1. Alex Bogacz, Geoff Krafft and Timofey Zolkin
Betatron Motion with Coupling of Horizontal and Vertical Degrees of Freedom – Part II
Alex Bogacz, Geoff Krafft and Timofey Zolkin
USPAS, Fort Collins, CO, June 10-21, 2013

2. Outline Practical Examples: Soft-edge Solenoid model
Vertex-to-plane adapter for electron cooling (Fermilab)
Spin Rotator for Figure-8 Collider ring
Ionization cooling channel for Neutrino Factory and Muon Collider
Generalized vertex-to-plane transformer insert
V. Lebedev, A. Bogacz, ‘Betatron Motion with Coupling of Horizontal and Vertical Degrees of Freedom’, 2000,
USPAS, Fort Collins, CO, June 10-21, 2013

3. ‘Hard-edge’ Solenoid
USPAS, Fort Collins, CO, June 10-21, 2013

4. Solenoid – ‘Hard-edge’ Model
Linear Transfer Matrix for infinitely long solenoid :
USPAS, Fort Collins, CO, June 10-21, 2013

5. ‘Soft-edge’ Solenoid
USPAS, Fort Collins, CO, June 10-21, 2013

6. Solenoid – ‘Soft-edge’ Model
Non-zero aperture - correction due to the finite length of the edge:
It introduces axially symmetric edge focusing at each solenoid end:
USPAS, Fort Collins, CO, June 10-21, 2013

7. ‘Soft-edge’ Solenoid – Off-axis Fields
Nonlinear focusing term DF ~ O(r2) follows from the scalar potential:
Solenoid B-fields
Nonlinear focusing included in particle tracking
USPAS, Fort Collins, CO, June 10-21, 2013

8. Axisymmetric Rotational Distribution
USPAS, Fort Collins, CO, June 10-21, 2013

9. Axisymmetric Rotational Distribution
USPAS, Fort Collins, CO, June 10-21, 2013

10. Axisymmetric Rotational Distribution
USPAS, Fort Collins, CO, June 10-21, 2013

11. Axisymmetric Rotational Distribution
USPAS, Fort Collins, CO, June 10-21, 2013

12. Spin Rotators for Figure-8 Collider Ring
total ring circumference ~1000 m
60 deg. crossing
USPAS, Fort Collins, CO, June 10-21, 2013

13. Spin Rotator - Ingredients…
320
230 15
0.15
-0.15
BETA_X&Y[m]
DISP_X&Y[m]
BETA_X
BETA_Y
DISP_X
DISP_Y
Arc end
4.4 0
8.8 0
Spin rotator ~ 46 m
BL = 11.9 Tesla m
BL = 28.7 Tesla m
USPAS, Fort Collins, CO, June 10-21, 2013

14. Locally Decoupled Solenoid Pair 15 5
BETA_X&Y[m]
BETA_1X
BETA_2Y
BETA_1Y
BETA_2X
BL = 28.7 Tesla m
solenoid m
solenoid m
decoupling quad insert
M = C
- C
Hisham Sayed, PhD Thesis ODU, 2011
USPAS, Fort Collins, CO, June 10-21, 2013

15. Locally Decoupled Solenoid Pair 15 1 -1
BETA_X&Y[m]
DISP_X&Y[m]
BETA_X
BETA_Y
DISP_X
DISP_Y
BL = 28.7 Tesla m
solenoid m
solenoid m
decoupling quad insert
M = C
- C
Hisham Sayed, PhD Thesis ODU, 2011
USPAS, Fort Collins, CO, June 10-21, 2013

16. Universal Spin Rotator - Optics
5 GeV
374
288 30 1 -1
BETA_X&Y[m]
DISP_X&Y[m]
BETA_X
BETA_Y
DISP_X
DISP_Y
4.4 0
8.8 0
Spin rotator ~ 46 m
BL = 11.9 Tesla m
BL = 28.7 Tesla m
USPAS, Fort Collins, CO, June 10-21, 2013

17. Ionization Cooling in an Axially Symmetric Channel
USPAS, Fort Collins, CO, June 10-21, 2013

18. Ionization Cooling in an Axially Symmetric Channel
USPAS, Fort Collins, CO, June 10-21, 2013

19. Principle of Ionization Cooling
Each particle loses momentum by ionizing a low-Z absorber
Only the longitudinal momentum is restored by RF cavities
The angular divergence is reduced until limited by multiple scattering
USPAS, Fort Collins, CO, June 10-21, 2013

20. Ionization Cooling in an Axially Symmetric Channel
USPAS, Fort Collins, CO, June 10-21, 2013

21. Canonical vs Geometric Variables
USPAS, Fort Collins, CO, June 10-21, 2013

22. Ionization Cooling in an Axially Symmetric Channel
USPAS, Fort Collins, CO, June 10-21, 2013

23. Ionization Cooling in an Axially Symmetric Channel
USPAS, Fort Collins, CO, June 10-21, 2013

24. Ionization Cooling in an Axially Symmetric Channel
USPAS, Fort Collins, CO, June 10-21, 2013

25. Ionization Cooling in an Axially Symmetric Channel
USPAS, Fort Collins, CO, June 10-21, 2013

26. Ionization Cooling in an Axially Symmetric Channel
USPAS, Fort Collins, CO, June 10-21, 2013

27. Axially Symmetric FOFO Cell
USPAS, Fort Collins, CO, June 10-21, 2013

28. Periodic Cell - Optics ‘inward half-cell’ ‘outward half-cell’
betatron phase adv/cell (h/v) = p/2p
USPAS, Fort Collins, CO, June 10-21, 2013

29. Periodic Cell – Magnets
‘inward half-cell’
‘outward half-cell’
betatron phase adv/cell (h/v) = p/2p
solenoids:
L[cm] B[kG]
quadrupoles:
L[cm] G[kG/cm]
dipoles:
$L=20; => cm
$B= ; => kGauss
$Ang=$L*$B/$Hr; => rad
$ang=$Ang*180/$PI; => deg
USPAS, Fort Collins, CO, June 10-21, 2013

30. ‘Snake’ Cooling Channel
entrance cell
periodic cells
exit cell
USPAS, Fort Collins, CO, June 10-21, 2013

31. Muon Cooling Channel - Optics
beam extension
disp. anti-suppr.
disp. suppr.
beam extension
RF cavity
skew quad
n periodic PIC/REMEX cells (n=2)
USPAS, Fort Collins, CO, June 10-21, 2013

32. Vertex-to-plane Transformer Insert
USPAS, Fort Collins, CO, June 10-21, 2013

33. Vertex-to-plane Transformer Insert
USPAS, Fort Collins, CO, June 10-21, 2013

34. Vertex-to-plane Transformer Insert
USPAS, Fort Collins, CO, June 10-21, 2013

35. Vertex-to-plane Transformer Insert 1 5
BETA_X&Y[m]
BETA_1X
BETA_2Y
BETA_1Y
BETA_2X -1
Betatron size X&Y[cm]
AlphaXY[-1, +1] Ax Ay
AlphaXY
0.5
PHASE/(2*PI)
Q_1
Q_2
Teta1
Teta2
USPAS, Fort Collins, CO, June 10-21, 2013

36. Summary
USPAS, Fort Collins, CO, June 10-21, 2013