Transverse Beam Dynamics or how to keep all particles inside beam chamber

1. Transverse Beam Dynamicsor how to keep all particles inside beam chamber Piotr Skowronski In large majority based on slides of B.Holzerhttps://indico.cern.ch/event/173359/contribution/9/material/0/0.pdf F.Chautardhttp://cas.web.cern.ch/cas/Holland/PDF-lectures/Chautard/Chautard-final.pdf W.Herr http://zwe.home.cern.ch/zwe/ O.Brüning http://bruening.web.cern.ch/bruening/ Y.Papaphilippouhttp://yannis.web.cern.ch/yannis/teaching/

2. Betatron & Cyclotron • Particles move in magnetic field

3. Betatron & Cyclotron • Geometric focusing • All particles are confined inuniform magnetic field • The beam size depends onthe initial spread • What about focusing in vertical plane ?

4. Vertical focusing in betatron & cyclotronThe weak focusing • For vertical stability value Bx must grow in vertical direction to provide restoring force • That is deflect particles back to the center plane • But this implies decreasing field and defocusing in horizontal plane because • This defocusing can not be stronger from geometric focusing

5. Equation of motion in uniform magnetic field • We choose coordinate system around the reference orbit • Restoring force linear dependence in vicinity of ref. trajectory • For convenience define magnetic field index • Rate that field changes in space normalized to Bz0 • ;

6. Vertical focusing in cyclotron and betatronThe weak focusing • Both coefficients n and (1-n) must be positive so both equations are of harmonic oscillator: 0 < n < 1

7. Betatron motion • Harmonic oscillator in both planes with • revolution frequency • Changing the gradientincreases oscillation frequency in one plane and decreases in another • Perfect isochronism is not possible since field changes radially

8. Azimuthally Varying Filed Cyclotron • Cyclotron works only to limited energies since increasing particle mass breaks isochronism • To restore it magnetic field must increase radially • This contradicts vertical stability condition • Solution: sectors with different fields and gradients • Orbit is not a circle, trajectory is not perpendicular to sector edge  Not uniformity of the field on the edge gives vertical focusing(we talk about it later)

9. Further optimizations • Separated sectors • Spiralled sectors increase the edge crossing angle (and foc. strength) • RF cavities instead of Dee’s TRIUMF, Vancouver, Canada, during construction ~1972 520MeV protons 18m diameter Still in operation

10. Synchrocyclotron • Another solution is to modulate RF frequency • This limits big advantage of cyclotron that produces almost continues beam: cyclotron produces one bunch per RF cycle • But it is able to reach GeV range

11. Synchrotron • Constant radius machine • Magnetic field and RF frequency is modulated • For the really strong guys where v/c is close to 1 also RF freq. constant

12. Separate function elements • Bends • Combined function bends • Bend and quad together • Quadrupoles • Accelerating Cavities • Sextupoles, Skew Quadruples, Octupoles, Decapoles … • Kickers, dumpers, oscillation modulators, … • Measurement devices, collimators • A single Hamiltonian or force can not be defined for a whole accelerator

13. Quadrupoles • Focusing with quadrupoles • Element acting as a lens • Deflection is linearly proportional to position • Magnetic field increases linearly with position

14. Focusing with quadrupoles • A magnet that fulfills requirement has 4 poles with parabolic shape • Due to nature of magnetic field we can not have magnet that focuses in both planes • When it focuses in one it defocuses in the other one

15. Focusing with quadrupoles • Still, focusing and defocusing lenses can be adjusted to confine the particles within finite space in both planes

16. FODO cell • The easiest configuration is FODO Focus (F) – Drift (O) – Defocus (D) – Drift (O)

17. Coordinate SystemCanonical Variables • Use position along accelerator s as time like variable • x : deviation from reference trajectory • Same for y • Canonical momentum x’ is • Longitudinal position is z = β c dt • where dt deviation in time from reference trajectory • Longitudinal momentum is

18. Pseudo-harmonic SolutionHill’s equation • Equation of motion for a separate function elements can be written in form of a pseudo oscillator • The force k(s) is different in each magnet and depends on position along the accelerator • For convince k normalized to beam rigidity • Takes the beam energy out of equation • For bending magnets k = B/(p/q) = B/(Br) = 1/r • For quadrupoles k = g/(Br), where g is the field gradient • Useful equations to remember =

19. Solution of Hills Equation • We guess that solution is of the form • Recall: s is our time-like variable s=bct, f(s)=ω(s)·s • It is a harmonic oscillator with time varying amplitude and frequency • It is the same as • A weight on a spring that changes it strength in time • A ball rolling in a gutter that has different radius along it • The solution can be expressed in matrix form • Any solution can be represented of linear combination of • pure cosine-like solution (x’i=0) • and pure sine-like solution (xi=0)

20. Solution of Hills Equation • Insert cos like solution into Hills equation

21. How to find the solution? • OK, we know how the solution looks like, but how to get it for my particular machine • Or usually inverse: we want a given solution, how to distribute our elements • We start from simplest cases: find solutions for each element where k=const

22. Focusing quadrupole • k=const>0, it is harmonic oscillator with a1, a2 and f0 depend on initial condition, i.e. coordinates of the particle entering the quadrupole, so we rewrite Focusing quadrupole of length l transforms coordinates

23. Transfer matrix of focusing quadrupole

24. Defocusing quadrupole • Negative k gives exponents as solution • The same way as for focusing quadrupole rewrite it using hyperbolic functions

25. Quadrupole transfer matrix • Together for horizontal and vertical planesFocusing:

26. Quadrupole transfer matrix • Together for horizontal and vertical planesDefocusing

27. Drift • No force, a free particle motion

28. Accelerator Line • Accelerator is a sequence of elements • Each one has its transfer matrix • Transfer matrix of accelerator line is product of its elements transfer matrices M=MDrift3MQuadF2MDrift2MBendMDrift1MQuadD1 This matrix describes motion of a single particle (green line)not the envelope (red line)! QF1 QF2 QF3 QD1 QD2 QD3

29. Twiss parameters • The one turn map, the map that is obtained for a whole ring, is solution of the Hills equation • If we take one particle and follow it for several turns Phase space ellipse

30. Phase space ellipse • Every particle follows an ellipse of the same shape • “Radius” (properly called action) and starting point depend on particle initial conditions • Of course most of the particles are in the centre and less and less towards outside

31. Beam size σx • For a whole bunch it gives distribution is space • Distribution of particles usually close to Gaussian σy

32. Twiss parameters and emittance • Take the ellipse that corresponds to 1 σ • Define beam sizes as • Where Ɛ is area of the ellipse, “temperature” of the beam • What for? Because conservative forces do not change volume of the phase space (Liouville theorem) Ɛ=const • Beta function shows how big the beam is at a given point of accelerator • γfor momentum • α is tilt of ellipse and it is anti-proportional to derivative of 

33. Envelope • Every bunch of particles has some initial spread in position and angle • Each starts with some different initial condition • The function describing beam size along accelerator line is called beam envelope

34. Hills equation • Rewrite solution of the Hills equation (or one turn map) for a ring using these 3 new parameters • What is μ? (often also referred as Q)Total phase advance for one turn. • Since multiple of 2π are not importantwe care only about fractional part:tune LHC: total phase advance (at collisions)64.31 and 59.32 (hor. and vert.)LHC tunes: 0.312π and 0.322π Ring special case: Periodic solution! μ Tune is a constant of a machine

35. Meaning of Twiss parameters • β(s) describes beam size at given point of accelerator • γ(s) describes spread (size) in momentum • det[M]=1  • As smaller beam size as bigger momentum spread • Periodicity implies • Alpha is proportional to derivative of the beta function • That is divergence of the beam size (with minus sign) • Putting above 2 relations the general solution

36. Equation of the phase space ellipse

37. Solution of Hills Equation • Insert cos like solution into Hills equation

38. Emittance • b function describes how envelope evolves along accelerator • The beam size , where is emittance • Emittance definition: phase space area occupied by the beam • You can think of it as the beam temperature • Emittance is constant (Liouville theorem) • It is conservation of energy • In transverse there is no energy dissipation or gain, no friction, no heating

39. Stability condition for a ring • The map obtained for a whole circular accelerator is called One Turn Map • For each turn that a particle makes around a ring its coordinates are modified according to the map zi+1= M zi • We want the beam to be stable => it means that coordinates of zn= Mnz0 must be finite when n going to infinity • Necessary and sufficient condition is that Mn is also finite • Consider eigenvectors Y and eigenvalues l of the one turn map MY=lI, (I is identity matrix) • For arbitrary M made of a,b,c,d, det(M-lI)=0  l2 + l(a+d) + (ad-bc)=0; (ad – bc)==det(M)=1 • Solution for l exist if (a+d)/2=tr(M)/2 < 1; l1l2=1 • What means that total phase advance must be real: tr(M)=2cos(μ)

40. Hills equation solution fora line (transfer line or linac)

41. Hills equation solution fora line (transfer line or linac)

42. Twiss parameter propagation • Emittance is constant • Coordinates transforms as • Insert to above and compare coefficients

43. Twiss parameter propagation • Propagation can be defined via transfer matrix elements • And in a matrix notation • Having matrix to propagate a single particle we obtain matrix to propagate the ensemble

44. Adiabatic damping • Phase space area is conserved only for a given energy • Is Louiville wrong? NO! • It is coordinate system that we use: we defined canonical momentum as • When particle accelerates • p0 increases • x’ decreases • emittance shrinks as 1/γrel • And the beam size also shrinks as • Normalized emittance that is conserved εN=ε*γrel • To avoid confusion ε is referred as geometric emittance

45. Example optics 3 8 7 6 5 2 1 4 • So how we design optics? • Every accelerator is different • Its optics is optimized for • a given job • to circumvent given problems • Example: LHC • In the arcs there is regular smooth optics: FODO cell • FODO is one of the easiest and permissive solutions, use it whenever the beam needs to be simply transported through and there are no other constraints • Prepare the beam for collisions: focus it as much as possible to maximize collision cross section (luminosity) • ATLAS(1) and CMS (5) need smallest beams, ALICE (2) and LHCb (8) less demanding • Other straight sections: accelerating cavities (4), ejection to dump (6), halo cleaning (7), off momentum cleaning (3)

46. FODO cell • The easiest configuration Focus (F) – Drift (O) – Defocus (D) – Drift (O) - …

47. FODO cell matrix

48. FODO cell matrix

49. FODO CELL parameters Min β at μ=~76° Stability diagram max,min

50. Reaching small beam sizes for collisions • Beta function along drift is parabolic • If we start from minimum point α0 = 0 and γ0=1/β0 • It is called “waist”