1. Betatron Motion with Coupling of Horizontal and Vertical Degrees of Freedom – Part II
S.A. Bogacz, G.A. Krafft, S. DeSilva and R. Gamage
Jefferson Lab and Old Dominion University
USPAS, Fort Collins, CO, June 13-24, 2016

2. Outline Practical Examples: Spin Rotator for Figure-8 Collider ring
Vertex-to-plane adapter for electron cooling (Fermilab)
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 13-24, 2016

3. Axisymmetric rotational distribution - Twiss functions
USPAS, Fort Collins, CO, June 13-24, 2016

4. Axisymmetric rotational distribution - Twiss functions
USPAS, Fort Collins, CO, June 13-24, 2016

5. Axisymmetric rotational distribution - Twiss functions
USPAS, Fort Collins, CO, June 13-24, 2016

6. Axisymmetric rotational distribution - Twiss functions
USPAS, Fort Collins, CO, June 13-24, 2016

7. Spin rotators for Figure-8 Collider Ring
total ring circumference ~1000 m
60 deg. crossing
USPAS, Fort Collins, CO, June 13-24, 2016

8. 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 13-24, 2016

9. 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 13-24, 2016

10. 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 13-24, 2016

11. 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 13-24, 2016

12. Ionization Cooling in an Axially Symmetric Channel
USPAS, Fort Collins, CO, June 13-24, 2016

13. Ionization Cooling in an Axially Symmetric Channel
USPAS, Fort Collins, CO, June 13-24, 2016

14. Ionization Cooling in an Axially Symmetric Channel
USPAS, Fort Collins, CO, June 13-24, 2016

15. Hamiltonian formulation - equations of motion
USPAS, Fort Collins, CO, June 13-24, 2016

16. Ionization Cooling in an Axially Symmetric Channel
USPAS, Fort Collins, CO, June 13-24, 2016

17. Ionization Cooling in an Axially Symmetric Channel
USPAS, Fort Collins, CO, June 13-24, 2016

18. Ionization Cooling in an Axially Symmetric Channel
USPAS, Fort Collins, CO, June 13-24, 2016

19. Ionization Cooling in an Axially Symmetric Channel
USPAS, Fort Collins, CO, June 13-24, 2016

20. Ionization Cooling in an Axially Symmetric Channel
USPAS, Fort Collins, CO, June 13-24, 2016

21. Ionization Cooling in an Axially Symmetric Channel
USPAS, Fort Collins, CO, June 13-24, 2016

22. Ionization Cooling in an Axially Symmetric Channel
USPAS, Fort Collins, CO, June 13-24, 2016

23. Beta functions in axially symmetric FOFO cell
USPAS, Fort Collins, CO, June 13-24, 2016

24. Periodic Cell - Optics ‘inward half-cell’ ‘outward half-cell’
betatron phase adv/cell (h/v) = p/2p
USPAS, Fort Collins, CO, June 13-24, 2016

25. 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 13-24, 2016

26. ‘Snake’ cooling channel – Dispersion suppression
entrance cell
periodic cells
exit cell
USPAS, Fort Collins, CO, June 13-24, 2016

27. 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 13-24, 2016

28. Vertex-to-plane transformer insert
USPAS, Fort Collins, CO, June 13-24, 2016

29. Vertex-to-plane transformer insert
USPAS, Fort Collins, CO, June 13-24, 2016

30. Vertex-to-plane transformer insert
USPAS, Fort Collins, CO, June 13-24, 2016

31. 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 13-24, 2016

32. Summary
USPAS, Fort Collins, CO, June 13-24, 2016