DAFNE wiggler optimization Simona Bettoni Frascati, 5th March 2007

Outline The wigglers in the main ring of DAFNE Analysis of the present situation (from experimental data) Method to reduce the integrated multipoles Modeling and optimization Conclusions

The wigglers of DAFNE Number of poles 5 fulls+2 halves Period (cm) 64 Magnetic field in the gap (T) ~1.8 Gap (cm) 3.7 Particles beams e+ e- Beam energy (MeV) 511 Excursion of ±1.3 cm with respect to the axis of the wiggler The integrated multipoles influence the beam dynamics. How to reduce them?

Previous attempts * Improvement of the field uniformity along the horizontal transverse dimension (±40 mm with respect to the axis of the magnet) A not negligible integrated octupole with respect to the beam trajectory still persists! * The modified wiggler of the DAFNE main ring, DAFNE Technical note MM-34

Other methods to reduce the integrated octupole CURVED POLE Reduction of the octupole around the beam trajectory in the region of the poles Proposed by P. Raimondi MOVING MAGNETIC AXIS Compensation of the integrated octupole in each semiperiod New method

Multipolar expansion of the field with respect to the beam trajectory Determination of the beam trajectory starting from the measured data Fit of By between -3 cm and +3 cm by a 4º order polynomial in x centered in xT(z) = xT xT +3 cm Beam trajectory (xT) xT -3 cm

Analysis of the present situation The integrated multipoles calculated with respect to the beam trajectory in the central semiperiod (z from -16.28 cm to 16.28 cm, z = 0 cm is the wiggler center) as: FC -3 HC -1 FC -2 FC 2 FC 1 FC 3 HC 1 zin zfin z I0 (T.m) I1 (T) I2 (T/m) I3 (T/m2) I4 (T/m3) ±16.28 cm -0.40 -0.09 0.80 -26.5 1448.7 Complete 0.00 -3.35 -139.7 1772.2 ↑ ↑ ↑ ↑ ↑ Dipole Quadrupole Sextupole Octupole Decapole

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

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

Optimization: how to proceed Optimization made on the central semiperiod of the wiggler : Determination of the position of the magnetic axis to minimize the integrated octupole in the central semiperiod Reduction of the dependence of the integrated octupole on a the beam displacement by means of pole shims Re-optimization of the magnetic axis in the configuration with the shims Optimization on the complete wiggler: Determination of the integrated multipoles in the other semiperiods and in the complete wiggler

Optimization: the magnetic axis HC -1 FC -3 FC -2 FC 1 FC 2 FC 3 HC 1 First guess → magnetic axis at x = ±1 cm

Optimization of the pole of the wiggler Goals Reduce as less as possible the magnetic field in the gap Maintain the left-right symmetry FC1-like FC2-like FC 1 FC 2

Optimization: cut with respect to ±1 cm (central semiperiod) Present configuration: I3 = -26.5 T/m2 After the movement of the magnetic axis: I3 = -1.4 T/m2 Reduction of about a factor 20 of the integrated octupole with respect to the present situation Taglio1 e la traeittoria corretta

Optimization of the central semiperiod Magnetic model of the complete wiggler Optimization of the beam trajectory Calculation of the integrated multipoles along the trajectory in the central semiperiod I0 (T.m) I1 (T) I2 (T/m) I3 (T/m2) I4 (T/m3) Present -0.40 -0.09 0.80 -26.5 1448.7 Partial model 0.43 -0.05 -7.62 -1.4 -3141.6 Complete model 0.42 -7.29 -2.6 -3290.8 Reduction of about a factor 10 of the integrated octupole with respect to the present

Optimization: dependence on the beam displacement The integrated multipoles in the central semiperiod are calculated simulating a beam displaced in x with respect to the nominal orbit Displacement of ±2 mm cancels the effect of the optimization!

Optimization: the shims Cut poles plane x (cm) FC -3 HC -1 FC -2 FC 2 FC 1 FC 3 HC 1 FC1 FC2 FC3 HC1 z (cm) Shims plane Reduce b4, b6 risp asse FC2 HC1 FC1 FC3 x (cm) z (cm) 50

Optimization: effects of the shims (central semiperiod) Complete wiggler model Optimization of the beam trajectory Calculation of the integrated multipoles along this new trajectory Only cut: I3 = 2.6 T/m2 Cut + shims: I3 = 6.7 T/m2 Present configuration: I3 = -26.5 T/m2 CONCLUSIONS Strongly reduced the dependence of the solution from the positioning of the wiggler The shims modify the optimal position of the magnetic axis

Optimization: re-optimization of the position of the magnetic axis in the configuration with the shims Moved the axis from x = ±1.00 cm to x = ± 1.15 cm Re-optimized the beam trajectory Calculated the integrated octupole along this trajectory in the central semiperiod Axis x = ±1 cm: I3 = 6.70 T/m2 Axis x = ± 1.15 cm: I3 = -0.29 T/m2 Present configuration: I3 = -26. 5T/m2 Reduction of 2 orders of magnitude of the integrated octupole with respect to the present situation

Re-optimization: complete wiggler FC1 -0.29 T/m2 FC2 -0.47 T/m2 MODEL I0 (T.m) I1 (T) I2 (T/m) I3 (T/m2) I4 (T/m3) FC1 0.42 -0.07 -8.04 -0.29 921.8 FC2 -0.42 8.07 -0.47 -904.1 FC3 -0.06 -8.03 -0.16 918.99 HC1 -0.21 -0.03 -2.99 -1.53 -657.1 Complete 0.00 -0.36 -1.67 -4.13 -327.1 FC3: -0.16 T/m2 HC1: -2.79 T/m2 PRESENT I0 (T.m) I1 (T) I2 (T/m) I3 (T/m2) I4 (T/m3) Complete 0.00 -0.09 -3.35 -139.7 1772.2

Conclusions The proposed configuration allows to reduce the integrated octupole by a factor 30 with respect to the present configuration I0 (T.m) I1 (T) I2 (T/m) I3 (T/m2) I4 (T/m3) Present 0.00 -0.09 -3.35 -139.7 1772.2 Model -0.36 -1.67 -4.1 -327.1 In particular: The integrated octupole in each semiperiod has been reduced by cutting the pole The dependence of the wiggler positioning has been reduced by introducing pole shims This modification will be tested on a spare wiggler and after implemented in DAFNE