collective effects in particle accelerators

1. collective effectsin particle accelerators Frank Zimmermann Bodrum Summer School September 2007

2. what is an accelerator? • charged particles moving in electromagnetic fields • these fields can be - static or time dependent - externally applied or beam generated - linear (restoring force proportional to displacement) or nonlinear • motion can be classical or quantum mechanical • combinations of all these occur and can be important examples

3. beam particles are like elephants… • they have good memory • they won’t forgive you • easily perturbed and mistakes add up

4. … and they are not alone! particles do not move independently; many of the limits of accelerator performance arise from interactions between beam particles = collective effects

5. various types of collective effects • beam instabilities (coherent, motion of many particles is correlated), due to • self-interaction [space charge] • interaction with vacuum chamber [impedance] • interaction with other beam [beam-beam effects] • interaction with “foreign” particles [ions or electrons] • poor beam lifetime or emittance growth (incoherent), due to • scattering of individual particles off each other [intrabeam scattering, Touschek effect] • motion of individual particles in the nonlinear electromagnetic field generated by one of the four interactions above [space charge, beam-beam, incoherent electron cloud,…]

6. cartoon - in reality transverse electric (and magnetic) fields increase for relativstic particles space charge 2 particles at rest or traveling many charged particles traveling in an unbunched beam with circular cross-section K.-H. Schindl

7. magnet space charge K.-H. Schindl space-charge force is defocusing in both x & y direction, unlike a quadrupole

8. consider as example: uniform round continuous charged beam of radius a & current I space-charge force:

9. interlude: betatron motion schematic of betatron oscillation around storage ring betatron tune = number of oscillations per turn

10. y ring effective focusing force x space charge defocusing force space-charge effect on betatron motion the global force on each particle should be focusing! in storage rings this is always true. find a “jungle” of coherent and incoherent effects tunes are dependent on transverse (and longitudinal) position through the global Coulomb-force effect of the beam G. Franchetti

11. change of betatron tune due to defocusing force: from space charge force • space-charge tune shift • - proportional to intensity N • - inversely proportional to emittance e • proportional to beam brightness N/eN • it decreases like 1/g2 or 1/g3 • (important for low g only) • it does not depend on machine radius 100% (norm.) beam emittance for our example

12. interlude: emittance e x’ “area in phase space” occupied by the beam = p x e x rms emittance for Gaussian distribution erms ~ 39%, 4erms ~ 86%, 6erms ~95% of the beam

13. incoherent tune shift due to conducting walls K.H. Schindl J.R. Laslett a line charge l representing the particle beam between parallel conducting plates of distance 2h; electric field parallel to the conducting plates have to be zero; this is achieved by introducing negative image line charges (right)

14. incoherent tune shift due to conducting walls - 2 K.H. Schindl J.R. Laslett

15. incoherent tune shift due to conducting walls - 3 K.H. Schindl direct s.c. image • features: • electric image field is vertically defocusing but horizontally focusing • (typical for most vacuum beam pipes) • field is larger for smaller chamber height h • image effects decrease as 1/g, much weaker than 1/g3 for direct space charge; they are of some concern for high-energy p machines and e rings d.c. beam current is accompanied by dc magnetic field, which is not shielded by beam pipe, but influenced by ferromagnetic boundaries, like magnets, represented by mirror currents → incoherent tune shift due to magnetic images

16. incoherent and coherent tune shift K.H. Schindl coherent motion of the whole beam after having received a transverse kick the source of the direct space charge is now moving, individual particles still continue incoherent motion around the common coherent trajectory incoherent betatron motion of a particle inside a static beam with its center of mass at rest amplitude and phase are distributed at random over all particles

17. coherent tune shift due to conducting wall example: round, perfectly conducting beam pipe coherent oscillation of the beam of the beam inside a circular perfectly conducting beam pipe and its oscilllating image charge defocusing force

18. coherent tune shift due to conducting wall - 2 coherent force coherent tune shift for symmetry reasons the force is the same in x and y direction • features: • force is linear in → there is a coherent tune shift • 1/g dependence stems from the fact that the field is proportional to the number of charged beam particles (independent of mass), but their deflection is inversely proportional to their relativistic mass m0g • the coherent tune shift is never positive • effect of a thin vacuum chamber with finite coductivity are more subtle Features:

19. more realistic example: elliptical vacuum chamber, still unbunched beam with uniform density incoherent and coherent tune shifts given by “Laslett coefficients” e and x

20. most coefficients are larger vertically; coherent coefficients x all positive or 0

21. most rings store bunched beams; the s.c. tune shift for bunched beams changes with longitudinal coordinate z → tune spread K.H. Schindl

22. space-charge tune spread in a storage ring, beam makes many turns (e.g. PS booster ~106 turns) particles with small deviations from the design orbit oscillate around this orbit in phase space integer tunes, ½ integer tunes etc. must be avoided since they lead to resonances and beam loss (particles will “sum up” all machine / magnet imperfection resonances turn by turn) the space charge reduces the tune, and also leads to a tune spread DQ in the beam (for a real non-uniform and bunched beam particles at large transverse and longitudinal amplitudes will see less tune shift) once DQ becomes too big there will always be some particles on resonance and these will be lost this is the major problem at low energy hadron accelerators particle M. Benedikt

23. Example for space-charge limited synchrotron: betatron tune diagram and areas covered by direct tune spread at injection, intermediate energy, and extraction, for the CERN Proton Synchrotron Booster. During acceleration, acceleration gets weaker and the “necktie” area shrinks, enabling the external machine tunes to move the “necktie” to a region clear of betatron resonances (up to 4th order) K.H. Schindl

24. nonlinear dynamics and space charge the problem is complex detuning resonance condition beam size growth particle amplitude growth particle loss G. Franchetti

25. nonlinear dynamics in a bunch x z bare tune resonance periodic crossing of a resonance G. Franchetti

26. trapping into resonances during synchrotron motion x z bare tune resonance periodic crossing of a resonance G. Franchetti

27. trapping into a resonanceduring synchrotron motion 1 synchrotron oscillation in 6000 turns G. Franchetti

28. Bare tune Bare tune Several particles remain on one side of the resonance increasing their amplitude single passage through a resonance role of transverse detuning when the stop-band is crossed Qy -0.15 +0.15 Qx = 0.1 Qx 3rd order resonance 3 Qx = 13

29. Full beam emittance Test particle N = 5 x103 turns Trapping 3rd order resonance Qy 0.03 Bare tune Qx Qx = 0.15 N = 103 turns “Scattering” particle trapping into a resonanceduring accumulation (at injection energy) Space charge increased in N turns G. Franchetti

30. If during 1 revolution around the fixed point the island moves less than its size than the particle can remain trapped Tune on the Fixed point Speed of the fixed point T << 1 characterizes the adiabatic regime A.W. Chao and Month NIM 121, 129 (1974). A. Schoch, CERN Report, CERN 57-23, (1958) A.I. Neishtadt, Sov. J. Plasma Phys. 12, 568 (1986) Size of the island adiabatic / non-adiabatic regimes condition for a particle to remain trapped G. Franchetti

31. how to overcome the ‘space-charge limit’? 1. raise the injection energy! FNAL booster K.H. Schindl CERN PS, BNL AGS

32. how to overcome the ‘space-charge limit’? 2. flatten the bunch distribution! maximum line density rms emittance transverse form factor Ftrans=1 for Gaussian Ftrans=1/2 for transversely uniform distribution “phase-space painting”, double harmonic rf,…

33. how to measure the incoherent tune shift/spread? K.H. Schindl

34. collective effects in particle accelerators • various types of collective effects; space charge • 2) wake fields, impedances, beam instabilities, Landau damping • 3) beam-beam effects • 4) ions and electron cloud effects

35. wake fields • the real vacuum chamber (beam pipe) is not a perfectly conducting pipe of constant aperture • a beam passing an obstacle radiates electromagnetic fields and excites the normal modes of the object • consequences: • beam loses energy • energy can be transferred from head to tail of a bunch • the head of the bunch can deflect the tail • energy and deflections can be transferred between bunches if the high Q (quality factor) normal modes • the wake fields characterize (“are”?) the beam induced energy losses and deflections Instabilities! }

36. calculation by T. Weiland

37. wake-field properties Particle of charge Q (=1) followed at a distance bct0 by q. Let Q travel on axis (RT=0) and let Ez be the longitudinal electric field at q. Longitudinal wake potential is

38. longitudinal wake field in the ultrarelativistic limit (b→1), Vd has simple properties: Vd t0 also, Vd is independent of rT

39. to get V(t0) for a beam, convolute Vd with the bunch shape

40. now, let Q travel off axis a on-axis wake potential modes that have Ez=0 on axis are excited

41. Vd’(t0) must have same qualitative behavior as Vd(t0) → deflections are possible transverse wake field from Maxwell”s equations:

42. longitudinal-transverse wake relation Panofsky- Wenzel theorem transverse wake is defocusing

43. emittance growth in linacs & linear colliders • 1st example of impact of wake fields • advantage of 2-particle model for getting insight single particle injected on axis travels down linac if injected off-axis, quadrupoles surrounding linac → oscillation about axis • cartoon • scales are • not correct!

44. 2nd particle follows 1st, wake from 1st deflects 2nd deflected outward, amplitude grows amplitude of second particle (bunch tail) grows, effective emittance growth multi-particle simulation by Karl Bane, large growth in effective emittance

45. possible solutions: • BNS damping (Balakin, Novokhatsky, Smirnov) • reduce wake fields BNS damping – analogy classical driven oscillator natural frequency response wdrive wdrive=whead, wnat=wtail

46. multi-particle simulation by Karl Bane, SLC with BNS damping

47. instabilities in circular accelerators • stick with 2-particle model • head produces wake that acts on the tail of charge q/2 each • head and tail interchange due to synchrotron oscillations WAKE FIELD TAIL HEAD 1/2 Ts later Dg/g Dg/g TAIL TAIL t t HEAD HEAD

48. example : “fast head-tail” instability • simplified transverse wake field W W t=0 t SHM for 0 < t < Ts/2: head tail driven harmonic oscillator

49. For Ts/2 < t < Ts: head tail look for solutions: • when 0<t<Ts/2: • Y1 is constant • when Ts/2<t<Ts: • Y2 is constant

50. a common technique for assessing stability of this and similar systems is to write as a matrix for N synchrotron periods: the motion is stable if all elements of MN remain bounded as N→infinity look at eigenvalues of M. They areexp(+/-i m) where cos m = ½ Trace(M) =1/2( 2-(WqTs/(8 wb))2)