Muon Acceleration in FFAG Rings Eberhard Keil CASA Seminar at JLab - PowerPoint PPT Presentation

CASA Seminar at JLab April 29, 2004 Muon Acceleration in FFAG Rings Eberhard Keil CASA Seminar at JLab 26 April 2004 My WWW home directory: http://keil.home.cern.ch/keil/ MuMu/Doc/JLab Apr04/talk.pdf E. Keil page 1

CASA Seminar at JLab April 29, 2004 Motivation • Neutrino factory studies in US and Europe assumed muon acceleration in recirculating linear accelerators ”similar” to CEBAF with – only 4 or 5 passes – 7 or 9 arcs – 4 spreaders and combiners – no kickers for injection and ejection – 37.5% and 20% of total cost of neutrino factory in studies I and II • FFAG rings promise – more passes – fewer arcs – no spreaders and combiners – fun with kickers for injection and ejection E. Keil page 2

CASA Seminar at JLab April 29, 2004 Styles of FFAG Accelerators • Scaling FFAG rings – have similar orbits at different momenta – have tunes independent of momentum – have nonlinear fields – radial or spiral sectors – are part of the Japanese neutrino factory design • Non-scaling FFAG rings – are essentially alternating-gradient lattices with small dispersion and controlled values of slip factors η 0 and η 1 – have tunes that vary with momentum – have linear fields – are considered for US neutrino factory design E. Keil page 3

CASA Seminar at JLab April 29, 2004 Actors and References • C.J. Johnstone and S. Koscielniak , Recent Progress on FFAGs for Rapid Acceleration, APS/DPF/DPB Summer Study on the Future of Particle Physics (Snowmass 2001) T508. • D. Trbojevic et al., Fixed Field Alternating Gradient Lattice Design without Opposite Bend, EPAC 2002, Paris, France, 1199. • C.J. Johnstone and S. Koscielniak, Recent Progress on FFAGs for Rapid Acceleration, EPAC 2002, Paris, France, 1261. • C. Johnstone and S. Koscielniak, FFAGS for Rapid Acceleration, accepted for publication in NIM-A Nov 2002. • E. Keil and A.M. Sessler , Muon Acceleration in FFAG Rings, PAC 2003, 414. • D. Trbojevic et al., FFAG Lattice for Muon Acceleration with Distributed RF, PAC 2003, 1816. • J.S. Berg and C. Johnstone, Design of FFAGs Based on a FODO Lattice, PAC 2003, 2216. • J.S. Berg et al., FFAGs for Muon Acceleration, PAC 2003, 3413. E. Keil page 4

CASA Seminar at JLab April 29, 2004 Longitudinal Dynamics • Longitudinal Hamiltonian for stationary buckets � � 2 πhβ 2 η 0 p 2 + η 1 p 3 0 E 0 + sin 2 πϕ t t H 1 ( p t , ϕ ) = + . . . eV N c 2 3 – p t momentum error relative to reference particle with total energy E 0 and speed β 0 c – ϕ phase measured in cycles with origin at stable fixed point and − 1 / 2 ≤ ϕ ≤ +1 / 2 – h harmonic number, V peak accelerating voltage, N c number of RF cavities • Consider 3 cases: – Linear motion with η 0 � = 0 and η 1 = η 2 = 0 – Nonlinear motion with η 0 � = 0 , η 1 � = 0 and η 2 = 0 – Motion near transition with η 0 = 0 , and η 1 � = 0 E. Keil page 5

CASA Seminar at JLab April 29, 2004 Linear Longitudinal Motion 1 • Measure momentum offset y in units of 0.5 half linear bucket height • For stationary buckets in FFAG rings 0 – Stable fixed point at ϕ = y = 0 – Unstable fixed points at ϕ = ± 1 / 2 and y = 0 -0.5 – Hamiltonian -1 H ( ϕ, y, a ) = y 2 + sin 2 πϕ -0.4 -0.2 0 0.2 0.4 Contour plot of Hamiltonian for linear motion. Muons move along level lines. E. Keil page 6

CASA Seminar at JLab April 29, 2004 Effect of η 1 � = 0 on Longitudinal Hamiltonian • a = η 1 p b /η 0 with half bucket height p b 1 • New stable fixed points at ϕ = ± 1 / 2 and y = − 1 /a 0.5 • New unstable fixed point at ϕ = 0 and y = 0 − 1 /a -0.5 • Ω -shaped trajectories start below fixed point at ϕ = ± 1 / 2 and y = − 1 /a , cir- -1 cle around fixed point at ϕ = 0 and y = 0 , and reach maximum y above it -1.5 • Acceleration in FFAG rings along light -2 -0.4 -0.2 0 0.2 0.4 blue Ω -shaped trajectories Contour plot of Hamiltonian at a = 1 . • Find limit on a for Ω -shaped trajectories E. Keil page 7

CASA Seminar at JLab April 29, 2004 Separatrices • Separatrices pass unstable fixed points 0.5 • 2 unstable fixed points and 2 separatrices when a � = 0 0.1 0.2 0.3 0.4 0.5 -0.5 • Find separatrices by solving for y : -1 -1.5 H ( ϕ, y, a ) = H ( − 1 / 2 , 0 , a ) a = 1 H ( ϕ, y, a ) = H (0 , − 1 /a, a ) 1 • Use symmetry and plot for 0 ≤ ϕ ≤ 1 / 2 0.1 0.2 0.3 0.4 0.5 -1 • Acceleration along trajectories in S-shaped channel be- -2 tween islands starts between separatrices in lower right -3 corner below y = − 3 / 2 a , and ends between separatri- a = 1 / 2 ces in upper left corner above y = 1 / 2 a 0.5 • At a = 1 / 2 regular bucket centred at ϕ = y = 0 blocks 0.1 0.2 0.3 0.4 0.5 -0.5 acceleration across y = 0 -1 √ -1.5 • At a = 1 / 3 buckets centred at ϕ = y = 0 and at -2 √ -2.5 ϕ = 1 / 2 and y = − 3 just touch, and channel of ac- √ celeration has width zero, agreeing with K.Y.Ng’s result a = 1 / 3 E. Keil page 8

CASA Seminar at JLab April 29, 2004 Longitudinal Motion Near Transition • Introduce scaled momentum variable y � 1 / 3 � 2 πβ 2 0 E 0 hη 1 y = p t 1 3 eV N c • Scaled Hamiltonian H 5 ( y, ϕ ) 0.5 H 5 ( y, ϕ ) = y 3 + sin 2 πϕ 0 • Acceleration in FFAG rings happens along light blue S -shaped trajectory, which starts at ϕ = 1 / 2 and y = − 1 , -0.5 and reaches maximum y = 1 at ϕ = 0 • Equation relates range ± p t and ring -1 parameters at y = ± 1 , cf. next page 0 0.1 0.2 0.3 0.4 0.5 • Discuss later two FFAG rings operat- Contour plot of H 5 ( y, ϕ ) ing near transition, doublet lattice for muons, and model for electrons E. Keil page 9

CASA Seminar at JLab April 29, 2004 Parameters and Scaling Laws • Calculate RF cavity voltage V from accelerating range p t and ring parameters: V = 2 πβ 2 0 E 0 � hη 1 � p 3 t 3 e N c • Scaling with energy E 0 in first term, with range p t in third term • Scaling with N lattice periods of length L in brackets: – h and circumference C at given RF frequency ∝ LN – N c ∝ N – η 1 ∝ 1 /N 2 derived analytically by K.Y. Ng for FODO lattice with N ≫ 1 ; I believe from numerical studies that it holds for any lattice style t /N 2 and N c V ∝ E 0 Lp 3 • V ∝ E 0 Lp 3 t /N • Assuming that cost of magnets, vacuum, tunnel is C M LN , that cost of RF cavities and power installation is C RF E 0 Lp 3 t /N yields cost optimum at eqal cost components � C M C RF E 0 p 3 C = 2 L t E. Keil page 10

CASA Seminar at JLab April 29, 2004 Johnstone-Koscielniak FODO Lattice JK • Focusing quadrupoles • Defocusing gradient dipoles Johnstone-Koscielniak FFAG lattice cell 6 to 20 GeV - apr07r Win32 version 8.51/15 27/03/04 15.45.43 3.8 0.30 • FODO lattice with Q x ≈ Q y β x β y 1/ 2 ) x (m) 1 / 2 1 / 2 D x 0.29 1/ 2 (m 3.6 D 0.28 • Number of cells N = 314 0.27 β 3.4 • Circumference C = 2041 m 0.26 3.2 0.25 • Space for two super- 0.24 3.0 0.23 conducting RF cavities 0.22 2.8 in cell 0.21 2.6 0.20 0.0 1. 2. 3. 4. 5. 6. 7. 8. • Accelerating voltage V = s (m) 2 . 5 MV E. Keil page 11

CASA Seminar at JLab April 29, 2004 Trbojevic Triplet Lattice T • Focusing gradient dipoles FFAG 15 Gev Lattice Dejan Trbojevic, APR 1, 2003 - mar28n • Defocusing gradient dipoles Win32 version 8.51/15 29/03/04 16.19.07 3.500 0.18 1/ 2 ) x (m) β x β y 1 / 2 1 / 2 D x 0.17 1/ 2 (m 3.275 • Triplet lattice with Q x � = Q y D 0.16 3.050 β 0.15 • Number of cells N = 60 2.825 0.14 0.13 2.600 • Circumference C = 318 m 0.12 2.375 0.11 2.150 0.10 • Space for super-conducting 0.09 1.925 RF cavity 0.08 1.700 0.07 1.475 • Accelerating voltage V = 0.06 1.250 0.05 10 MV 0.0 1.0 2.0 3.0 4.0 5.0 6.0 s (m) E. Keil page 12

CASA Seminar at JLab April 29, 2004 Keil-Sessler FODO Lattice KS-F • Focusing quadrupoles • Defocusing gradient dipoles FFAG cell 6-20 GeV - Lp=3.7m - apr28p • FODO lattice with Q x ≈ Q y Win32 version 8.51/15 29/04/04 00.40.11 2.90 0.22 β y β x 1/ 2 ) x (m) 1 / 2 1 / 2 D x 2.81 • FODO lattice with Q x ≈ Q y 1/ 2 (m 0.21 D 2.72 0.20 2.63 • Number of cells N = 2800 β 2.54 0.19 • Circumference C = 1036 m 2.45 0.18 2.36 • Space for two room- 2.27 0.17 2.18 temperature RF cavities 0.16 2.09 in cell 2.00 0.15 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 s (m) • Accelerating voltage V = δ 3 MV E. Keil page 13

CASA Seminar at JLab April 29, 2004 Keil-Sessler Doublet Lattice KS-D • Focusing gradient dipoles Doublet cell 10-20 GeV - Lp=4 m - mar22r • Defocusing gradient dipoles Win32 version 8.51/15 24/03/04 20.04.59 2.60 0.14 β y β x 1/ 2 ) x (m) 1 / 2 1 / 2 D x • FODO lattice with Q x ≈ Q y 2.47 0.13 1/ 2 (m D 2.34 0.12 • Number of cells N = 100 2.21 0.11 β 2.08 0.10 • Circumference C = 400 m 1.95 0.09 1.82 0.08 • Space for super-conducting 1.69 0.07 1.56 0.06 RF cavity 1.43 0.05 • Accelerating voltage V = 1.30 0.04 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 13 . 5 MV s (m) E. Keil page 14

CASA Seminar at JLab April 29, 2004 Tunes q x and q y vs. δp/p Trbojevic triplet lattice Johnstone-Koscielniak lattice 0.4 0.5 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 deltap deltap Keil-Sessler FODO lattice Keil-Sessler doublet lattice 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 deltap deltap E. Keil page 15