Georg.otaetter@Corell.eu USPAS Avace Accelerator Phic - ue 6 CESS & EPP CESS & EPP 56 Setupole caue oliear aic which ca be chaotic a utable. l M co i i co l i i co co i i co l l l l ta ta α l ta co i i co co ta l Noliear Motio
Georg.otaetter@Corell.eu USPAS Avace Accelerator Phic - ue 6 CESS & EPP CESS & EPP 57 co ta co i i ta i co l co i co ] [ M Tr The aitioal ie poit i utable! l ta co i i co -.5.5.5.75.5 -. -....55 The Daic Aperture
58 ξ ξ η l Setupole Aperture I the choraticit i correcte b a igle etupole: ta i η co l ξ co Ote the aic aperture i uch aller tha the ie poit iicate! - - -6-8 -. CESS & EPP N Whe a etupole are ue: ξ η l The u o all i the reuce to about The aic aperture i thereore greatl icreae whe itribute etupole are ue. -.5.5 5 7.5 l N l N η ξ Georg.otaetter@Corell.eu USPAS Avace Accelerator Phic - ue 6
59 Setupole Etractio CESS & EPP - - - - - -6-8. - -5. - -6 -.5.5 5 7.5 5 Due to the arrow regio o utable trajectorie etupole are ue or low particle etractio at a tue o /. The iterectio o table a utable aiol i a certai iicatio o chao. Georg.otaetter@Corell.eu USPAS Avace Accelerator Phic - ue 6
6 oocliic Poit CESS & EPP At itable ie poit there i a table a a itabile ivariat curve. Iterectio o thee curve hoocliic poit lea to chao. Georg.otaetter@Corell.eu USPAS Avace Accelerator Phic - ue 6
6 perio: rotatio arou the vertical t a α t r t i CESS & EPP t O the path ro Rotatio to ibratio arou the Spi-Orbit-Couplig i a trog chaotic regio. Georg.otaetter@Corell.eu USPAS Avace Accelerator Phic - ue 6
6 Tilt o the earth CESS & EPP Tial orce ro oo a u caue a tabilizatio o the rotatio ai. Solar raiatio Icliatio -Ma with a without oo M a Georg.otaetter@Corell.eu USPAS Avace Accelerator Phic - ue 6
Georg.otaetter@Corell.eu USPAS Avace Accelerator Phic - ue 6 CESS & EPP CESS & EPP 6 S a a K a α co i S S α Thi woul be a olutio with cotat a whe. Variatio o cotat: S S α i co with Perturbatio
Georg.otaetter@Corell.eu USPAS Avace Accelerator Phic - ue 6 CESS & EPP CESS & EPP 6 co i α a α co i a a co i α Correpo to Floquet Theore Sipliicatio o liear otio
65 Quai-perioic Perturbatio CESS & EPP co i co i New iepeet variable co i i The perturbatio are perioic i a i i approiatel For irratioal ν the perturbatio are quai-perioic. Georg.otaetter@Corell.eu USPAS Avace Accelerator Phic - ue 6
Georg.otaetter@Corell.eu USPAS Avace Accelerator Phic - ue 6 CESS & EPP CESS & EPP 66 -.5.5.5.75.5 -. -... i co The otio reai ailtoia i the perturbe cooriate! I there i a part i that oe ot epe o Tue hit The eect o other ter te to average out..55 Tue Shit with Aplitue
Georg.otaetter@Corell.eu USPAS Avace Accelerator Phic - ue 6 CESS & EPP CESS & EPP 67 Quarupole: Setupole: Octupole: i i!! e e i i 6!!!! i Tue Shit Eaple
68 co Noliear Reoace The eect o the perturbatio i epeciall trog whe co or i ha cotributio that harl chage i.e. the chage o i i i reoace with the rotatio agle. CESS & EPP Perioicit allow Fourier epaio: i[ ] e co Ψ Tue hit Georg.otaetter@Corell.eu USPAS Avace Accelerator Phic - ue 6
69 The Sigle Reoace Moel CESS & EPP i Ψ co Ψ Strog eviatio ro: Occur whe there i coherece betwee the perturbatio a the phae pace rotatio: Reoace coitio: tue i ratioal O reoace the itegral woul icreae ieiitel! Neglectig all but the ot iportat ter co Ψ Georg.otaetter@Corell.eu USPAS Avace Accelerator Phic - ue 6
7 Fie poit i Ψ CESS & EPP [ co Ψ ] Φ [ Ψ ] δ i Φ Φ δ co Φ δ co Φ Fie poit: i Φ Φ I δ ± ha a olutio. ± Φ Φ [ ± ] Stable ie poit or: [ ± ] < Georg.otaetter@Corell.eu USPAS Avace Accelerator Phic - ue 6
Georg.otaetter@Corell.eu USPAS Avace Accelerator Phic - ue 6 CESS & EPP CESS & EPP 7 Setupole: ] i ] [i[ i!!! co δ Sipliicatio: oe etupole co! or or 5 > Φ ± Φ δ δ A co Φ A All thee ie poit are itable ice > Thir Iteger Reoace
7 Octupole: Fourth Iteger Reoace!!8 CESS & EPP i!! [co[ ] co ] Sipliicatio: oe octupole Φ δ A A [ coφ] ± δ co or Either 8 ie poit: or oe or: δ < δ > [ ± ] < A [ ± ] < Stabilit or i.e. or the outer ie poit. Georg.otaetter@Corell.eu USPAS Avace Accelerator Phic - ue 6
7 Fie poit: Reoace With Stregth i Φ Φ CESS & EPP I δ ± ha a olutio. δ ha to avoi the regio δ ± ± δ or all particle. Sprea i the bea Auig that the tue hit a perturbatio are ootoou i : Thi tue regio ha the with a or trog reoace. I calle Reoace With Reoace Stregth or Stop-Ba With Georg.otaetter@Corell.eu USPAS Avace Accelerator Phic - ue 6
Georg.otaetter@Corell.eu USPAS Avace Accelerator Phic - ue 6 CESS & EPP CESS & EPP 7 i co i co co Ψ The itegral or ca be choe ice it i path iepeet. Thi i ue to the ailtoia ature o the orce: Sigle Reoace oel or two ieio ea retaiig ol the aplitue epeet tue hit a oe ter i the two ieioal Fourier epaio: For Couplig Reoace
Georg.otaetter@Corell.eu USPAS Avace Accelerator Phic - ue 6 CESS & EPP CESS & EPP 75 K ea that ocillatio i ca rive ocillatio i i co Ψ The reoace ter i the ailtoia the chage ol lowl: Dierece reoace lea to table otio ice: Su reoace lea to utable otio ice:. cot. cot Su a Dierece Reoace
76 Reoace Diagra K CESS & EPP ea that ocillatio i ca rive ocillatio i i ν All thee reoace have to be avoie b their repective reoace with. The poitio o a accelerator i the tue plae calle it Worig Poit. ν Georg.otaetter@Corell.eu USPAS Avace Accelerator Phic - ue 6