Advanced beam manipula/ons

Transcription

1 Advanced beam manipula/ons Beam manipula+ons involves the interac+on of the beam with external fields: laser tailored RF field (e.g. mul+ frequency) external beams The beam manipula+ons topics explored so far were aimed at tailoring the beam s phase spaces The same type of manipula+ons can be used to cool the beam 1

2 What is cooling? Reduc+on of the phase space volume Liouville s theorem states that in a well- behaved system (linear, non interac+ng, ) this cannot happen When collec+ve effects and aberra+ons are taken into account the phase space actually dilutes No magnet can provide such a phase space reduc+on (cooling) effect 2

3 Cooling principle most cooling schemes are based on an exchange of energy with an external system We already saw one type of cooling: radia+on cooling due to synchrotron- radia+on emission (t) =Ae Et cos( t ) (t) =Be Et sin( t ) 3

4 Type of cooling Radia+on cooling: light par+cles (electrons) cool themselves by emizng radia+on. Can also be performed with an external laser Electron cooling for low/medium- energy protons, an+protons and ions* Stochas+c cooling for medium/high- energy protons, an+protons and ions Laser cooling for a few kinds of atomic ions Ioniza+on cooling, soon, for muon cooling 4

5 Electron cooling Invented by Budker (1966) Cool hot ions beam using a cold electron beam Relies on the rela+ve fric+on force between ions and electrons Consider the Rutherford cross sec+on Z: ion charge state, and ( ) = µ m im e m i + m e Z 2 e 4 1 4(4 ") 2 µ 2 u 4 sin 4 /2 ion at rest 5

6 Fric/on force The change in e- longitudinal velocity is u z = u(1 cos ) =2usin 2 /2 so that the average change is h u z i = Z can integrate [Maxima]: where the Coulomb log is =2 u z Z max ( )ud min u z ( )u sin d h u z i = 4 Ze L c log sin( max/2) sin( min /2) 2 L c µ 2 u 2 6

7 Fric/on force Can be generalized to the ion velocity as (invoking momentum conserva+on in the ion s frame) to yield h v i i = 4 Ze corresponding to a force : F m i h v i i = 4 2 Ze 2 L c m e m i v i v e v i v e Lc m e v i v e v i v e 3 which can be average over the e- distribu+on 7

8 Prac/cal implementa/on velocity of ion and electron should match for prolonged interac+on 8

9 Stochas/c cooling Beam informa+on is picked- up manipulated and fed back to the beam Liouville s theorem does not apply Simple model a difference pick- up measures the bary- center of a beam slice A kicker correct for the measured displacement from H. Danared, 2005 CERN accelerator school, Zeegse, Netherlands 9

10 Stochas/c cooling The beam is sampled with a +me resolu+on of T 1/! where! is the pick- up bandwidth The cooling rate is es+mated as 1 = 1! TN S N where N s = N T s is the number of T ' N T! sample slices taken large cooling decrement requires large bandwidth 10

11 Experimental implementa/on FNAL pioneered many aspects of stochas+c cooling 11

12 Recent advances coherent electron cooling Phys. Rev. Lei. 102, (2009) Vladimir N. Litvinenko and Yaroslav S. Derbenev E < E h E h E < E h Hadrons Modulator Dispersion section ( for hadrons) E > E h E h Kicker E > E h λ l 1 High gain FEL (for electrons) l 2 Electrons op+cal stochas+c cooling Phys. Rev. Lei. 75, 4146 (1993) A. Mikhailichenko and M. Zolotorev 12

13 ioniza/on cooling par+cle passes through maier all momentum component are reduced longitudinal momentum is restored via accelera+ng cavi+es limited to few par+cle type: scaiering increases emiiance and there is an equilibrium (the beam cannot be cooled indefinitely) 13

14 laser cooling mainly used to cool non- rela+vis+c atoms and trapped ions laser with frequency slightly below a strong transi+on is used Doppler effects is used to provide a velocity- dependent force via the Doppler shil =! 0! k.v transi+on frequency laser frequency p =(~!/c, ~k) 14

15 what is the ul/mate cooling limit? in principle the beam can be cooled to the quantum limit " ~ the beam is then described by its wave func+on (cannot take the classical approach anymore) in prac+ce because of repulsive forces in mono- species beams there is an equilibrium between cooling force and space charge forma+on of crystals 15

16 Crystalline beams condi+on for forma+on: kine+c energy of par+cle << elec. poten+al in such a case par+cle are trapped in the electrosta+c poten+al and get cooled in an ordered fashion kine+c energy of par+cle is represented by emiiance 16

17 from A. Chao, hip:// 1D Crystalline beams Consider a lazce arranged as a 1D string with par+cle located as where par+cle mass and charge are M and Q the equa+on of mo+on of the nth par+cle is n a Coulomb force contribu+on from lel (not missing) 4 0 Coulomb force contribu+on from right (not missing)

18 from A. Chao, hip:// 1D crystalline beam previous equa+on can be linearized to yield eigenmode of this system of equa+on are 18

19 Example of crystalline beams simula+on of laser cooling in rings 19

20 Experimental forma/on of Coulomb realized in the lab: in ion storage rings in atomic molasse Crystals mostly achieved for species that have an op+cal transi+on (easy to reach with laser) 20