1. DAFNE wiggler optimization
Simona Bettoni
Frascati, 5th March 2007

2. 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

3. 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?

4. 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

5. 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

6. 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

7. Analysis of the present situation
The integrated multipoles calculated with respect to the beam trajectory in the central semiperiod (z from cm to 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

8. 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 →

9. 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

10. 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

11. 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

12. 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

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

14. 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
Complete model
0.42
-7.29
-2.6
Reduction of about a factor 10 of the integrated octupole with respect to the present

15. 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!

16. 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

17. 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 = T/m2
Cut + shims: I3 = T/m2
Present configuration: I3 = 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

18. 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 = T/m2
Present configuration: I3 = T/m2
Reduction of 2 orders of magnitude of the integrated octupole with respect to the present situation

19. Re-optimization: complete wiggler
FC T/m2
FC 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: T/m2
HC1: 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

20. 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