FYS4565 Physics and applications of accelerators and beams

1. FYS4565Physics and applications of accelerators and beams Lecture 4: Transverse dynamics II Erik Adli Department of Physics, University of Oslo, Norway Erik.Adli@fys.uio.no February 2019 v1.1

2. Transverse dynamicsCourant-Snyder framework Previous lecture: transfer matrices for single particle motion These lectures : generalized solution, analysis of a beam of particles

3. Particle motion: Hill's equation • We have calculated particle motion for a single particle by solving Hill’s equation piece-wise and multiplying transfer matrices M(s). We now seek a general solution of the the equation : • Reminder: solution of Hill’s equation with K(s) =K  harmonic oscillator

4. Reformulation of Hill's equation: beta function We will define a function varying along the lattice in the solution to the Hill’s equation; the beta function, b(s). The solution is a quasi-harmonic oscillator, with amplitude and phase-advance dependent on s. The variable amplitude given by solving the betatron equation, including initial conditions (b(0), b'(0)) Derivation: Wille 3.7.

5. Transverse phase-space • The transverse phase-spacein the horizontal plane is spanned by xand x‘: • One can eliminate the phase, f(s). Equation for an ellipsewith area pe : Defined : a(s)  -(1/2)b'(s), g(s)  (1+ a2(s))/ b(s), b(s), a(s), g(s): “Twiss” parameters Also called the Courant-Snyder parameters (US). e : single-particle emittance b(0) b0, a(0)  -(1/2)b’0 Derivation: Wille 3.8.

6. The phase-space ellipse Envelope of particle motion :

7. Description of a beam

8. (Background note: Multi-variate Gaussian distributions) The beam sigma matrix

9. Gaussian beams

10. Evolution of the beam sigma matrix • We may describe second order moments using the Twiss parameters. Before to describe the motion of a single particle. • For Gaussian beams, and linear optics, the Twiss parameters uniquely define the beam distribution along the lattice. Individual particles have differences phases, and different singe-particle emittance, but the same Twiss parameters

11. Evolution of a beam Rms beam size: Beam quality Evolves with lattice

12. Evolution of the Twiss parameters

13. Propagation sigma matrixusing M(s) • Evolution of single particles along an accelerator lattice using transfer matrices: x1= M10x0 • We can calculate the evolution of the sigma matrix using transfer matrices. Using we get xTS-1x= 1, and I.e. the beam Twiss parameters may be found by propagating S using M(s) Wille 3.10.

14. Twiss parameters: evolution For a non-circular lattice: need initial Twiss parameters(the initial beam), b(0) and a(0)=-1/2b’(0). One aims to match the incoming beam to design (lattice) Twiss. Example to the right: evolution of Twiss parameters where the initial magnets are adjusted to match the beam into a periodic focusing lattice of FODO-type. Example of beta function evolution (from the CLIC Test Facility at CERN) F D

15. M parameterized using Twiss • We may also express the transfer matrix between two locations in terms of the Twiss parameters at the locations. • Especially interesting when studying periodic lattices. • Using • and • we can derive :

16. Periodic lattices

17. Periodic lattices • Consider a periodicaccelerator lattice, withMss+L(s) from point s to point s+L. For example, one full turn of a circular accelerator may be the period. • The Twiss parameters must obey the following condition : • The Twiss parameters are now uniquely defined byMss+L(s) : Circular accelerators: periodic lattice by default. Wille 3.14. Existence of periodic solutions follows from Floquet Theory.

18. Twiss parameters: period lattice Start with the general parameterization : For a periodic lattice, with period L, this reduces to : where F = y(s2) – y(s1)is the phase-advance from s1 to s2. The lattice Twiss parameters may then be calculated from the periodic transfer matrix : In a periodic lattice, including a circular accelerator, the Twiss parameters is given by the ring lattice design.

19. Example: periodic FODO f L (thin-lens approx used)

20. Betatron oscillations and phase advance Solution of Hill’s equation : Example solution in a periodic lattice : • A particle will undergo betatron oscillations around the reference trajectory. b(s) defines the envelope for the particle motion • f(s): phase-advance from point s0 to point s1 in a lattice • In a FODO structure the beta function is at maximum in the middle of the F quadrupole and at minimum in the middle of the D quadrupole About 8 FODO cells. Phase-advance per FODO cell offcell 68º.

21. Periodic lattice; tune • Beta function is periodic • Particle motion is in general notperiodic • After one revolution the initial phase f0 is altered, since the particle has advanced by a certain phase,fturn. • Phase advance per turn may be given as the number of periodic cells, and the phase-advance per cell, fturn = Ncell fcell • . • Tune, Q: number of betatron oscillations per turn : • Q = Ncell fcell/ 2p (From M. Sands)

22. Tune diagram • In order to avoid driving resonant instabilities, the fractional part of the tune, Q, must not be mQx + nQy = N, where m,n,N may be 0,1,2,3….(number depending on the performance requirements). • These instabilities originate from imperfections of magnets. Beam undergoes small kicks, which if they add up coherently may blow up beam size. Integer and N-integer tunes must be avoided

23. Stability of a periodic lattice • Requirementfor the stable lattice structure : bounded motion • Let M(s) be the matrix for one periodic cell (or, for the whole acceleratorO). M(s) N corresponds to n turns in the accelerator/ • Stability requires that the elements of M(s) N remain bounded as N   • Applying this criterion on the FODO cell gives the condition : Liming case, L = 4 f

24. Emittance preservation • We have shown that the rms emittance is a preserved quantity for Gaussian beams, in drift and with linear magnetic lenses • More generally the phase-space area, emittance, is preserved if only conservative forces do work on the particles (Liouville’s theorem) • Particle acceleration by an RF-field is not conservative, the emittance will shrink if the beam is accelerated : • The emittance will shrink proportional to the beam energy increase, given by bg,. We define a normalized emittance, conserved under acceleration : eN,rms = gb erms Independent quantities in each plane, x, y. NB: b g:normalized velocity and the Lorentz factor. Not related to the Twiss parameters.

25. Summary: Transverse parameters • Betatron oscillations: particle oscillations in the transverse planes, due to the focusing, for example alternating gradient focusing • b(s):beta function  square of envelope of the particle motion • defined by the lattice • e : emittance, measurement of beam quality; conserved under conservative forces • Q = Ncell fcell / 2p : accelerator tune • Integer and half-integer tunes etc. will lead to instabilities and must be avoided • An alternative expression for luminosity:

26. Evolution of a beam Rms beam size: Beam quality Evolves with lattice

27. Off-energy motionDispersion and Chromaticity

28. Dispersion • So far, we have studied particles with reference momentum p = p0. • A dipole field (say By=const.) disperses particles according to their energy. • This introduces an x-E correlation in the beam. • Off-momentum : p=p0+Dp = p0(1+Dp/p0). Hill’s equation : • Solution gives a extra dispersion term to the homogenous solution xDp=0(s) D(s): “dispersion function” Wille Ch. 3.6

29. Dispersion suppression Rms beam size with dispersion : https://xkcd.com/964/ Todd Satogata srms(s) = √(ermsb(s) + (D(s) sp/p0)2 ) D = 0 D > 0 • Dispersion can be locally suppressed by lattice design. • Important when you want a very small beam size, for example at a collider interaction region.

30. Dispersion in linear lattices • Lattice elements may add or subtract to the dispersion in the beam (x-E correlation). Dispersion propagation through the lattice can be calculated : • Dispersion is affected by quadrupoles as well, and propagates along the lattice in the same manner as a particle. We may use the matrix framework for particle tracking to calculate dispersion. To add dispersion generated from dipoles and other elements to the framework, we add a third row : represents here a dipole "kick" D(s) > 0 D(s) = 0

31. Dispersion in rings • In rings, the design dispersion is uniquely defined by the lattice • In a constant bending field, the circumference would increase by Dp / p0 yielding a constant dispersion, D(s) = Dx(s) / (Dp/p0) = Dr/ (Dr/r) = r. • Quadrupole focusing modifies the dispersion function. We define the momentum compaction factor(cf. longitudinal dynamics) as : • The momentum compaction factor can be calculated by computer codes. The value is usually >0 << 1. “weak focusing” “strong focusing”

32. Chromaticity Linacs: • Particles with Dp ≠ 0 focuses differently in quadrupoles • Light optics analogy, “chromatic aberration” • Focal length: f = 1/kl, k a 1/ p, f a p=p0(1+ Dp/p0). • Detrimental effects on beam : • Focusing (beta function) depends on energy: “projected” emittance growth in lines • The accelerator tunes, Q, depends on energy; energy-spread -> tune-spread -> particles on resonances -> lost. Chromaticity must be compensated. Rings: Wille Ch. 3.16

33. Chromaticiy versus dispersion From D. Gamba Dispersion effect: linear Chromatic effect: non-linear

34. Chromaticity cancellation - Sextupoles

35. Magnet multipole expansion • We discussed earlier the normalized magnet strengths : Dipole Quadrupole • Generalize this concept to magnetic multipole components kn for a 2(n+1) - pole : • The kicks on a particle from a magnetic 2(n+1) - pole can be expressed as a combination of multipolecomponents : , with unit [m-(n+1)] • Can be derived by Laplace eq. for the B-field. • See Wille Ch. 3. • n=1: the quadrupole linear terms • n=2: sextupole terms : l: the effective length of the magnet along s knl : integrated magnet strength Wille Ch. 3

36. p = p0 Correction with sextupoles D > 0 quad sext P1 > p0 D > 0 Sextupole can be set in order to cancel chromaticity induced by the quadrupoles. d = Dp/p0

37. Non-linear terms • Sextupoles introduces higher order non-linear terms, f(x2, y2…). If not cancelled by –I transforms, these terms adds non-linear terms to the particle dynamics in the accelerator • Real magnets contain small amount of higher order multipole fields, which also adds to the non-linear terms • For circular accelerators, the orbit stability now becomes as non-linear problem • The part of the transverse phase-space which is stable can be studies by particle tracking for many, many turns. The resulting stable phase-space is called the dynamics aperture • The non-linear dynamics can also be studied analytically, using Hamiltonian dynamics Example of LHC dynamic aperture simulation study

38. Example: LHC lattice • We have studied the transverse optics of a circular accelerator and we have had a look at the optics elements, • the dipole for bending • the quadrupole for focusing • the sextupole for chromaticity correction • In LHC: also octupoles for controlling non-linear dynamics The periodic structure in the LHC arc section

39. Apochromatic chromaticity correction Can you also make apochromatic particle beam focusing, analogous to how achromatic camera lenses are made? The answer is yes, if the goal is to ensure energy-independence of the beam focusing (Twiss parameters). The answer is no, to ensure energy-independent phase-advance of the individual particles, and thus energy-independet tune. This means that sextupoles are required in rings to mitigate energy dependence of the tune. C. A. Lindstrøm and E. Adli, “Design of general apochromatic drift-quadrupole beam lines”, Phys. Rev. Accel. Beams 19, 071002 (2016)

40. Acknowledgements • Many derivations in these slides follow K. Wille (2000).

41. Exercises • Exercise 1 from exercise set 1 (terminology) • Exercise 3-4 from exercise set 1 (LHC parameters, LHC periodic FODO structure) • Exercise 1 from exercise set 2 (envelope equation and beta function)