

Long wiggler modeling Problem: very long running time (3D) because of the large number of conductors in the model Solution:  Build 2D models increasing number of periods until the field distribution of the first two poles from the center give the same field distribution (Np)  Build 3D model with a number of poles Np  “Build” the magnetic map from this