Accelerators Part 2 of 3: Lattice, Longitudinal Motion, Limitations Rende Steerenberg BE-OP CERN - Geneva Rende Steerenberg BND Graduate School 2 6 September 2017 CERN - Geneva
Topics A Brief Recap and Transverse Optics • Longitudinal Motion • Main Diagnostics Tools • Possible Limitations • Rende Steerenberg BND Graduate School 3 6 September 2017 CERN - Geneva
A brief recap and then we continue on transverse optics Rende Steerenberg BND Graduate School 4 6 September 2017 CERN - Geneva
Magnetic Element & Rigidity Dipole magnets Quadrupole magnets LB q = ( ) r B 𝑙 = 𝐿 𝐶𝜍 𝑛 :+ ⃑ × 𝐶 = 𝑛𝑤 + 𝐶𝜍 Tm = 𝑛𝑤 𝑟 = 𝑞 GeV c ⁄ 𝐶𝜍 = 3.3356 𝑞 𝐺 = 𝑟𝑤 𝑟 𝜍 Increasing the energy requires increasing the magnetic • field with B 𝜍 to maintain radius and same focusing The magnets are arranged in cell, such as a FODO lattice • Rende Steerenberg BND Graduate School 5 6 September 2017 CERN - Geneva
� � � Hill’s Equation Hill’s equation describes the horizontal and vertical betatron oscillations • d x 2 + = K ( s ) x 0 ds 2 Position: Angle: 𝑦 G = −𝛽 𝜁 𝛾 𝜁 𝛾 cos 𝜒 − sin (𝜒)𝜒 𝑦 𝑡 = 𝜁𝛾 ? cos (𝜒 𝑡 + 𝜒) J J • 𝜁 and 𝜒 are constants determined by the initial conditions 𝛾 (s) is the periodic envelope function given by the lattice configuration • +U = 1 𝑒𝑡 𝑅 N O 2𝜌 S ⁄ 𝛾 N O ⁄ (𝑡) V Q x and Q y are the horizontal and vertical tunes : the number of oscillations • per turn around the machine Rende Steerenberg BND Graduate School 6 6 September 2017 CERN - Geneva
Betatron Oscillations & Envelope The 𝜸 function is the envelope function within which all particles oscillate • The shape of the 𝜸 function is determined by the lattice • Rende Steerenberg BND Graduate School 7 6 September 2017 CERN - Geneva
FODO Lattice & Phase Space Q D Q F • Calculating a single FODO Lattice and repeat it several time • Make adaptations where you have insertion devices e.g. experiment, x ’ x ’ x ’ x ’ injection, extraction etc. x x ’ • Horizontal and vertical phase space e . g e / b • Q h = 3.5 means 3.5 horizontal − α ε / γ − α ε / β betatron oscillations per turn around the machine, hence 3.5 turns on the x phase space ellipse • Each particle, depending on it’s initial e / g conditions will turn on it’s own ellipse e . b in phase space Rende Steerenberg BND Graduate School 8 6 September 2017 CERN - Geneva
Let’s continue…. Rende Steerenberg BND Graduate School 9 6 September 2017 CERN - Geneva
Momentum Compaction Factor The change in orbit length for particles with different • momentum than the average momentum This is expressed as the momentum compaction factor, 𝛃 p , • where: ∆𝑠 ∆𝑞 𝑠 = 𝛽 [ 𝑞 𝛃 p expresses the change in the radius of the closed • orbit for a particle as a a function of the its momentum Rende Steerenberg BND Graduate School 10 6 September 2017 CERN - Geneva
Dispersion A particle beam has a momentum spread that in a homogenous dipole field will translate • in a beam position spread at the exit of a magnet ∆𝑦 𝑦 = 𝐸(𝑡) ∆𝑞 𝑞 ∆𝑞 LB ∆𝑦 q = ( ) 𝑞 r B 𝑦 The beam will have a finite horizontal size due to it ’ s momentum spread, unless we • install and dispersion suppressor to create dispersion free regions e.g. for experiments Rende Steerenberg BND Graduate School 11 6 September 2017 CERN - Geneva
Chromaticity • The chromaticity relates the tune spread of the transverse motion with the momentum spread in the beam. ∆𝑅 ] ^ ⁄ ∆𝑞 ⁄ = 𝜊 ] ^ 𝑅 ] ^ 𝑞 ⁄ A particle with a higher momentum p > p 0 as the central momentum will be deviated less in the quadrupole and p 0 will have a lower betatron tune A particle with a lower momentum p < p 0 as the central momentum will be deviated more in the quadrupole QF and will have a higher betatron tune Rende Steerenberg BND Graduate School 12 6 September 2017 CERN - Geneva
Q1: How to Measure Chromaticity Looking at the formula for Chromaticity, could • you think about how to measure the actual chromaticity in you accelerator ? ∆𝑅 ] ^ ⁄ ∆𝑞 ⁄ = 𝜊 ] ^ 𝑅 ] ^ 𝑞 ⁄ What beam parameter would you change ? • Any idea how ? • What beam parameter would you observe ? • Any idea how ? • Rende Steerenberg BND Graduate School 13 6 September 2017 CERN - Geneva
Q1: How to Measure Chromaticity Looking at the formula for Chromaticity, could you think about how to • measure the actual chromaticity in you accelerator ? ∆𝑅 ] ^ ⁄ ∆𝑞 ⁄ = 𝜊 ] ^ 𝑅 ] ^ 𝑞 ⁄ What beam parameter would you change ? • Change the average momentum of the beam and you beam will move • coherently as a single particle with a different momentum Any idea how ? • Add an offset to the RF system to slightly increase the beam momentum at a • constant magnetic field What beam parameter would you observe ? • You would need to observe the change in beam tune for a change in • beam momentum Any idea how ? • Measuring the beam position over many turns and make an FFT that will show • the change in frequency Rende Steerenberg BND Graduate School 14 6 September 2017 CERN - Geneva
Chromaticity Correction (Quadrupole) B y = K q x B y Final “ corrected ” B y x (Sextupole) B y = K s x 2 4𝜌 𝑚𝛾(𝑡) 𝑒 + 𝐶 O ∆𝑅 𝑅 = 1 𝐸(𝑡) ∆𝑞 𝑒𝑦 + 𝐶𝜍 𝑅 𝑞 Chromaticity Control through sextupoles Rende Steerenberg BND Graduate School 15 6 September 2017 CERN - Geneva
Longitudinal Motion Rende Steerenberg BND Graduate School 16 6 September 2017 CERN - Geneva
Motion in the Longitudinal Plane What happens when particle momentum increases in a constant • magnetic field? Travel faster (initially) • Follow a longer orbit • Hence a momentum change influence on the revolution frequency • 𝑒𝑔 𝑔 = 𝑒𝑤 𝑤 − 𝑒𝑠 𝑠 ∆𝑠 ∆𝑞 𝑠 = 𝛽 [ From the momentum compaction factor we have: • 𝑞 𝑒𝑔 𝑔 = 𝑒𝑤 𝑒𝑞 𝑤 − 𝛽 [ Therefore: • 𝑞 Rende Steerenberg BND Graduate School 17 6 September 2017 CERN - Geneva
Revolution Frequency - Momentum 𝑒𝑔 𝑔 = 𝑒𝑤 𝑒𝑞 𝑒𝑤 𝑤 = 𝑒𝛾 𝛾 ⟺ 𝛾 = 𝑤 𝑤 − 𝛽 [ 𝑑 𝑞 𝑞 = 𝐹 V 𝛾𝛿 From the relativity theory: 𝑑 𝑒𝑤 𝑤 = 𝑒𝛾 𝛾 = 1 𝑒𝑞 We can get: 𝛿 + 𝑞 Resulting in : 𝑒𝑔 𝑔 = 1 𝑒𝑞 𝛿 + − 𝛽 [ 𝑞 Rende Steerenberg BND Graduate School 18 6 September 2017 CERN - Geneva
Transition 𝑒𝑔 𝑔 = 1 𝑒𝑞 𝛿 + − 𝛽 [ 𝑞 1 > a Low momentum ( 𝛾 << 1 & 𝛿 is small) à • p g 2 1 High momentum ( 𝛾 ≈ 1 & 𝛿 >> 1) à < a • p g 2 1 = a Transition momentum à • p g 2 Rende Steerenberg BND Graduate School 19 6 September 2017 CERN - Geneva
Frequency Slip Factor 𝑒𝑔 𝑔 = 1 𝑒𝑞 𝑞 = 1 𝛿 + − 1 𝑒𝑞 𝑞 = 𝜃 𝑒𝑞 𝛿 + − 𝛽 [ + 𝑞 𝛿 gh 1 𝛿 + > 𝛽 [ ⟹ 𝜃 > 0 • Below transition: 1 • Transition: 𝛿 + = 𝛽 [ ⟹ 𝜃 = 0 1 𝛿 + < 𝛽 [ ⟹ 𝜃 < 0 • Above transition: • Transition is very important in hadron machines • CERN PS: 𝛿 tr is at ~ 6 GeV/c (injecting at 2.12 GeV/c à below) • LHC : 𝛿 tr is at ~ 55 GeV/c (injecting at 450 GeV/c à above) • Transition does not exist in lepton machines, why …..? Rende Steerenberg BND Graduate School 20 6 September 2017 CERN - Geneva
RF Cavities Variable frequency cavity (CERN – PS) Super conducting fixed frequency cavity (LHC) Rende Steerenberg BND Graduate School 21 6 September 2017 CERN - Geneva
RF Cavity • Charged particles are accelerated by a longitudinal electric field • The electric field needs to alternate with the revolution frequency Rende Steerenberg BND Graduate School 22 6 September 2017 CERN - Geneva
Low Momentum Particle Motion Lets see what a low energy particle does with • this oscillating voltage in the cavity V V time 1 st revolution period 2 nd revolution period Lets see what a low energy particle does with • this oscillating voltage in the cavity Rende Steerenberg BND Graduate School 23 6 September 2017 CERN - Geneva
Longitudinal Motion Below Transition V A time B 1 st revolution period Rende Steerenberg BND Graduate School 24 6 September 2017 CERN - Geneva
….after many turns… V A time B 100 st revolution period Rende Steerenberg BND Graduate School 25 6 September 2017 CERN - Geneva
….after many turns… V A time B 200 st revolution period Rende Steerenberg BND Graduate School 26 6 September 2017 CERN - Geneva
….after many turns… V A time B 400 st revolution period Rende Steerenberg BND Graduate School 27 6 September 2017 CERN - Geneva
….after many turns… V A time B 500 st revolution period Rende Steerenberg BND Graduate School 28 6 September 2017 CERN - Geneva
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