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

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, http://dx.doi.org/10.1088/1748-0221/5/10/P10010 USPAS, Fort Collins, CO, June 10-21, 2013

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

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

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

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

‘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

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

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

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

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

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

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

Locally Decoupled Solenoid Pair 17.9032 15 5 BETA_X&Y[m] BETA_1X BETA_2Y BETA_1Y BETA_2X BL = 28.7 Tesla m solenoid 4.16 m solenoid 4.16 m decoupling quad insert M = C - C Hisham Sayed, PhD Thesis ODU, 2011 USPAS, Fort Collins, CO, June 10-21, 2013

Locally Decoupled Solenoid Pair 17.9032 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 4.16 m solenoid 4.16 m decoupling quad insert M = C - C Hisham Sayed, PhD Thesis ODU, 2011 USPAS, Fort Collins, CO, June 10-21, 2013

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

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

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

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

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

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

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

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

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

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

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

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

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

Periodic Cell – Magnets ‘inward half-cell’ ‘outward half-cell’ betatron phase adv/cell (h/v) = p/2p solenoids: L[cm] B[kG] 22 105 22 105 quadrupoles: L[cm] G[kG/cm] 8 1.79754 8 -0.3325 dipoles: $L=20; => 20 cm $B= 25; => 25 kGauss $Ang=$L*$B/$Hr; => 0.4996 rad $ang=$Ang*180/$PI; => 28.628 deg USPAS, Fort Collins, CO, June 10-21, 2013

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

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

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

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

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

Vertex-to-plane Transformer Insert 2.62647 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

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