[PPT] - A new Space-Charge Model for high-intensity Particle Beams PowerPoint Presentation

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

A new Space-Charge Model for high-intensity Particle Beams

Michael Holz

FREIA, Uppsala University

14. September 2017

14/09/17 FREIA, Uppsala University

1 -
M. Holz

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

Outline

1

Introduction

2

Basic Accelerator Physics

3

Space Charge Model

4

Beam Core Dynamics

5

Spectator Particle Dynamics

6

Conclusion

14/09/17 FREIA, Uppsala University

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M. Holz

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

Introduction to High Power Accelerators

European Spallation Source (Lund) neutron source High-intensity proton beam Consequences of beam loss: heat load (cryogenics) radiation (maintanance) Life- and Material Sciences 5 MW rms beam power loss threshold: 1 W

m

intensity-dependent effects

14/09/17 FREIA, Uppsala University

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M. Holz

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

Space Charge

same-charged particles ⇒ Space Charge non-linear electric forces (e.g. Gauss) particles pushed to higher amplitudes ⇒ halo halo is prone to be lost

0.03
0.02
0.01

0.01 0.02 0.03 x / [m]

150
100
50

50 100 150 Field / [a.u.]

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

Research Questions

What does the new space charge model address? Understand beam dynamics at high intensity Evolution of beam size Mechanics of beam halo formation Limits in beam current The new model works generally, but now: Rings.

14/09/17 FREIA, Uppsala University

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M. Holz

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

Propagation of Particles

Particle Transport ¯ x ¯ x

′

=
R11

R12 R21 R22 x x

′

QF

L

s

Linear optics (drift space, dipoles, quadrupoles) R: lattice transfer matrix Rdrift = 1 L 1

Distribution Transport

σ = R11 R12 R21 R22 σ11 σ12 σ21 σ22 R11 R12 R21 R22 T ; σ = RσRT σ = x2 xx′ xx′ x′2

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

Visualize Particle Oscillations

phase space angle vs. position Particle trajectory in ring stable: follows ellipse normalized phase space particle action J =

˜ x2+ ˜ x′2 2

has equivalent for distributions Twiss Parameters α, β, γ β: amplitude J: action variable Distribution ǫ = J; emittance σx = √ǫβ; size

14/09/17 FREIA, Uppsala University

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

Beam Focusing and Tune

Quadrupole magnets focus the beam Tune: number of oscillations per turn equilibrium ⇒ stable motion perturb equilibrium ⇒ oscillations Harmonic Oscillator Tune Q for stability: fractional tune Effects of space-charge: defocusing reduces tune Space Charge Tune Shift

14/09/17 FREIA, Uppsala University

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

Existing Simulation Methods

1 Envelope equations

d2 a ds + aκ − Fsc a − ǫ2 a3 = 0

2 Particle in Cell (PIC), multi-particle

tracking ⇒ computationally expensive We complement these methods with a space-charge model that is valid for transverse, Gaussian beams includes non-linear space charge forces is fully analytic

Eq. of Motion

a(s): envelope amplitude Fsc: space charge

x density F x density F

14/09/17 FREIA, Uppsala University

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

Basis of the Model

Electric field of a Gaussian particle distribution: (Bassetti & Erskine, 1980)

Ex − iEy = −i Q 2ǫ

2π
σ2

x − σ2 y



  w    x + iy

2(σ2

x − σ2 y )

   − e

− x2

2σ2 x + y2 2σ2 x

w

   x σy

σx + iy σx σy

2(σ2

x − σ2 y )

      

w: complex error function

nly valid for upright Gaussian beam

⇒ we use a covariant form F0 = f3 + if1 to include tilt angles ⇒ cross-plane coupling

y x y x

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

Self-Interaction of the Beam

1 particle distribution → Ψ(

x) =

1 (2π)2√ det σe− 1

2 σ−1 mn xmxn

2 electric field → F0(x1, x3, σ) = f3 + if1

⇒ beam matrix σ is identical for space charge Particle experiences a kick: ¯ x1 = x1 ¯ x2 = x2 + f1 ¯ x3 = x3 ¯ x4 = x4 + f3 What happens to a beam distribution?

x, x

′, y, y ′

= [x1, x2, x3, x4]

14/09/17 FREIA, Uppsala University

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

Space Charge Map

Averaging over all particles σ = x2 xx′ xx′ x′2

¯

σ11 = ¯ x1¯ x1 = x1x1 = σ11 ¯ σ12 = ¯ x1¯ x2 = x1(x2 + f1) = σ12 + x1f1 ¯ σ13 = ¯ x1¯ x3 = x1x3 = σ13 ¯ σ14 = ¯ x1¯ x4 = x1(x4 + f3) = σ14 + x1f3 ¯ σ22 = ¯ x2¯ x2 = (x2 + f1)(x2 + f1) = σ22 + 2x2f1 + f 2

1

¯ σ23 = ¯ x2¯ x3 = x3(x2 + f1 = σ23 + x3f1 ¯ σ24 = ¯ x2¯ x4 = (x2 + f1)(x4 + f3) = σ24 + x2f3 + x4f1 + f1f3 ¯ σ33 = ¯ x3¯ x3 = x3x3 = σ33 ¯ σ34 = ¯ x3¯ x4 = x3(x4 + f3) = σ34 + x3f3 ¯ σ44 = ¯ x4¯ x4 = (x4 + f3)(x4 + f3) = σ44 + 2x4f3 + f 2

3

linear terms xkF0 quadratic terms f 2

1 , f 2 3 , f1f3

f 2

3 = ℜ{F0}2 =

1

2(F0 + ¯

F0) 2 = 1

4

F 2

0 + ¯

F 2

0 + 2F0 ¯

F0

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

Linear Terms

xkF0 =NK

∞

−∞

d4x xkΨ( x)F0(x1, x3, σ); k = 1, 2, 3, 4 General Ansatz: w (z) = 2 √π

∞

dα e−α2+2iαz

∞

−∞

dx xne− x2

2σ2 =

∂ ∂B n

∞

−∞

dx eBx− x2

2σ2

B=0

General solution to linear terms: xkF0 = iNK 2(σ11 − σ33 + 2iσ13)   σk1 + iσk3 −    σk1(σ33 − iσ13) + iσk3(σ11 + iσ13)

σ11σ33 − σ2

13

     

N,K: scaling Complex error fct integral form completing squares parametric differentiation

14/09/17 FREIA, Uppsala University

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

Quadratic Terms

F0 ¯ F0 = (NK)2

∞

−∞

d4x Ψ( x)F0 ¯ F0 . . . fast-forward several weeks of work . . .

f 2

3 = ℜ{F0}2 =

1

2 (F0 + ¯

F0) 2 =

1 4

F 2

0 + ¯

F 2

0 + 2F0 ¯

F0

w (z) = w (ax1 + bx3)

S (σ, a, b)

F0 ¯ F0 = (−NK)2 π

2(σ11 − σ33 + 2iσ13)
2(σ11 − σ33 − 2iσ13)

·

P (σ, 1, a, ¯

a) − P

σ, 2, a, ¯

b

− P (σ, 2, ¯

a, b) + P

σ, 3, b, ¯

b

with

P (σ, n, a, b) = 2 nπ 1

S11S22 − S2

12

π

2 − arctan

S12
S11S22 − S2

12

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

Test Application: Simple Ring

FODO-cell with 22.5◦ bending angle → 16 cells periodic structure

QF QD B B

Turn-by-turn data: (equilibrium) beam sizes emittance growth space charge tune shifts

14/09/17 FREIA, Uppsala University

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

Beam motion with Space Charge

1 Slice machine lattice 2 Advance beam through a slice 3 Calculate space charge contribution Tsc in

that slice

4 update beam matrix by adding the

inhomogeneous term

Rs: sliced transfer matrix Damping D = I e−

1 δturn

Excitation E Synchrotron radiation, quantized process, random

One slice: σs,2 = Rs,1σs,1RT

s,1 + Tsc,1

One turn: σn+1 = · · · Rs,3

Rs,2
Rs,1σnRT

s,1 + Tsc,1

RT

s,2 + Tsc,2

RT

s,3 + Tsc,3 · · ·

Effect of synchrotron radiation⇒ defines equilibrium ¯ σn+1 = Dσn+1DT + E ⇒ reach equilibrium

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

Turn-by-Turn Simulation

Turn-by-Turn data as function of number of particles in the beam during ’injection’:

500 1000 1500 2000 2500 3000 Turns 2.45 2.5 2.55 2.6 2.65 2.7 2.75 2.8

x 2

10-6 zero current 1e8 p+ 3e8 p+ 5e8 p+ 8e8 p+

rms beam size squared σ2

x

scale of effect ≈ 10 %

500 1000 1500 2000 2500 3000 Turns 3.6 3.65 3.7 3.75 3.8 3.85 3.9 3.95

x / [m rad]

10-7 zero current 1e8 p+ 3e8 p+ 5e8 p+ 8e8 p+

beam emittance

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

Space Charge Tune Shift

2 4 6 8 10 Number of Particles 108 0.68 0.7 0.72 0.74 0.76 0.78 0.8 0.82 Qx Qsc Q0

for a round, uniform beam: ∆Q = − Nr0L 2 (2π)

3 2 σzǫβ2γ3

We obtain identical ∆Q by: making the beam as round as possible linearize the electric field

0.05 0.1 0.15 Q N 3.6 3.7 3.8 3.9 4 4.1

x / [m rad]

10-7

emittance vs. tune shift ⇒ current Quantifies space charge effect ǫ vs. ∆Q machine-’invariant’?

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

Spectator Particle Dynamics

How do the particles move to the tails of the Gaussian? Apply mismatch to equilibrium beam core Oscillating core with varying electric field Passive spectator particles ride

n beam core

Matched: Bmag = 1 Mismatch Bmag

2.5
2
1.5
1
0.5

0.5 1 1.5 2 2.5 X / [m] 10-3

4
3
2
1

1 2 3 4 x' / [rad] 10-4 Equilibirum mismatched

Method of investigation Poincar´ e-maps: dynamics of single particles parameters to vary: current, mismatch, coupling

14/09/17 FREIA, Uppsala University

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

Poincar´ e-Maps

place single particles along position x

bserve particle coordinates
n turn-by-turn basis

dynamic behavior of single particles display in normalized phase space

3
2
1

1 2 3 X / [m] 10-3

300
200
100

100 200 300 a.u. 0.5 1 1.5 2 2.5 3

el. field

What kind of effects happen? amplitude-dependent tune shift resonance islands unaffected particles draw circles

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

N= 4 × 108 protons per bunch

2

2 10-3

3
2
1

1 2 3 10-3 1 Matched 0.5 1 1.5 2 2.5 3

2

2 10-3

3
2
1

1 2 3 10-3 3 Coupled

2

2 10-3

3
2
1

1 2 3 10-3 2 Bmag= 1.08

1

almost perfect circles

2

disturbance between

the 1σ- & 2σ-particle

heavy displacement (to

center)

3

clear structures
action exchange?

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

N= 10 × 108 protons per bunch

2

2 10-3

3
2
1

1 2 3 10-3 1 Matched

2

2 10-3

3
2
1

1 2 3 10-3 3 Coupled

2

2 10-3

3
2
1

1 2 3 10-3 2 Bmag= 1.4

1

smeared out circles
bigger amplitudes

2

heavy disturbance
heavy displacement (to

fringe)

3

clear structures
action exchange?

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

Analysis of Action Variables

Particle Action J =

˜ x2+ ˜ x′2 2

Plot (Je − Jinit) vs. Jinit for mismatched cases standard deviation: measure of disturbance

0.2 0.4 0.6 0.8 1 Jinit 10-5

1.5
1
0.5

0.5 1 1.5 Displacement J 10-6 stdx Jx stdy Jy

N= 4 × 108 protons per bunch

0.2 0.4 0.6 0.8 1 1.2 Jinit 10-5

4
2

2 4 Displacement J 10-6 stdx Jx stdy Jy

N= 10 × 108 protons per bunch

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

Summary

1 Analytic space charge model for transverse Gaussian beams 2 Turn-by-turn beam core evolution in a test ring 3 Analysis of single particles, riding on beam cores

Beam Core Observations: Space charge tune shift of the core Machine performance (ǫ vs. ∆Q) Single Particle Observations: Beam mismatch appears to be a major contributor to beam halo Beam current serves as ’lever arm’ Analysis of particle actions quantifies chaos

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Introduction Basic Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Particle Dynamics Conclusion

Future Investigations

Include longitudinal plane Simulate planned machines, e.g. ESS accumulator ring (ESSnuSB) Effects of injection mismatch Magnet ripple (periodic disturbance, non-linear harmonic

scillator)

stability and robustness of machines Linear accelerators

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