Accelerator Physics Coupling Control. G. A. Krafft Old Dominion University Jefferson Lab Lecture 7

Transcription

1 Aeerator Phyi Couping Contro G. A. Krafft Od Dominion Univerity Jefferon Lab Leture 7 Graduate Aeerator Phyi Fa 07

2 Caia Mirotron: Veker (945) 6 xtration Magneti Fied y RF Cavity x Graduate Aeerator Phyi Fa 07

3 Synhrotron Phae Stabiity dwin MMian diovered phae tabiity independenty of Veker and ued the idea to deign firt arge eetron ynhrotron. V (t) h / frf f RF t t / f RF h Lf / RF Harmoni number: # of RF oiation in a revoution Graduate Aeerator Phyi Fa 07

4 Tranition nergy Beam energy where peed inrement effet baane path ength hange effet on aeerator revoution frequeny. Revoution frequeny independent of beam energy to inear order. We wi auate in a few week Beow Tranition nergy: Partie arriving ARLY get e aeeration and peed inrement, and arrive ater, with repet to the enter of the bunh, on the next pa. Appie to heavy partie ynhrotron during firt part of aeeration when the beam i non-reativiti and aeeration ti produe veoity hange. Above Tranition nergy: Partie arriving ARLY get more energy, have a onger path, and arrive ater on the next pa. Appie for eetron ynhrotron and heavy partie ynhrotron when approah reativiti veoitie. A een before, Mirotron operate here. Graduate Aeerator Phyi Fa 07

5 Graduate Aeerator Phyi Fa 07 Phae Stabiity Condition Synhronou eetron ha Phae o ev o Differene equation for differene after paing through avity pa + : Beaue for an eetron paing the avity before after ev o o M ev 0 in 0 56

6 Graduate Aeerator Phyi Fa 07 Phae Stabiity Condition ) / ( 56 0 o / D M d d ev ev in 4 in 4 / i i K,,0 o / 0 x p i i i D

7 Phae Stabiity Condition Have Phae Stabiity if Tr M ev in ev frfev frf in o tan tan fm f i.e., 0 tan Graduate Aeerator Phyi Fa 07

8 Phae Stabiity Condition Have Phae Stabiity if Tr M i.e., 0 tan Graduate Aeerator Phyi Fa 07

9 Synhrotron Two bai generaization needed Aeeration of non-reativiti partie Differene equation deribing per turn dynami beome a differentia equation with oution invoving a new frequeny, the ynhrotron frequeny RF Cavity Graduate Aeerator Phyi Fa 07

10 Aeeration of non-reativiti partie For mirotron, raetrak mirotron and other poytron, eetron peed i at the peed of ight. For non-reativiti partie the reiruation time ao depend on the ongitudina veoity v z = β z. t L / reir L p L t p p z p z t M p M p p 56 z 56 treir L p z L p p z Graduate Aeerator Phyi Fa 07

11 Momentum Compation t p M56 t p L reir 56 L / L / p / p M / L p t p t pp z reir z Tranition nergy: nergy at whih the hange in the one around time beome independent of momentum (energy) M 56 0 t L No Phae Fouing at thi energy! Graduate Aeerator Phyi Fa 07

12 quation for Synhrotron Oiation L 0 z ev in 0 L z L ev in ev in z Aume momentum owy hanging (adiabati aeeration) Phae advane per turn i L L o ev in in ev z z Graduate Aeerator Phyi Fa 07

13 So hange in phae per unit time i L T T p 0 0 z ev in yieding ynhrotron oiation with frequeny h ev rev in p where the harmoni number h = L / β z λ, give the integer number of RF oiation in one turn Graduate Aeerator Phyi Fa 07

14 Phae Stabe Aeeration At energie beow tranition, η > 0. To ahieve aeeration with phae tabiity need 0 h ev rev p in At energie above tranition, η < 0, whih orrepond to the ae we re ued to from eetron. To ahieve aeeration with phae tabiity need 0 h ev rev in p Graduate Aeerator Phyi Fa 07

15 Large Ampitude ffet Can no onger inearize the energy error equation. L z ev o o d dt T p 0 d ev o o dt T T d ev o o dt pt Graduate Aeerator Phyi Fa 07

16 Contant of Motion (Longitudina Hamitonian ) d d d ev o o dt dt pt0 dt d ev in o C dt pt 0 H, T T ev in o 0 0 pt0 Graduate Aeerator Phyi Fa 07

17 quation of Motion If neget the ow (adiabati) variation of p and T 0 with time, the equation of motion approximatey Hamitonian d 0 dt H T 0 d T dt H In partiuar, the Hamitonian i a ontant of the motion Kineti nergy Term Potentia nergy Term V ev T p T 0 in o Graduate Aeerator Phyi Fa 07

18 No Aeeration / V ev o d in dt Better known a the rea penduum. Graduate Aeerator Phyi Fa 07

19 Separatrie and Buket Graduate Aeerator Phyi Fa 07 USPAS, Fort Coin, CO, June 0-, 03

20 With Aeeration d o o dt in d in o C dt in quation for eparatrix yied fih diagram in phae pae. Fixed point at o o 0, Graduate Aeerator Phyi Fa 07

21 Separatrie and Buket Graduate Aeerator Phyi Fa 07 USPAS, Fort Coin, CO, June 0-, 03