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Professor Philip Burrows John Adams Institute for Accelerator Science Oxford University ACAS School for Accelerator Physics January 2014 Longitudinal Dynamics I 1
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Acknowledgements 2 Ted Wilson Ken Peach Emmanuel Tsesmelis
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Lecture 4 outline 3 Longitudinal motion Momentum compaction Transition Dispersion Radio frequency (RF) systems Longitudinal phase space Chromaticity
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Longitudinal motion (1) 4 So far assumed motion in accelerator with constant speed (momentum) in the direction of the orbit, s Neglected: differences in momentum between particles acceleration (change in speed/momentum) as particles traverse accelerator
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5 If particle momentum increases: Particles travel faster, until ultra-relativistic when v ≈ c In a fixed dipole bend field the orbit will get longer Longitudinal motion (2)
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6 How does revolution frequency, f, change with momentum? df / f=dv / v-dr / r frequencyspeedradius If p increases, the length of the orbit will change (for a given magnetic lattice) Can define dr / r = α p dp / p α p is the momentum compaction factor Change in radius of closed orbit for a change in momentum So df / f=dv / v-α p dp / p Longitudinal motion (3)
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7 So df / f=dv / v-α p dp / p Longitudinal motion (4)
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8 So df / f=dv / v-α p dp / p Recall β = v / c, so dv / v = dβ / β Longitudinal motion (4)
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9 So df / f=dv / v-α p dp / p Recall β = v / c, so dv / v = dβ / β p = γ m 0 v = γ m 0 β c= m 0 c β / √ (1 - β 2 ) Longitudinal motion (4)
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10 So df / f=dv / v-α p dp / p Recall β = v / c, so dv / v = dβ / β p = γ m 0 v = γ m 0 β c= m 0 c β / √ (1 - β 2 ) Exercise: show that dp / dβ = m 0 c γ 3 = γ 2 p / β Longitudinal motion (4)
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11 So df / f=dv / v-α p dp / p Recall β = v / c, so dv / v = dβ / β p = γ m 0 v = γ m 0 β c= m 0 c β / √ (1 - β 2 ) Exercise: show that dp / dβ = m 0 c γ 3 = γ 2 p / β dβ / β= 1 / γ 2 dp / p Longitudinal motion (4)
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12 So df / f=dv / v-α p dp / p Recall β = v / c, so dv / v = dβ / β p = γ m 0 v = γ m 0 β c= m 0 c β / √ (1 - β 2 ) Exercise: show that dp / dβ = m 0 c γ 3 = γ 2 p / β dβ / β= 1 / γ 2 dp / p df / f= ( 1 / γ 2 - α p ) dp / p Longitudinal motion (4)
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13 df / f= ( 1 / γ 2 - α p ) dp / p momentum lattice Transition
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14 df / f= ( 1 / γ 2 - α p ) dp / p momentum lattice For a particle in a constant magnetic bend field: Low p: β α p f ↑ as p ↑ Transition
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15 df / f= ( 1 / γ 2 - α p ) dp / p momentum lattice For a particle in a constant magnetic bend field: Low p: β α p f ↑ as p ↑ High p:β 1, γ >> 1: 1 / γ 2 < α p f ↓ as p ↑ Transition
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16 df / f= ( 1 / γ 2 - α p ) dp / p momentum lattice For a particle in a constant magnetic bend field: Low p: β α p f ↑ as p ↑ High p:β 1, γ >> 1: 1 / γ 2 < α p f ↓ as p ↑ Transition energy: 1 / γ 2 = α p Transition
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17 df / f= ( 1 / γ 2 - α p ) dp / p = η dp / p frequency slip factor Frequency slip factor
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18 df / f= ( 1 / γ 2 - α p ) dp / p = η dp / p frequency slip factor Below transition (low p): η > 0 Transition:η = 0 Above transition (high p):η < 0 Frequency slip factor
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19 df / f= ( 1 / γ 2 - α p ) dp / p = η dp / p frequency slip factor Below transition (low p): η > 0 Transition:η = 0 Above transition (high p):η < 0 Proton Synchrotron (CERN): transition energy ≈ 6 GeV Frequency slip factor
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20 df / f= ( 1 / γ 2 - α p ) dp / p = η dp / p frequency slip factor Below transition (low p): η > 0 Transition:η = 0 Above transition (high p):η < 0 Proton Synchrotron (CERN): transition energy ≈ 6 GeV Not applicable to electron accelerators – discuss! Frequency slip factor
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21 So far assumed beam has no energy/momentum spread: Δ E / E = 0 = Δ p / p Different momentum different radius of curvature r in main dipole magnets Particles pass through quadrupoles at different positions Quadrupoles bend differently, depending on position Closed orbits different, depend on momentum Horizontal displacement characterised by dispersion function D(s): depends on s Dispersion (1)
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22 Local radial displacement at position s caused by momentum spread: Δ x (s) = D (s) Δ p / p D (s) is a property of the magnetic lattice, units meters Non-zero Δ p / p non-zero Δ x Momentum spread causes finite beam size Normally no vertical dipoles: D = 0 in vertical plane Dispersion (2)
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23 Magnetic fields deflect, but do not accelerate, charged particles For acceleration need electric field Early days: V fixed Modern accelerators: use an oscillating voltage of ‘radio frequency’ (RF) Acceleration
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24 Consider single particle in oscillating RF longitudinal electric field produced by a RF cavity Set oscillation frequency = period of revolution of the particle RF electric field (1)
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25 Add a second particle of lower energy, which arrives later in time, with respect to the first particle, at the RF cavity On first turn: B arrives late w.r.t. A sees a higher voltage and gets accelerated w.r.t. A Second turn: still late, but not quite as late, accelerated again RF electric field (2)
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26 Turn 1
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27 Turn 100
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28 Turn 200
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29 Turn 400
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30 Turn 500
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31 Turn 600
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32 Turn 700
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33 Turn 800
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34 Turn 900
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Synchrotron oscillations 35 In this toy example, after 900 turns particle B has made one full oscillation around particle A Motion due to energy difference is ‘synchrotron oscillation’ Amplitude of oscillation depends on initial energy difference, i.e. initial ‘phase difference’
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36 Potential well (1) A B Cavity voltage Potential
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37 Potential well (2) A B Cavity voltage Potential
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38 Potential well (3) A B Cavity voltage Potential
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39 Potential well (4) A B Cavity voltage Potential
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40 Potential well (5) A B Cavity voltage Potential
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41 Potential well (6) A B Cavity voltage Potential
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42 Potential well (7) A B Cavity voltage Potential
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43 Potential well (8) A B Cavity voltage Potential
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44 Potential well (9) A B Cavity voltage Potential
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45 Potential well (10) Cavity voltage Potential A B
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46 Potential well (11) Cavity voltage Potential A B
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47 Potential well (12) Cavity voltage Potential A B
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48 Potential well (13) Cavity voltage Potential A B
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49 Potential well (14) Cavity voltage Potential A B
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50 Potential well (15) Cavity voltage Potential A B
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Longitudinal phase space 51 By analogy with transverse phase space (x, x’) and (y, y’) define longitudinal phase space in terms of conjugate variables E and t, or the relative variables ΔE and Δt:
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Motion in phase space (1) 52 Define zero to be at phase space position of particle A Particle B oscillates around particle A:
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Motion in phase space (2) 53 Define zero to be at phase space position of particle A Particle B oscillates around particle A:
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Motion in phase space (3) 54 Define zero to be at phase space position of particle A Particle B oscillates around particle A:
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Motion in phase space (4) 55 Define zero to be at phase space position of particle A Particle B oscillates around particle A:
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Motion in phase space (5) 56 Define zero to be at phase space position of particle A Particle B oscillates around particle A: synchrotron oscillations
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Chromaticity 57 Recall normalised focussing strength of quadrupoles: k=1/Bρ dB y /dx
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Chromaticity 58 Recall normalised focussing strength of quadrupoles: k=1/Bρ dB y /dx andB ρ~p
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Chromaticity 59 Recall normalised focussing strength of quadrupoles: k=1/Bρ dB y /dx andB ρ~p hence Δk / k=- Δp / p
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Chromaticity 60 Recall normalised focussing strength of quadrupoles: k=1/Bρ dB y /dx andB ρ~p hence Δk / k=- Δp / p but tune shift ΔQ / Q~ Δk / k
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Chromaticity 61 Recall normalised focussing strength of quadrupoles: k=1/Bρ dB y /dx andB ρ~p hence Δk / k=- Δp / p but tune shift ΔQ / Q~ Δk / k so ΔQ / Q= ξ Δp / p
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Chromaticity 62 Recall normalised focussing strength of quadrupoles: k=1/Bρ dB y /dx andB ρ~p hence Δk / k=- Δp / p but tune shift ΔQ / Q~ Δk / k so ΔQ / Q= ξ Δp / p ξ is chromaticity
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Interpretation of chromaticity 63 Chromaticity relates tune spread of transverse motion to the momentum spread of the beam: ΔQ / Q= ξ Δp / p
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Calculation of chromaticity 64 RecallΔQ = 1/4π β Δk ds and Δk / k = - Δp/p HenceΔQ / Q = - 1/4π β k/Q ds Δp/p comparing ΔQ / Q = ξ Δp/p ξ = - 1/4π β k/Q ds To correct the tune spread ΔQ need to: increase k for higher momentum particles decrease k for lower momentum particles
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This is physically easy to see 65 Chromaticity relates tune spread of transverse motion to the momentum spread of the beam: ΔQ / Q= ξ Δp / p
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Sextupole magnets 66 Used to compensate for chromatic aberrations in strongly- focussing magnetic structures Particle motion in horizontal plane is coupled to vertical plane Field along x-axis:
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Sextupole magnets: examples 67 LEP sextupole: electromagnet, 1m long, few 100 kg LHC correction sextupole: SC magnet, 11cm long, 10 kg 500A @ 2K 1630 T/m 2
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Summary 68 RF accelerating system forms a ‘potential well’ within which particles oscillate: synchrotron oscillation Describe motion using longitudinal phase space: energy vs. time (or phase) Works for particles below ‘transition’ Oscillation is non-linear motion Chromaticity and sextupole corrections
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