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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

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SLIDE 1
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SLIDE 2

Accelerators

Part 2 of 3: Lattice, Longitudinal Motion, Limitations

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 2

Rende Steerenberg BE-OP CERN - Geneva

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SLIDE 3

Topics

  • A Brief Recap and Transverse Optics
  • Longitudinal Motion
  • Main Diagnostics Tools
  • Possible Limitations

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 3

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SLIDE 4

A brief recap and then we continue

  • n transverse optics

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 4

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SLIDE 5

Magnetic Element & Rigidity

  • Increasing the energy requires increasing the magnetic

field with B𝜍 to maintain radius and same focusing

  • The magnets are arranged in cell, such as a FODO lattice

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 5

𝐺 = π‘Ÿπ‘€ ⃑ Γ— 𝐢 = 𝑛𝑀+ 𝜍 𝐢𝜍 Tm = 𝑛𝑀 π‘Ÿ = π‘ž GeV c ⁄ π‘Ÿ 𝐢𝜍 = 3.3356 π‘ž Dipole magnets Quadrupole magnets

( )

r q B LB =

𝑙 = 𝐿 𝐢𝜍 𝑛:+

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SLIDE 6

Hill’s Equation

  • Hill’s equation describes the horizontal and vertical betatron oscillations

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 6

) (

2 2

= + x s K ds x d

𝑦 𝑑 = πœπ›Ύ?

  • cos

(πœ’ 𝑑 + πœ’) 𝑦G = βˆ’π›½ 𝜁 𝛾 J

  • cos πœ’ βˆ’

𝜁 𝛾 J

  • sin

(πœ’)πœ’ Position: Angle:

  • 𝜁 and πœ’ are constants determined by the initial conditions
  • 𝛾(s) is the periodic envelope function given by the lattice configuration

𝑅N O

⁄

= 1 2𝜌 S 𝑒𝑑 𝛾N O

⁄ (𝑑) +U V

  • Qx and Qy are the horizontal and vertical tunes: the number of oscillations

per turn around the machine

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SLIDE 7

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 CERN - Geneva BND Graduate School 6 September 2017 7

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SLIDE 8

FODO Lattice & Phase Space

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 8

x’ x’ x x’ x’

QF QD

  • Calculating a single FODO Lattice

and repeat it several time

  • Make adaptations where you have

insertion devices e.g. experiment, injection, extraction etc.

x’ x b e / g e.

g e /

βˆ’Ξ± Ξ΅ / Ξ²

b e. βˆ’Ξ± Ξ΅/Ξ³

  • Horizontal and vertical phase space
  • Qh = 3.5 means 3.5 horizontal

betatron oscillations per turn around the machine, hence 3.5 turns on the phase space ellipse

  • Each particle, depending on it’s initial

conditions will turn on it’s own ellipse in phase space

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SLIDE 9

Let’s continue….

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 9

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SLIDE 10

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, 𝛃p,

where:

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 10

βˆ†π‘  𝑠 = 𝛽[ βˆ†π‘ž π‘ž

  • 𝛃p expresses the change in the radius of the closed
  • rbit for a particle as a a function of the its momentum
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SLIDE 11

Dispersion

  • 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 CERN - Geneva BND Graduate School 6 September 2017 11

βˆ†π‘ž π‘ž βˆ†π‘¦ 𝑦

( )

r q B LB =

βˆ†π‘¦ 𝑦 = 𝐸(𝑑) βˆ†π‘ž π‘ž

  • 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

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Chromaticity

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 12

  • The chromaticity relates the tune spread of the transverse motion

with the momentum spread in the beam.

p0

A particle with a higher momentum as the central momentum will be deviated less in the quadrupole and will have a lower betatron tune A particle with a lower momentum as the central momentum will be deviated more in the quadrupole and will have a higher betatron tune

p > p0 p < p0

QF

βˆ†π‘…] ^

⁄

𝑅] ^

⁄

= 𝜊] ^

⁄ βˆ†π‘ž

π‘ž

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SLIDE 13

Q1: How to Measure Chromaticity

  • Looking at the formula for Chromaticity, could

you think about how to measure the actual chromaticity in you accelerator ?

  • What beam parameter would you change ?
  • Any idea how ?
  • What beam parameter would you observe ?
  • Any idea how ?

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 13

βˆ†π‘…] ^

⁄

𝑅] ^

⁄

= 𝜊] ^

⁄ βˆ†π‘ž

π‘ž

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SLIDE 14

Q1: How to Measure Chromaticity

  • Looking at the formula for Chromaticity, could you think about how to

measure the actual chromaticity in you accelerator ?

  • 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 CERN - Geneva BND Graduate School 6 September 2017 14

βˆ†π‘…] ^

⁄

𝑅] ^

⁄

= 𝜊] ^

⁄ βˆ†π‘ž

π‘ž

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SLIDE 15

Chromaticity Correction

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 15

x By Final β€œcorrected” By By = Kqx (Quadrupole) By = Ksx2 (Sextupole)

βˆ†π‘… 𝑅 = 1 4𝜌 π‘šπ›Ύ(𝑑) 𝑒+𝐢O 𝑒𝑦+ 𝐸(𝑑) 𝐢𝜍 𝑅 βˆ†π‘ž π‘ž Chromaticity Control through sextupoles

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SLIDE 16

Longitudinal Motion

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 16

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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

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 17

𝑒𝑔 𝑔 = 𝑒𝑀 𝑀 βˆ’ 𝑒𝑠 𝑠 βˆ†π‘  𝑠 = 𝛽[ βˆ†π‘ž π‘ž 𝑒𝑔 𝑔 = 𝑒𝑀 𝑀 βˆ’ 𝛽[ π‘’π‘ž π‘ž

  • From the momentum compaction factor we have:
  • Therefore:
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Revolution Frequency - Momentum

From the relativity theory:

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 18

𝑒𝑔 𝑔 = 𝑒𝑀 𝑀 βˆ’ 𝛽[ π‘’π‘ž π‘ž

𝑒𝑀 𝑀 = 𝑒𝛾 𝛾 ⟺ 𝛾 = 𝑀 𝑑 π‘ž = 𝐹V𝛾𝛿 𝑑 𝑒𝑀 𝑀 = 𝑒𝛾 𝛾 = 1 𝛿+ π‘’π‘ž π‘ž

Resulting in : 𝑒𝑔

𝑔 = 1 𝛿+ βˆ’ 𝛽[ π‘’π‘ž π‘ž

We can get:

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SLIDE 19

Transition

  • Low momentum (𝛾 << 1 & 𝛿 is small) Γ 
  • High momentum (𝛾 β‰ˆ 1 & 𝛿 >> 1) Γ 
  • Transition momentum Γ 

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 19

𝑒𝑔 𝑔 = 1 𝛿+ βˆ’ 𝛽[ π‘’π‘ž π‘ž

p

a g >

2

1

p

a g <

2

1

p

a g =

2

1

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SLIDE 20

Frequency Slip Factor

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 20

𝑒𝑔 𝑔 = 1 𝛿+ βˆ’ 𝛽[ π‘’π‘ž π‘ž = 1 𝛿+ βˆ’ 1 𝛿gh

+

π‘’π‘ž π‘ž = πœƒ π‘’π‘ž π‘ž

1 𝛿+ > 𝛽[ ⟹ πœƒ > 0

  • Below transition:

1 𝛿+ = 𝛽[ ⟹ πœƒ = 0

  • Transition:

1 𝛿+ < 𝛽[ ⟹ πœƒ < 0

  • Above transition:
  • Transition is very important in hadron machines
  • CERN PS: 𝛿tr is at ~ 6 GeV/c (injecting at 2.12 GeV/c Γ  below)
  • LHC : 𝛿tr is at ~ 55 GeV/c (injecting at 450 GeV/c Γ  above)
  • Transition does not exist in lepton machines, why …..?
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SLIDE 21

RF Cavities

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 21

Variable frequency cavity (CERN – PS) Super conducting fixed frequency cavity (LHC)

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SLIDE 22

RF Cavity

BND Graduate School 6 September 2017 22 Rende Steerenberg CERN - Geneva

  • Charged particles are accelerated by a longitudinal electric field
  • The electric field needs to alternate with the revolution frequency
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SLIDE 23

Low Momentum Particle Motion

  • Lets see what a low energy particle does with

this oscillating voltage in the cavity

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 23

1st revolution period V time 2nd revolution period V

  • Lets see what a low energy particle does with

this oscillating voltage in the cavity

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SLIDE 24

Longitudinal Motion Below Transition

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 24

1st revolution period V time A B

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….after many turns…

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 25

100st revolution period V time A B

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….after many turns…

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 26

200st revolution period V time A B

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….after many turns…

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 27

400st revolution period V time A B

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SLIDE 28

….after many turns…

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 28

500st revolution period V time A B

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….after many turns…

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 29

600st revolution period V time A B

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….after many turns…

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 30

700st revolution period V time A B

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SLIDE 31

….after many turns…

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 31

800st revolution period V time A B

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….after many turns…

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 32

900st revolution period V time A B

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SLIDE 33

….after many turns…

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 33

900st revolution period V time A B

  • Particle B has made 1 full oscillation around particle A
  • The amplitude depends on the initial phase
  • This are Synchrotron Oscillations
  • Phase Stability: β€œoff-momentum” particles are contained
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SLIDE 34

Stationary Bunch & Bucket

  • Bucket area = longitudinal Acceptance [eVs]
  • Bunch area = longitudinal beam emittance = 𝜌.βˆ†E.βˆ†t/4 [eVs]

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 34

βˆ†E βˆ†t (or 𝛸)

βˆ†E βˆ†t Bunch Bucket

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SLIDE 35

What About Beyond Transition

  • Until now we have seen how things look like

below transition

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 35

Higher energy ] faster orbit ] higher Frev ] next time particle will be earlier. Lower energy ] slower orbit ] lower Frev ] next time particle will be later.

  • What will happen above transition ?

Higher energy ] longer orbit ] lower Frev ] next time particle will be later. Lower energy ] shorter orbit ] higher Frev ] next time particle will be earlier.

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SLIDE 36

Longitudinal Motion Beyond Transition

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 36

βˆ†E βˆ†t (or 𝛸) V

Phase w.r.t. RF voltage

𝛸

Synchronous particle RF Bucket Bunch

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SLIDE 37

βˆ†E βˆ†t (or 𝛸) V

Longitudinal Motion Beyond Transition

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 37

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SLIDE 38

βˆ†E βˆ†t (or 𝛸) V

Longitudinal Motion Beyond Transition

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 38

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SLIDE 39

βˆ†E βˆ†t (or 𝛸) V

Longitudinal Motion Beyond Transition

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 39

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SLIDE 40

βˆ†E βˆ†t (or 𝛸) V

Longitudinal Motion Beyond Transition

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 40

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SLIDE 41

βˆ†E βˆ†t (or 𝛸) V

Longitudinal Motion Beyond Transition

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 41

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SLIDE 42

βˆ†E βˆ†t (or βˆ†) V

Longitudinal Motion Beyond Transition

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 42

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SLIDE 43

βˆ†E βˆ†t (or 𝛸) V

Longitudinal Motion Beyond Transition

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 43

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SLIDE 44

βˆ†E βˆ†t (or 𝛸) V

Longitudinal Motion Beyond Transition

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 44

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SLIDE 45

Before & Beyond Transition

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 45

Before transition

Stable, synchronous position

E βˆ†t (or 𝛸) After transition E βˆ†t (or 𝛸)

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SLIDE 46

Synchrotron Oscillation

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 46

  • On each turn the phase, 𝛸, of a particle w.r.t. the RF

waveform changes due to the synchrotron

  • scillations.

rev

f h dt d D = p f 2

Change in revolution frequency Harmonic number

E dE f df

rev rev

h

  • =

rev

f dE E h dt d Γ— Γ—

  • =

\ h p f 2 dt dE f E h dt d

rev Γ—

Γ—

  • =

h p f 2

2 2

  • We know that
  • Combining this with the above
  • This can be written as:
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SLIDE 47

Synchrotron Oscillation

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 47

  • So, we have:

dt dE f E h dt d

rev Γ—

Γ—

  • =

h p f 2

2 2

  • Where dE is just the energy gain or loss due to the RF system during

each turn

𝛸 V

Synchronous particle dE = zero

V βˆ†t (or 𝛸)

dE = V.sin𝛸

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SLIDE 48

Synchrotron Oscillation

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 48

  • If 𝛸 is small then sin𝛸=𝛸

dt dE f E h dt d

rev Γ—

Γ—

  • =

h p f 2

2 2

f sin V dE = f sin V f dt dE

rev

=

and

f h p f sin . 2

2 2 2

V f E h dt d

rev Γ—

Γ—

  • =

2

2 2 2

= Γ· ΓΈ ΓΆ Γ§ Γ¨ Γ¦ Γ— Γ— + f h p f V f E h dt d

rev

  • This is a SHM where the synchrotron oscillation

frequency is given by:

rev

f E V h Γ— Γ· Γ· ΓΈ ΓΆ Γ§ Γ§ Γ¨ Γ¦ h p 2

Synchrotron tune Qs

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SLIDE 49

Acceleration

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 49

  • Increase the magnetic field slightly on each turn.
  • The particles will follow a shorter orbit. (frev < fsynch)
  • Beyond transition, early arrival in the cavity causes a gain in energy

each turn.

  • We change the phase of the cavity such that the new synchronous

particle is at 𝛸s and therefore always sees an accelerating voltage

  • Vs = Vsin𝛸s = Vπ›₯ = energy gain/turn = dE

𝛸 V

dE = V.sin𝛸s

βˆ†t (or 𝛸)

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SLIDE 50

Accelerating Bucket

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 50

fs βˆ†E βˆ†t (or 𝛸)

Stationary synchronous particle accelerating synchronous particle

βˆ†t (or 𝛸)

Stationary RF bucket Accelerating RF bucket

V

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SLIDE 51

Accelerating Bucket

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 51

  • The modification of the RF bucket reduces the acceptance
  • The faster we accelerate (increasing sin 𝛸s ) the smaller the

acceptance

  • Faster acceleration also modifies the synchrotron tune.
  • For a stationary bucket (𝛸s = 0) we had:
  • For a moving bucket (𝛸s β‰  0) this becomes:

rev

f E h Γ— Γ· Γ· ΓΈ ΓΆ Γ§ Γ§ Γ¨ Γ¦ h p 2

s rev

f E h f h p cos 2 Γ— Γ· Γ· ΓΈ ΓΆ Γ§ Γ§ Γ¨ Γ¦

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SLIDE 52

𝑔

hp = β„Ž Γ— 𝑔 hr^

Harmonics & Buckets

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 52

  • 150
  • 100
  • 50

50 100 150 @ degD

  • 200000
  • 100000

100000 200000@ VoltD

  • 150
  • 100
  • 50

50 100 150

f@

degD

  • 2
  • 1

1 2 dpΓͺ p

1 Trev

  • We will have: h buckets
  • Doing this dynamically, we can perform

bunch splitting

Harmonic number Frequency of cavity voltage Variable for b < 1

Split in four at flat top

25 ns

26 GeV/c

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SLIDE 53

72 bunches Eject 36 or

h = 7 or 9 h = 21 h = 84

Eject 24 or 48 bunches

Controlled blow-ups

gtr

Split in four at flat top

25 ns

26 GeV/c

BCMS (8 PSB b.) Standard (6 PSB b.) 8b4e (7 PSB b.) 80 bunches (7 PSB b.)

Bunch Splitting

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 53

Standard: 72 bunches @ 25 ns BCMS: 48 bunches @ 25 ns The PS defines the longitudinal beam characteristics

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SLIDE 54

RF Beam Control

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 54

Radial Position regulation Phase regulation Beam phase and position data Cavity voltage and phase (frequency) data

Beam

Beam Position Monitor Radio frequency Cavity

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SLIDE 55

Main Diagnostics Tools

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 55

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SLIDE 56

Beam Current & Position

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 56

Beam intensity or current measurement:

  • Working as classical transformer
  • The beam acts as a primary winding

Beam position/orbit measurement: Correcting orbit using automated beam steering

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SLIDE 57

Transverse Beam Profile Monitor

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 57

  • Transverse beam profile/size measurement:
  • Secondary EMission Grids (SEM-Grid)
  • Based on integration of induced current
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SLIDE 58

Transverse Beam Profile Measurement

  • (Fast) wire scanner
  • Uses photo multipliers to

measure scintillator light produced by secondary particles

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 58

x’ x

b e. b e /

x’ x

b e. b e /

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SLIDE 59

Wall Current Monitor

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 59

  • A circulating bunch creates an image current

in vacuum chamber.

  • +

+ + + + +

bunch vacuum chamber induced charge

Β§ The induced image current is the same size but has the

  • pposite sign to the bunch current.

resistor Insulator (ceramic)

+ +

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SLIDE 60

Longitudinal TomoScope

  • Make use of the synchrotron motion that turns the β€œpatient”

in the Wall Current monitor

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 60

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SLIDE 61

Possible Limitations

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 61

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SLIDE 62

Space Charge

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 62

  • Between two charged particles in a beam we have

different forces:

Coulomb repulsion Magnetic attraction I=ev

𝛾

𝛾=1 + + magnetic coulomb force total force

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SLIDE 63

Space Charge

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 63

  • At low energies, which means Ξ²<<1, the force is mainly

repulsive β‡’ defocusing

  • It is zero at the centre of the beam and maximum at the

edge of the beam

++++++ +++++ +++ +++++ +++++ ++ +++ + + ++ + + ++ ++++ ++++ ++++ ++ + ++++++ + +++ + +++ + + + + +++ + + + ++++ ++ +++++

x x

Linear Non-linear Defocusing force Non-uniform density distribution

y

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SLIDE 64

Laslett Tune Shift

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 64

3 2 , ,

4 g b pe

v h v h

N r Q

  • Β»

D

  • For the non-uniform beam distribution, this non-linear

defocusing means the Ξ”Q is a function of x (transverse position)

  • This leads to a spread of tune shift across the beam
  • This tune shift is called the β€˜LASLETT tune shift’
  • This tune spread cannot be corrected and does get very large at high

intensity and low momentum

Relativistic parameters Beam intensity Transverse emittance

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SLIDE 65

Imperfections & Resonances

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 65

  • Same phase and frequency for

driving force and the system can cause resonances and be destructive

  • Machines and elements cannot be built and aligned with

infinite perfection

  • We have to ask ourselves:
  • What will happen to the betatron oscillations due to the different

field errors.

  • Therefore we need to consider errors in dipoles, quadrupoles,

sextupoles, etc…

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SLIDE 66

Phase Space & Betatron Tune

  • If we unfold a trajectory of a particle that makes one turn in our machine

with a tune of Q = 3.333, we get:

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 66

2𝜌 y

𝛾x’ x

2Ο€q

Normalised phase space

  • This is the same as going 3.333 time around
  • n the circle in phase space
  • The net result is 0.333 times around the

circular trajectory in the normalised phase space

  • q is the fractional part of Q
  • So here Q= 3.333 and q = 0.333
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SLIDE 67

Resonance

  • If the phase advance per turn is 120ΒΊ then the betatron oscillation will repeat

itself after 3 turns.

  • This could correspond to Q = 3.333 or 3Q = 10
  • But also Q = 2.333 or 3Q = 7
  • The order of a resonance is defined as:

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 67

π‘œ Γ— 𝑅 = π‘—π‘œπ‘’π‘“π‘•π‘“π‘ 

1st turn 2nd turn 3rd turn

2Ο€q = 2Ο€/3

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SLIDE 68

Quadrupole (defl.∝ position)

  • For Q = 2.50: Oscillation induced by the quadrupole kick grows on each

turn and the particle is lost (2nd order resonance 2Q = 5)

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 68

Q = 2.50

1st turn 2nd turn 3rd turn 4th turn

Q = 2.33

  • For Q = 2.33: Oscillation is cancelled out every third turn, and therefore

the particle motion is stable.

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SLIDE 69

A more rigorous approach (1)

  • Let us try to find a mathematical expression for the amplitude

growth in the case of a quadrupole error:

Rende Steerenberg CERN - Geneva BND Graduate School 6 September 2017 69

y’b y a Dby’ Da 2Ο€DQ ΞΈ ΞΈ

2Ο€Q = phase angle over 1 turn = ΞΈ Δβy’ = kick a = old amplitude Ξ”a = change in amplitude 2πΔQ = change in phase y does not change at the kick

y = a cos(πœ„)

In a quadrupole Ξ”y’ = lky Only if 2πΔQ is small So we have:

Ξ”a = Ξ²Ξ”y’ sin(πœ„) = lΞ² sin(πœ„) a k cos(πœ„)

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SLIDE 70

A more rigorous approach (1)

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  • So we have: βˆ†a = l·𝛾·sin(πœ„) aΒ·kΒ·cos(πœ„)

) 2 sin( 2 q bk a a ! = D \

  • Each turn ΞΈ advances by 2Ο€Q
  • On the nth turn ΞΈ = ΞΈ + 2nΟ€Q
  • So, for q = 0.5 the phase term, 2(ΞΈ + 2nΟ€Q) is constant:
  • Over many turns:

( ) ( )

Γ₯

Β₯ =

+ = D

1

2 2 sin 2

n

Q n k a a p q b !

( ) ( )

Β₯ = +

Γ₯

Β₯ =1

2 2 sin

n

Q np q

Β₯ = D a a

and thus:

Sin(ΞΈ)Cos(ΞΈ) = 1/2 Sin (2ΞΈ) This term will be β€˜zero’ as it decomposes in Sin and Cos terms and will give a series of + and – that cancel

  • ut in all cases where the fractional tune q β‰  0.5
  • So, resonance for q = 0.5
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SLIDE 71

Resonances & Multipoles

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  • Quadrupoles excite 2nd order resonances (q = 0.5)
  • Sextupoles excite 1st and 3rd order resonances (q = 0.0 & q = 0.33)
  • Octupoles excite 2nd and 4th order resonances (q = 0.25 & q = 0.5)
  • This is true for small amplitude particles and low

strength excitations

  • However, for stronger excitations higher order

resonance’s can be excited which can be highly non- linear

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SLIDE 72

Resonance & Tune Diagram

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4.0 5.0 4.1 4.2 4.3 4.4 4.5

QH QV

5.1 5.2 5.3 5.4 5.5 5.6 5.7

3Qv=17 Injection Ejection 3Qv=16 2Qv=11 3Qh=13 Qh-2Qv=-6 Qh-Qv= -1 Qh-2Qv= -7 2Qh-Qv= -3 Qh+Qv=10 2 Q h + Q v = 1 4 Q h + 2 Q v = 1 5

During acceleration we change the horizontal and vertical tune to a place where the beam is the least influenced by resonances.

injection ejection

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SLIDE 73

A Measured Tune Diagram

  • Move a large emittance low intensity beam around in this tune

diagram and measure the beam losses

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SLIDE 74

Collective Effects

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  • Induced currents in the vacuum chamber (impedance) can result in

electric and magnetic fields acting back on the bunch or beam

Coupled Bunch Instabilities Head-Tail Instabilities

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SLIDE 75

Cures for Collective Effects

  • Ensure a spread in betratron/synchrotron

frequencies

  • Increase Chromaticity
  • Apply Octupole magnets (Landau Damping)
  • Reduce impedance of your machine
  • Avoid higher harmonic modes in cavities
  • Apply transverse and longitudinal feedback

systems

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SLIDE 76