Kwang-Je Kim University of Chicago and Argonne National Laboratory. MUTAC Review Lawrence Berkeley National Laboratory October 18-19, 2001

Transcription

1 Kwang-Je Kim Universit of Chicago and Argonne National Laborator MUTAC Review Lawrence Berkele National Laborator October 8-9,

2 ANL & UofC Effort on Ionization Cooling Theor KJK & CXW apers: - Formulas for transverse ionization cooling in SFC RL 85(4) 7, - Recursive solution for beam dnamics stud of ICC RE 64(5), (KJK & CXW) - Beam parameterization and invariants in a period SC (CXW & KJK) (CXW & KJK) - Magnetic field epansion for particle tracking in a bent-solenoidel channel (CXW & Lee C. Teng) - Linear theor of transverse & longitudinal IC in quadrupole channel - Linear theor of 6- ionization cooling Snowmass roceedings to be published (CXW & KJK)

3 Ionization Cooling Theor in Linear Approimation Similar in principle to radiation damping in electron storage rings, but needs to take into account: - Solenoidal focusing and angular momentum - Emittance echange Mess unless guided b fundamental principles! - Orthonormal basis of moments for Hamiltonian sstem, - issipation and fluctuation leading to an equilibrium - Slow evolution near equilibrium can be described b five Hamiltonian invariants

4 Emittance Echange ipole (bend) δ p µbeam -δ p ipole introduces dispersion (η) >o η δp/p Wedge Absorber reduces energ spread

5 Lab frame Hamiltonian Under Consideration Solenoid dipole quadrupole RF absorber Goal: theoretical framework and possible solution solenoid dipole quadrupole r.f. rotating frame with smmetric focusing,

6 Model for Ionization rocess in Larmor Transverse: Longitudinal: d ds Frame η( κe ) z ξ M.S. wedge η dδ η η η ξ δ ds straggling : Average loss replenished b RF η ( u, v), η

7 Equations for ispersion Functions In Larmor frame ispersion function decouples the betatron motion and dispersive effect

8 Equation for 6- hase Space Variables β δ, β δ z z β - ispersion vanishes at cavities rop suffi β ) ( V V ( ) I(s) z ) v (u v) (u V(s)z ( ) \ ( ) [

9 hase space vector: Moment matri Equation for Moments d ds X ( K A) X K JH (Hamiltonian motion) A issipation Fluctuation Rij ij (6 6) dr ds (K A) R, ( T T ) A R K X z ecitation Assume the non-hamiltonian effects are small.

10 Moment Evolution Near Equilibrium (F. Ruggiero, E. icasso, & L.A. Radicati, Ann. of hsics 97, 396 (99)) Sstem approaches an equilibrium under damping & ecitation Near the equilibrium, the moment matri R changes slowl (weak dissipation) R can be epanded b a complete set of orthonormal basis matrices: R σ α are constructed from tensor products of the eigenvectors of the Hamiltonian motion. Most of σ α var rapidl i( ± q ) (s l) ε α (s) σ α (s) e ( l lattice period, µ p phrase advance per period (p,, z) Coefficients ε α of the rapidl varing σ α are small and can be neglected The sstem near equilibrium can be represented b a special set of matrices, with coefficients which are invariants of the Hamiltonian motion p ± (s)

11 Hamiltonian Invariants Three Courant-Snder invariants: z z z z z ( ) (γ, α, β), (γ z, α z, β z ); Twist parameters for and z, Two more invariants when µ µ : L ( )

12 Hamiltonian Invariants (Contd.) Inverse relationship leads to moment parametrization:,, ( ), ( z),, ( ) L

13 Evolution of Invariants Under Non-Hamiltonian Force ε i, i, 5, var slowl due to dissipation and ecitation Equation for ε i is obtained b inserting the C-S parameterization into the moment equation The calculation is manageable since ε α, α > 5, are neglected in view of the RR analsis!

14 IC and Emittance Echange Equation in Linear Approimation dε ds 5 i G ij j ε j χ i G ij : L z A B η A B η A A η L B B η L z η z A η ηl η ( u v ) ( u v ), B κβη [( α β ) u ( α β ) v] η η - u, η η - u, η z u v,

15 The Ecitations χ β χ χ δ H χ β χ χ δ H ( ) [ ] χ δ χ β γ χ z z z χ δ χ H ( ) δ χ χ L ( ), β α γ H ( ) β α γ H

16 Remarks The result generalizes the previous analsis 6- phase spare area ε 6 d ds ε ε z ε ε ε ε L ( η ) η ηz ε6 η ε6 6 Choose u v η η η η L η Averaged over one period, χ L esign eamples are being worked out

17 Transverse Cooling Theor case

18 esign rinciples Update good transverse cooling channels Maintain periodic structure, esp. beta function Create localized dispersion in desired periods closed dispersion bump Maimum dispersion at minimum beta Keep smmetric focusing No dispersion in RF ispersion section to be st order achromat

19 A Mawell-Happ Solution Ma. dipole field ~.7 T, dispersion ~ cm b h h kfoc ksol cos q sin q kr at, 5, 5 MeV

20 A Mawell-Happ Solution b h h kfoc ksol cos q sin q kr