Betatron Motion with Coupling of Horizontal and Vertical

Betatron Motion with Coupling of Horizontal and Vertical Degrees of Freedom – Part II S. A. Bogacz, G. A. Krafft, S. De. Silva and R. Gamage Jefferson Lab and Old Dominion University Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 1

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, http: //dx. doi. org/10. 1088/1748 -0221/5/10/P 10010 Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 2

Axisymmetric rotational distribution - Twiss functions Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 3

Axisymmetric rotational distribution - Twiss functions Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 4

Axisymmetric rotational distribution - Twiss functions Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 5

Axisymmetric rotational distribution - Twiss functions Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 6

Spin rotators for Figure-8 Collider Ring total ring circumference ~1000 m 60 deg. crossing Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 7

0 -0. 15 DISP_X&Y[m] BETA_X&Y[m] 0. 15 15 Spin Rotator - Ingredients… 230 BETA_X BETA_Y DISP_X Arc end DISP_Y 320 BL = 11. 9 Tesla m 8. 8 0 BL = 28. 7 Tesla m 4. 4 0 Spin rotator ~ 46 m Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 8

15 5 Locally decoupled solenoid pair 0 0 0 BETA_X&Y[m] BL = 28. 7 Tesla m BETA_1 X BETA_2 Y solenoid 4. 16 m BETA_1 Y BETA_2 X 17. 9032 solenoid 4. 16 m decoupling quad insert M= C 0 0 -C Hisham Sayed, Ph. D Thesis ODU, 2011 Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 9

15 1 Locally decoupled solenoid pair 0 -1 DISP_X&Y[m] BETA_X&Y[m] BL = 28. 7 Tesla m 0 BETA_X BETA_Y solenoid 4. 16 m DISP_X DISP_Y 17. 9032 solenoid 4. 16 m decoupling quad insert M= C 0 0 -C Hisham Sayed, Ph. D Thesis ODU, 2011 Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 10

Universal Spin Rotator - Optics 1 0 -1 DISP_X&Y[m] BETA_X&Y[m] 30 5 Ge. V 288 BETA_X BETA_Y DISP_X DISP_Y 374 BL = 11. 9 Tesla m 8. 8 0 BL = 28. 7 Tesla m 4. 4 0 Spin rotator ~ 46 m Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 11

Ionization Cooling in an Axially Symmetric Channel Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 12

Ionization Cooling in an Axially Symmetric Channel Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 13

Ionization Cooling in an Axially Symmetric Channel Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 14

Hamiltonian formulation - equations of motion Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 15

Ionization Cooling in an Axially Symmetric Channel Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 16

Ionization Cooling in an Axially Symmetric Channel Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 17

Ionization Cooling in an Axially Symmetric Channel Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 18

Ionization Cooling in an Axially Symmetric Channel Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 19

Ionization Cooling in an Axially Symmetric Channel Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 20

Ionization Cooling in an Axially Symmetric Channel Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 21

Ionization Cooling in an Axially Symmetric Channel Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 22

Beta functions in axially symmetric FOFO cell Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 23

Periodic Cell - Optics ‘inward half-cell’ ‘outward half-cell’ betatron phase adv/cell (h/v) = p/2 p Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 24

Periodic Cell – Magnets ‘inward half-cell’ ‘outward half-cell’ betatron phase adv/cell (h/v) = p/2 p solenoids: L[cm] 22 22 B[k. G] 105 quadrupoles: dipoles: L[cm] 8 8 8 $L=20; => 20 cm $B= 25; => 25 k. Gauss $Ang=$L*$B/$Hr; => 0. 4996 rad $ang=$Ang*180/$PI; => 28. 628 deg G[k. G/cm] 1. 79754 -0. 3325 1. 79754 Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 25

‘Snake’ cooling channel – Dispersion suppression entrance cell periodic cells exit cell Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 26

Muon Cooling Channel - Optics beam extension disp. anti-suppr. n periodic PIC/REMEX cells (n=2) disp. suppr. beam extension RF cavity skew quad Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 27

Vertex-to-plane transformer insert Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 28

Vertex-to-plane transformer insert Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 29

Vertex-to-plane transformer insert Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 30

Ax Ay Alpha. XY BETA_2 X 2. 62647 5 1 0 Alpha. XY[-1, +1] PHASE/(2*PI) -1 0 Betatron size X&Y[cm] 0 0 0 2. 62647 0 BETA_1 X BETA_2 Y BETA_1 Y BETA_2 X Q_1 Q_2 Teta 1 Teta 2 2. 62647 0 0. 5 BETA_1 Y 0 BETA_2 Y 1 BETA_1 X 1 0 0. 5 0 BETA_X&Y[m] 5 0 1 Vertex-to-plane transformer insert 2. 62647 Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 31

Summary Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 9 - Coupled Betatron Motion II USPAS, Fort Collins, CO, June 13 -24, 2016 32