Lecture 8 Beams and Imperfections Professor Emmanuel Tsesmelis
Lecture 8 Beams and Imperfections Professor Emmanuel Tsesmelis Visiting Professor, University of Oxford Directorate-General Unit, CERN Graduate Accelerator Physics Course John Adams Institute for Accelerator Science 5 November 2015
Contents – Lecture 8 Resonant Conditions üClosed-orbit Distortion üGradient Errors üChromaticity Correction ü
Resonant Conditions ü There are many things in machine that will excite resonances: ü The magnets themselves ü Unwanted higher-order field components in magnets ü Tilted magnets ü Experiment solenoids (LHC experiments) ü The trick is to reduce and compensate these effects as much as possible and then find some point in the tune diagramme where the beam is stable. 3
What is a Resonance? ü After a certain number of turns around the machine the phase advance of the betatron oscillation is such that the oscillation repeats itself. ü For example: If the phase advance per turn is 120º then the betatron oscillation will repeat itself after 3 turns. ü This could correspond to Q = 3. 333 or 3 Q = 10 ü But also Q = 2. 333 or 3 Q = 7 ü ü The order of a resonance is defined as ‘n’ n x Q = integer 4
Q = 3. 333 in more detail 1 st turn 2 nd turn 3 rd turn Third order resonant betatron oscillation 3 Q = 10, Q = 3. 333, q = 0. 333 5
Q = 3. 333 in Phase Space ü Third order resonance on a normalised phase space plot 2 nd turn 3 rd turn 1 st turn 6 2πq = 2π/3
Machine Imperfections ü It is not possible to construct a perfect machine. Magnets can have imperfections ü The alignment in the machine has non-zero tolerance. ü etc… ü ü So, have to ask: What will happen to the betatron oscillation due to the different field errors. ü And therefore consider errors in dipoles, quadrupoles, sextupoles, etc… ü ü Have a look at the beam behaviour as a function of ‘Q’. ü How is it influenced by these resonant conditions? 7
Dipole (deflection independent of position) y’b Q = 2. 00 1 st turn y’b Q = 2. 50 2 nd turn y y 3 rd turn ü For Q = 2. 00: Oscillation induced by the dipole kick grows on each turn and the particle is lost (1 st order resonance Q = 2). ü For Q = 2. 50: Oscillation is cancelled out every second turn, and therefore the particle motion is stable. 8
Quadrupole (deflection position) Q = 2. 50 1 st turn Q = 2. 33 2 nd turn 3 rd turn 4 th turn ü For Q = 2. 50: Oscillation induced by the quadrupole kick grows on each turn and the particle is lost (2 nd order resonance 2 Q = 5) ü For Q = 2. 33: Oscillation is cancelled out every third turn, and therefore the particle motion is stable. 9
Sextupole (deflection position 2) Q = 2. 33 1 st turn Q = 2. 25 2 nd turn 3 rd turn 4 th turn 5 th turn ü For Q = 2. 33: Oscillation induced by the sextupole kick grows on each turn and the particle is lost (3 rd order resonance 3 Q = 7) ü For Q = 2. 25: Oscillation is cancelled out every fourth turn, and therefore the particle motion is stable. 10
More Rigorous Approach (1) ü Let us try to a mathematical expression for amplitude growth in the case with a quadrupole: y’ 2πQ = phase angle over 1 turn = θ Δβy’ = kick a = old amplitude Δa = change in amplitude 2πΔQ = change in phase θ a y does not change at the kick y θ y’ 2π Q a So we have: y = a cos(θ) In a quadrupole Δy’ = lky Only if 2πΔQ is small Δa = βΔy’ sin(θ) = lβ sin(θ) a k cos(θ) 11
More Rigorous Approach (2) ü So have: a = l· ·sin( ) a·k·cos( ) ü Each turn θ advances by 2πQ ü On the nth turn θ = θ + 2 nπQ Sin(θ)Cos(θ) = 1/2 Sin (2θ) a 0 ü Over many turns: This term will be ‘zero’ as it decomposes in Sin and Cos terms and will give a series of + and – that cancel out in all cases where the fractional tune q ≠ 0. 5 ü For q = 0. 5 the phase term, 2(θ + 2 nπQ) is constant: and thus: 12
More Rigorous Approach (3) ü In this case the amplitude will grow continuously until the particle is lost. ü Therefore, conclude as before that: quadrupoles excite 2 nd order resonances for q=0. 5 ü Thus, for Q = 0. 5, 1. 5, 2. 5, 3. 5, …etc…… 13
More Rigorous Approach (4) ü Study phase θ: y’ 2πQ = phase angle over 1 turn = θ Δβy’ = kick a = old amplitude Δa = change in amplitude 2πΔQ = change in phase θ a y does not change at the kick y s 2π Q θ y = a cos(θ) In a quadrupole Δy’ = lky y’ a 2πΔQ << Therefore Sin(2πΔQ) ≈ 2πΔQ 14
More Rigorous Approach (5) ü So have: ü Since: can rewrite this as: , which is correct for the 1 st turn ü Each turn θ advances by 2πQ ü On the nth turn θ = θ + 2 nπQ ü Over many turns: ‘zero’ ü Averaging over many turns: 15
Stopband ü , which is the expression for the change in Q due to a quadrupole… (fortunately !!!) ü But note that Q changes slightly on each turn Related to Q Max variation 0 to 2 ü Q has a range of values varying by: ü This width is called the stopband of the resonance ü So even if q is not exactly 0. 5, it must not be too close, or at some point it will find itself at exactly 0. 5 and ‘lock on’ to the resonant condition. 16
Sextupole Kick ü Can apply the same arguments for a sextupole: ü For a sextupole and thus ü Get : ü Summing over many turns gives: 1 st order resonance term 3 rd order resonance term ü Sextupole excite 1 st and 3 rd order resonance q=0 17 q = 0. 33
Octupole Kick ü Can apply the same arguments for an octupole: ü For an octupole and thus ü We get : 4 th order resonance term 2 nd order resonance term ü Summing over many turns gives: a 2(cos 4( +2 pn. Q) + cos 2( +2 pn. Q)) Amplitude squared q = 0. 5 q = 0. 25 ü Octupolar errors excite 2 nd and 4 th order resonance and are very important for larger amplitude particles. 18 Can restrict dynamic aperture
Intermediate Summary ü Quadrupoles excite 2 nd order resonances ü Sextupoles excite 1 st and 3 rd order resonances ü Octupoles excite 2 nd and 4 th order resonances ü This is true for small amplitude particles and low strength excitations. ü However, for stronger excitations sextupoles will excite higher order resonances (non-linear). 19
Coupling ü Coupling is a phenomena that converts betatron motion in one plane (horizontal or vertical) into motion in the other plane. ü Fields that will excite coupling are: 20 ü Skew quadrupoles, which are normal quadrupoles, but tilted by 45º about their longitudinal axis. ü Solenoidal (longitudinal magnetic field).
Skew Quadrupole Magnetic field S N N Like a normal quadrupole, but tilted by 45º S 21
Solenoid; Longitudinal Field (1) Particle trajectory Magnetic field Beam axis Transverse velocity component 22
Solenoid; Longitudinal Field (2) Above: The LPI solenoid that was used for the initial focusing of the positrons. It was pulsed with a current of 6 k. A for some 7 s, it produced a longitudinal magnetic field of 1. 5 T. At right: the somewhat bigger CMS solenoid 23
Coupling and Resonance ü This coupling means that one can transfer oscillation energy from one transverse plane to the other. ü Exactly as for linear resonances (single particle) there are resonant conditions. n. Qh m. Qv = integer ü If meet one of these conditions, the transverse oscillation amplitude will again grow in an uncontrolled way. 24
General Tune Diagramme Qv Qh - Qv = 0 2. 75 2. 5 Qh + Q v = 5 2. 25 2 2 2. 25 2. 33 25 2. 75 Qh 2. 66
Resonant Conditions n n Change in tune or phase advance resulting from errors. q Steer Q away from certain fractional values which can cause motion to resonate and result in beam loss. Resonance takes over and walks proton out of the beam for: where is resonance order and p is azimuthal frequency that drives it. SPS Working Diagramme
P. S. Booster Tune Diagramme injection During acceleration change the horizontal and vertical tune to a place where the beam is least influenced by resonances. ejection 27
Closed-orbit Distortion n As current is slowly raised in dipole: q The zero-amplitude betatron particle follows distorted orbit. q Distorted orbit is closed. q Particle still obeys Hill’s Equation. q Except at the kink (dipole) it follows a betatron oscillation. q Other particles with finite amplitudes oscillate about this new closed orbit.
Betatron Motion Re-visited Locus at F quadrupoles
Circle Diagramme
Closed Orbit in Circle Diagramme n n Tracing a closed orbit for one turn in the circle diagramme with a single kick. The path is ABCD
Sources of Closed-orbit Distortion
Uncorrelated Errors n n Historic measurement from FNAL Main Ring Each bar is position at quadrupole. +/- 100 is width of vacuum chamber. Q = 19. 25
Gradient Errors I n Quadrupoles also have errors. q n Understanding effects of gradient errors leads to study of non-linear errors. Consider ring of magnets as a circle with a small gradient error afflicting a quadrupole.
Gradient Errors II n n n Very useful result. Surprising that change is independent of the phase of the perturbation. To a first approximation, it is sufficiently accurate to explain resonant phenomena.
Physics of Chromaticity
Measurement of Chromaticity n n Steer beam to different mean radius and different momentum by changing RF frequency. Measure Q
Chromaticity Correction (1) Final “corrected” By By By = Kq. x (Quadrupole) x (Sextupole) By = Ks. x 2 ü Vertical magnetic field versus horizontal displacement in a quadrupole and a sextupole. 38
Chromaticity Correction (2) ü The effect of the sextupole field is to increase the magnetic field of the quadrupoles for the positive ‘x’ particles and decrease the field for the negative ‘x’ particles. ü However, the dispersion function, D(s), describes how the radial position of the particles change with momentum. ü Therefore, the sextupoles will alter the focusing field seen by the particles as a function of their momentum. ü Can use sextupoles to compensate the natural chromaticity of the machine. 39
Sextupole & Chromaticity (1) ü In a sextupole for y = 0 have a field By = C. x 2 ü Now calculate ‘k’, the focusing gradient as for a quadrupole: ü ‘k’ is no longer constant, it depends on ‘x’ Displacement due to dp/p 40
Sextupole & Chromaticity (2) ds = length of the sextupole Remember B = C. x 2 ü This term shows the effect of a sextupole, of length l, on the tune, Q, of a particle as a function of its Δp/p. ü If can make this term exactly balance the natural chromaticity, then will have solved our problem. 41
Sextupole & Chromaticity (3) ü There are two chromaticities ξh, ξv ü However, the effect of a sextupole depends on β(s) and this varies around the machine. ü Two types of sextupoles are used to correct the chromaticity. ü One (SF) is placed near QF quadrupoles where βh is large and βv is small, this will have a large effect on ξh ü Another (SD) placed near QD quadrupoles, where βv is large and βh is small, will correct ξv ü Also sextupoles should be placed where D(s) is large, in order to increase their effect, since Δk is proportional to D(s). 42
Bibliography n n The Physics of Particle Accelerators – An Introduction, Klaus Wille, OUP, 2000 An Introduction to Particle Accelerators, Edmund Wilson, OUP, 2001 Engines of Discovery, Andrew Sessler & Edmund Wilson, World Scientific, 2006 Rende Steerenberg, AXEL Accelerator Course, CERN
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