Accelerator School Transverse Beam Dynamics-3. V. S. Pandit

Transcription

1 Accelerator School 8 Tranvere Bea Dnaic-3 V S Pandit

2 Cloed or olution Equation o otion K i a or o Hill equation arge accelerator and tranport line are equipped with an dipole or delection and quadrupole or ocuing with drit pace in between agnet and drit pace are placed in identical ection o length called the lattice or cell to or the accelerator The ocuing unction K becoe periodic : K K k k K K K ρ p

3 Cloed or olution K loquet Theore: tate that there eit two unction periodic and in ter o which the general olution o can be epreed The general olution i co[ δ ] Aw co[ δ ] and δ are contant o integration relecting the initial condition and are not independent i called the achine phae hit

4 ] co[ δ co in in co 4 in co 3 δ δ δ δ δ δ Cloed or olution K and to be independent o δ coeicient o ine and coine ter ut vanih individuall K

5 Cloed or olution The phae unction d co[ δ ] Knowing one can calculate the phae unction can be obtained b dierential equation 4 K Beta unction depend onl on K Writing w, we have 4 w Envelope w K w K 3 w equation w w w C-S invariant

6 Bea dnaic uing beta unction Rewrite the olution a A co B in A B co A B in et u deine a new paraeter We have A B co[ δ ] Initial condition at,,,, /

7 Bea dnaic uing beta unction We have the tranoration atri or to a in co co in in in co The periodicit and Traner atri or one period J in I co in co in in in co I J, and are reerred to a Courant-Snder paraeter

8 Bea dnaic uing beta unction or N equal period in one revolution N N I co J in I co N I co J in J in N The eleent o the above atri reain bounded i i real ie - co - Tr or cople or iaginar, con and inn increae eponentiall and the eleent o N do the ae and otion becoe unbounded N i the phae hit per revolution o betatron ocillation The betatron tune Q i given b Q N π N π d

9 Coputation o CS paraeter The ot general olution o the Hill equation i a peudo-haronic ocillation d the phae change o ocillation ro to The traner atri or the repeat period i K co δ co in in in co in or tabilit o otion - Tr, and all are periodic with

10 Coputation o CS paraeter, To copute the Courant-Snder paraeter irt we ultipl all the individual atrice o variou eleent o the repeat period to get a reultant atri, co in in in co in B coparing the two atrice co in in in R In Tranport R R R R

11 Beta unction A regular ODO lattice o ocuing and deocuing lene Beta unction Coine like trajector or Sine like trajector or

12 achine Ellipe The olution o betatron ocillation i co δ co δ in δ B eliinating -δ ter and uing contant i a contant o otion and i called Courant-Snder invariant At an point in the achine the invariant or o equation decribe an ellipe

13 achine Ellipe Area π π Bea Envelope a Bea Divergence a or, the bea envelope ha a local iniu wait or a local aiu and the ellipe will be upright

14 Tranoration o Ellipe Phae ellipe at an point i characterized b the paraeter,, and the eittance Since i an invariant quantit, the phae ellipe at a reerence point and at an other point, have ae area π, but the dier in hape and orientation Y Y Y Y

15 Tranoration o Ellipe

16 Tranoration o Ellipe In atri or achine Ellipe T Y Y T Y T Y Y T T T

17 ODO attice ot coon building block ued in the deign o achine lattice and bea line Cae : when the odule begin and end at the centre o the len ± Upper ign, plane ower ign, plane

18 ODO attice ± in co in in in co / in, co Coparing two atrice in / in, ± / / in / in in a

19 Cae : when the odule begin ro the iddle o the drit ODO attice ± ± c c 3 4 / in, co / in in, in / in, Sae propert a poeed b thin len QP

20 ODO attice: λ /4 tranorer when π ± ± / / Here - and both change ign in and plane Ueul cell or phae atching

21 Parallel to point Iaging or parallel to Point iaging hould not depend on ie in co in co in in in co Phae hit π/ or tan Siilar or point to parallel iaging

22 3 -D Telecope Ste Siultaneou point to point and parallel to parallel tranoration ie Wait to wait tranoration in co in in in co / / Phae hit π a To achieve telecope in both plane at leat two quadrupole needed to iulate each len agniication a be dierent in each plane / agniication

23 Thank