Introduction to Particle Accelerators Pedro F. Tavares – MAX IV Laboratory

CAS – Vacuum for Particle Accelerators Örenäs Slott – Glumslöv, Sweden June 2017

June 2017 CERN Accelerator School – Vacuum for Accelerators Introduction to Particle Accelerators

Pre-requisites: classical mechanics & electromagnetism + matrix algebra at the undergraduate level. No specific knowledge of accelerators assumed. Objectives – Provide motivations for developing and building particle accelerators – Describe the basic building blocks of a particle accelerator – Describe the basic concepts and tools needed to understand how the vacuum system affects accelerator performance.

Caveat: I will focus the discussion/examples on one type of accelerator, but most of the discussion can be translated into other accelerator models.

June 2017 CERN Accelerator School – Vacuum for Accelerators Outlook ●Why Particle Accelerators ? – Why Synchrotron Light Sources ?

●Storage Ring Light Sources: accelerator building blocks

●Basic Beam Dynamics in Storage Rings. – Transverse dynamics: twiss parameters, betatron functions and tunes, chromaticity. – Longitudinal dynamics: RF acceleration, synchrotron tune – Synchrotron light emission, radiation damping and emittance

●How vacuum affects accelerator performance.

June 2017 CERN Accelerator School – Vacuum for Accelerators Why Particle Accelerators ?

Nuclear Physics Physics

Chemistry

Materials Particle Biology Accelerators Science Physics

Environmental Sciences

Archeology Medical Cultural Heritage Industrial Applications

June 2017 CERN Accelerator School – Vacuum for Accelerators Beams for Materials Research

PhotonPhoton Neutron NeutronIon SourcesSourcesSources SourcesSources

MAX IV Storage FELs RingsStorage FELs Rings

ESS

June 2017 CERN Accelerator School – Vacuum for Accelerators What is Synchrotron Light ?

Properties: Wide band High intensity/Brightness Polarization Time structure

Picture: https://universe-review.ca/I13-15-pattern.png June 2017 CERN Accelerator School – Vacuum for Accelerators Why Synchrotron Light ?

A.Malachias et ak, 3D Composition of Epitaxial Nanocrystals by Anomalous X-Ray Diffraction, PRL 99, 17 (2003)

OLIVEIRA, M. A. et al. Crystallization and preliminary X-ray diffraction analysis of an oxidized state of Ohr from Xylella fastidiosa. Acta Crystallographica. Section D, Biological Crystallography, v. D60, p. 337-339, 2004

Photo: Vasa Museum

Sandstrom, M. et al. Deterioration of the seventeenth-century warship vasa by J.Lindgren et al, Molecular preservation of the pigment melanin Juneinternal formation2017 of sulphuric acid. CERNNature 415, Accelerator893 - 897 (2002) School – Vacuumin fossil melanosomes for Accelerators, Nature Communications DOI: 10.1038/ncomms1819 (2012) Building Blocks of a SR based Light Source

Insertion Device

Linear Accelerator

Electron Gun

Storage Ring Beamlines

Annika Nyberg, MAX IV-laboratoriet, 2012

June 2017 CERN Accelerator School – Vacuum for Accelerators Insertion Devices Undulator

Periodic arrays of magnets cause the beam to “undulate”

Photo E.Wallen Wiggler

www.lightsources.org June 2017 CERN Accelerator School – Vacuum for Accelerators Electron Sources

Photo: Di.kumbaro

June 2017 CERN Accelerator School – Vacuum for Accelerators Injector Systems

Linear Accelerator LINAC + Booster

Photo by S.Werin

LNLS - MAX IV Full Energy Injector LINAC Brazil

June 2017 CERN Accelerator School – Vacuum for Accelerators Storage Ring Subsystems

Accumulate and maintain particles circulating stably for many turns

● Magnet System : Guiding and Focussing – DC – Pulsed

● Radio-Frequency System: Replace lost energy

● Diagnostic and Control System: Measure properties, feedback if necessary

● Vacuum System: Prevent losses and quality degradation

June 2017 CERN Accelerator School – Vacuum for Accelerators The magnet Lattice

Guiding - Dipoles

MAX IV Magnet block. Photo: M.Johansson

Photo: LNLS

Photo: LNLS

Note: Many functions can be integrated into a single magnet block Focussing - Quadrupoles

Correction of Chromatic aberrations Sextupoles Photo: LNLS

Icons adapted from BESSY Junewww.helmholz 2017 -berlin.de CERN Accelerator School – Vacuum for Accelerators The Radio-Frequency System

100 MHz Copper Cavities Solid State UHF Amplifiers

June 2017 CERN Accelerator School – Vacuum for Accelerators The Vacuum System

June 2017 CERN Accelerator School – Vacuum for Accelerators Optical Diagnostics Diagnostics and Controls

Photo:L. Isaksson B320B diagnostic beamline σx = 20.86 ± 0.14 µm (fit uncertainty) (visible SR radiation) σy = 15.70 ± 0.15 µm (fit uncertainty) Electrical Diagnostics and Controls

Fig:J. Ahlbäck

RF BPMs

June 2017 CERN Accelerator School – Vacuum for Accelerators Outlook ●Why Particle Accelerators ? – Why Synchrotron Light Sources ?

●Storage Ring Light Sources: accelerator building blocks

●Basic Beam Dynamics in Storage Rings. – Transverse dynamics: twiss parameters, betatron functions and tunes, chromaticity. – Longitudinal dynamics: RF acceleration, synchrotron tune – Synchrotron light emission, radiation damping and emittance

●How vacuum affects accelerator performance.

June 2017 CERN Accelerator School – Vacuum for Accelerators Storage Ring Beam Dynamics Goals: To determine necessary conditions for the beam to circulate stably for many turns, while optimizing photon beam parameters – larger intensity and brilliance. We want to study motion close to an ideal or reference orbit: Only small deviations w.r.t this reference are considered. Understand the behaviour of a system composed of a large number (∼ 1010 particles) of non-linear coupled oscillators governed by both classical and quantum effects.

June 2017 CERN Accelerator School – Vacuum for Accelerators Symmetry conditions for the Field

B (x,q,z) = B (x,q,-z) Only transverse z z components (no edge effects) Bx(x,q,z) = - Bx(x,q,-z) B (x,q,z) = 0 Only vertical q component on the symmetry plane

Bz(x,q,z) = B0 - g x First order expansion for the field close to the design orbit Bx(x,q,z) = - g z

June 2017 CERN Accelerator School – Vacuum for Accelerators Equations of Motion    F  e v  B  0 dv   Lorentz Force m  e v  B dt 0    r(t)  rur  zu z     v(t)  ru  rqu  zu  r q z dv     (r  rq 2 )u  (2rq  rq)u  zu dt r q z        v  B  rqBzur  (zBr  rBz )uq  rqBr u z       rq (B0  gx)ur  (zgz  r(B0  gx))uq  rqgzu z

June 2017 CERN Accelerator School – Vacuum for Accelerators Paraxial Approximation r    x Azimuthal velocity >> transverse velocity x   Small deviations p  p  p Independent variable t => s 0

p  p0

e0 g 2 1 p K  x(s)  1/ (s)  K(s)x(s)  p  p 0 0 p z(s)  K(s)z(s)  0   0 e0 B0 K(s) periodic s  q Oscillatory (stable) solutions June 2017 CERN Accelerator School – Vacuum for Accelerators How to guarantee stability ?

Weak focussing

Azimuthally symmetric machine: 푦′′ 푠 + 퐾푦 푠 = 0 Oscillatory requires K>0 1 K   K  0 1 x  2 0  K   2

K z  K  0 z S Combined function magnets  x

N June 2017 CERN Accelerator School – Vacuum for Accelerators Weak Focusing Limitations

Magnet apertures scale with machine energy and become impractical SOLUTION Alternating Gradient Courant/Snyder

Eliminate azimuthal symmetry and alternate field gradients of opposite signs

June 2017 CERN Accelerator School – Vacuum for Accelerators On-Energy - General Solution p On-energy particles x(s)  x0C(s)  x0S(s)  0 p0 C(0) 1 S(0)  0 C(0)  0 S(0)  1 Particular solutions

 x   C(s) S(s)  x         x C(s) S(s) x   s   0 Matrix Solution Combining elements means  x   x     M (s)  multiplying matrices  x  x   s  0

June 2017 CERN Accelerator School – Vacuum for Accelerators Transfer Matrices - Examples

x(s)  0 C(s)  1 Field-Free Straight section S(s)  s

1 L M    0 1 

′ 푥 퐿 = 푥0 + 푥0퐿

June 2017 CERN Accelerator School – Vacuum for Accelerators Transfer Matrices - Examples Focussing Quad x(s)  Kx(s)  0

C(s)  cos( K s) 1 S(s)  sin( K s) K  1   cos( K L) sin( K L) M   K     K sin( K L) cos( K L) 

 1 0 L  0 Thin Lens M   1  1  1 KL  Approximation  f  f

June 2017 CERN Accelerator School – Vacuum for Accelerators Stability Analysis – Periodic Systems

Transfer Matrix for a full a b  period M (s)     c d   x   x  Stability  matrix elements remain    M N   bounded  x  x   N  0

June 2017 CERN Accelerator School – Vacuum for Accelerators Alternating Gradient: Stability

F D F

L/2

 1 0 L  1 0 L  1 0  1  1   M  1 2 1 2 1  1  1  1  2 f 0 1  f 0 1  2 f   L2  L   1 L1   8 f 2  4 f  M     L  L  L2   1  1   2   2   4 f  4 f  8 f 

June 2017 CERN Accelerator School – Vacuum for Accelerators Stability Analysis 1 0  L2  L  I         1 2 L 1 0 1  8 f  4 f    M     cos()I  sin()J  L  L  L2   1  1   0    2   2     4 f  4 f  8 f  J  1 2   L  0  cos()  1  8 f 2   2 L  L  L  I  I sin()  1 1     2 2 f  4 f  4 f  J  I  L  2 1  M  cos(2)I  sin(2)J  4 f    2 f   n  L  M  cos(n)I  sin(n)J 1   4 f  L2 L Stable if  real 1  1 f  8 f 2 4

June 2017 CERN Accelerator School – Vacuum for Accelerators Off-Energy Particles Non-homogenuous term 1 p D(s)  1/ (s)2  K(s)D(s)   p0

D(s) can be obtained from the solution to the homogeneous eqs.

푠 ′ 푠 ′ 푑푠 ′ 푑푠 ′ 퐷 푠 = 푆(푠) න ′ 퐶 푠 − 퐶(푠) න ′ 푆 푠 0 휌 푠 0 휌 푠

풙 푥 ′ ′ 퐶(푠) 푆(푠) 퐷(푠) 풙 푥 ′ ′ ′ Matrix Solution ∆풑 = 푀(푠) ∆풑 푀 = 퐶 (푠) 푆 (푠) 퐷 (푠) 0 0 1 풑ퟎ 푠 풑ퟎ 푠=0

June 2017 CERN Accelerator School – Vacuum for Accelerators Example: Sector Dipole Magnet

푠 퐶 푠 = cos 푠 퐾 푠 = 0 휌0 퐷 푠 = 휌0 1 − cos 푠 휌0 ρ s = 휌0 푆 푠 = 휌0sin 휌0

푠2 1 푠 퐶(푠) 푆(푠) 퐷(푠) 2휌0 푀 = 퐶′(푠) 푆′(푠) 퐷′(푠) ≈ 푠 0 1 0 0 1 휌0 0 0 1

Small bending angle

June 2017 CERN Accelerator School – Vacuum for Accelerators General pseudo-harmonic solution

1 p x(s)  1/ (s)2  K(s)x(s)   p0 z(s)  K(s)z(s)  0 p Pseudo-harmonic solution x(s)  (s) cos((s)  ) (s) 0 p s ds (s)  Betatron Dispersion Betatron Phase Advance  Function Function 0  (s)  (L) Periodic Betatron Tune Q   2 2 1 ′ 1 2 Equation for Betatron 훽 푠 훽′ 푠 − 훽′ 푠 + 훽2 푠 퐾 푠 = 1 Function 2 4 June 2017 CERN Accelerator School – Vacuum for Accelerators Twiss Parameters 훽 푠

1 Figure E.Willson, CAS 2003 훼 푠 = − 훽′(푠) 2 1 + 훼2(푠) 훾 푠 = 훽(푠)

Courant Snyder Invariant

휀 = 훾 푠 푥2 푠 + 2훼 푠 푥 푠 푥′ 푠 + 훽(푠)푥′2(푠)

These are properties of the ring, defined by how the focussing is distributed along the accelerator and give us a conveniente way to describe any trajectory (in linear approximation) June 2017 CERN Accelerator School – Vacuum for Accelerators Twiss Parameters and Beam sizes

Equilibrium beam parameters: Emittance, Energy Spread 휖푥, 휖푦, 휎훿

2 2 휎푥 푠 = 휖푥훽푥 푠 + 휎훿 휂 푠

2 ′ 2 휎푥′ 푠 = 휖푥훾푥 푠 + 휎훿 휂 푠

휎푦 푠 = 휖푦훽푦 푠

휎푦′ 푠 = 휖푦훾푦 푠

June 2017 CERN Accelerator School – Vacuum for Accelerators Twiss Parameters and Perturbations

 Localized dipole error (휽) – perturbation of the closed orbit (periodic solution)

훽(푠)훽 휃 cos(휙 푠 − 휋푄) Δ푥 (푠) = 0 0 푐.표. 2sin(휋푄)

 Localized quadrupole error (Δ퐾퐿) – perturbation of the tune and beta function

Δ훽(푠) 훽 = 0 cos(2휙 푠 − 2휋푄)Δ퐾퐿 훽(푠) 2 sin 2휋푄

June 2017 CERN Accelerator School – Vacuum for Accelerators Perturbations Some freqs. (tunes) must be avoided to prevent resonances. mQx+nQy=p m,n,p integer

Resonance Diagram for the MAX IV 3 GeV Ring

June 2017 CERN Accelerator School – Vacuum for Accelerators Twiss Parameters MAX IV 3 GeV Ring

June 2017 CERN Accelerator School – Vacuum for Accelerators Non-linear perturbations 2 Bz (x)  Sx G(x)  2Sx Chromaticity: quad strength varies with

energy. Correction of chromatic aberration with sextupoles Photo LNLS

A sextupole produces a position dependent focussing

Sextupoles are non- liear elements and QF S introduce perturbations

June 2017 CERN Accelerator School – Vacuum for Accelerators Non-Linear Perturbations and Dynamic Aperture

MAX IV DDR, 2010

June 2017 CERN Accelerator School – Vacuum for Accelerators Longitudinal Dynamics: Phase Stability

Synchrotron Oscillations Particles with different energies have different revolution periods

1.5

Vrf1 0.5 U0 0 0 5 10 15 -0.5

-1 -1.5 MAX IV 100 MHz RF Cavity

For small amplitudes: simple harmonic motion Larger amplitudes: non-linearities (like a pendulum) 2 s  0

June 2017 CERN Accelerator School – Vacuum for Accelerators Brief Recap – Beam Dynamics 1 p x(s)  1/ (s)2  K(s)x(s)   p Transverse Plane: 0 z(s)  K(s)z(s)  0

2 Longitudinal Plane: s  0 The beam is a collection of many 3D – oscillators. If parameters are properly chosen (magnet lattice, RF system), stable oscillations are realized in all planes. Non-linearities cause distortions that may reduce the available stable area in phase space: reduction of the dynamic aperture. Linear Oscillations – Twiss Parameters 푄, 훽 푠 , 훼 푠 , 훾 푠 Are a property of the lattice (the whole accelerator). Provide a convenient way to summarize all about the linear behaviour of the accelerator: trajectories, sizes, sensitivity to errors June 2017 CERN Accelerator School – Vacuum for Accelerators Outlook ●Why Particle Accelerators ? – Why Synchrotron Light Sources ?

●Storage Ring Light Sources: accelerator building blocks

●Basic Beam Dynamics in Storage Rings. – Transverse dynamics: twiss parameters, betatron functions and tunes, chromaticity. – Longitudinal dynamics: RF acceleration, synchrotron tune – Synchrotron light emission, radiation damping and emittance

●How vacuum affects accelerator performance.

June 2017 CERN Accelerator School – Vacuum for Accelerators How Vacuum Systems affect SR Performance

Quantity Quality Examples

limits max. current Wakefields bunch dimensions

Coherent lifetime Ion Trapping tune shifts emittance growth transverse stability Incoherent lifetime : need to Scattering top-up more often Elastic/Inelastic

June 2017 CERN Accelerator School – Vacuum for Accelerators Elastic Scattering Illustration from http://hyperphysics.phy-astr.gsu.edu Rutherford Scattering cross-section

2 2 푑휎푒푙 1 푍푒0 1 = 4 푑Ω 4휋휀0 2푝0푐 휃 sin 2 Particles are lost if scattered by angles larger than: # electrons Gas density

2 Aperture limitation 퐴 1 1 푑푁 휋 푑휎 ൗ훽 around the whole ring 푒푙 2 푚푖푛 = − = 2휋푐푛 න sin 휃 푑휃 휃푚푎푥 = 휏푒푙 푁 푑푡 휃 푑Ω 훽0 푚푎푥 Beta function where the collision occured Watch out for: Assuming Nitrogen • low energy 10.25퐸[퐺푒푉]2휖 [푚푚푚푟푎푑] • small apertures 휏 [ℎ푟] = 퐴 푒푙 훽 푚 푃[푛푡표푟푟] • high pressure at high beta locations • High Z June 2017 CERN Accelerator School – Vacuum for Accelerators Inelastic Scattering (bremsstrahlung)

• Particle lose energy through radiation emission in collision with nuclei and electrons. • If energy loss is larger than acceptance, particle is lost

2 2 2 푑휎 훼푓4푍 푟푒 4 훿 훿 183 1 훿 퐵푆 = 1 − + ln + 1 − 푑훿 훿 3 퐸 퐸 1 9 퐸 푍3

Particles are lost if they lose energy larger than the acceptance: 훿푎푐푐

1 1 푑푁 Ilustration https://de.wikipedia.org = − 휏퐵푆 푁 푑푡 퐸 푑휎 4 퐸 5 183 1 퐸 = 푐푛 න 퐵푆 푑훿 = 푐푛4훼 푍2푟2 ln − ln + ln − 1 푑훿 푓 푒 3 훿 8 1 9 훿 푎푐푐 3 푎푐푐 훿푎푐푐 푍 Watch out for high Z 훼푓 Fine structure constant Weak dependence on energy and

푟푒 Classical electron radius energy acceptance June 2017 CERN Accelerator School – Vacuum for Accelerators Ion Trapping • circulating electrons collide with residual gas molecules producing positive ions that can be captured (trapped) by the beam

• Reduces beam lifetime : increased local pressure. • Tune –shifts, Tune spreads • Emittance Growth • Coherent Collective instabilities (multi-bunch)

This had some nearly catastrophic effects on some early low energy injection machines

June 2017 CERN Accelerator School – Vacuum for Accelerators Lifetime contributions at the MAX IV 3 GeV Ring

Tautot = 36 h or I*Tautot = 2.5 Ah Slide courtesy Åke Andersson

βyscr = 3.8 m Ay = 2.2 mm.mrad (less than physical: Ay ~ 4 mm.mrad, but larger than future ID-chambers: Ay ~ 1 mm.mrad)

 Taugas = 60 h or I*Taugas = 4.2 Ah

Vertcal Scraper Measurements  TauTou = 90 h or I*TauTou = 6.2 Ah by Jens Sundberg

June 2017 CERN Accelerator School – Vacuum for Accelerators Transverse collective instabilities driven by ions

Increased stability by adding a gap Transverse beam blow up due to ion trapping to the bunch train

Ion Clearing ON Ion Clearing OFF

LNLS 1.37 GeV electron storage rig

R.H.A.Farias et at at: Optical Beam Diagnostics for the LNLS Synchrotron Light Source,EPAC98, p.2238. 2016/07/06: MAX IV 3 GeV Ring Early commissioning

June 2017 CERN Accelerator School – Vacuum for Accelerators Thank you for your attention

June 2017 CERN Accelerator School – Vacuum for Accelerators References H. Wiedemann, Particle Accelerator Physics I and II, Springer Verlag. M.Sands, The Physics of Electron Storage Rings D.A.Edwards and M.J.Syphers, An Introduction to the Physics of High Energy Accelerators, Wiley CAS – CERN Accelerator Schools (Basic and Advanced)

June 2017 CERN Accelerator School – Vacuum for Accelerators Back up slides

June 2017 CERN Accelerator School – Vacuum for Accelerators Why Synchrotron Light ?

Image: Lawrence Berkely Lab

June 2017 CERN Accelerator School – Vacuum for Accelerators Lattice Design for Low Emittance Rings General Problem Statement – Scaling Laws 퐻 훾2 3 휌 ′2 2 휀0 = 퐶푞 퐻 푠 = 훽 푠 휂 푠 + 2훼 푠 훽 푠 + 훾(푠)휂 (푠) 퐽푥 1 휌2 휂(푠) (ׯ 1 + 2휌2 푠 푘(푠 휌3 푠 풟 = 퐽 = 1 − 풟 푑푠 푥 ׯ 휌2(푠)

2 H  dip 0  Cq isomagnetic J x 

June 2017 CERN Accelerator School – Vacuum for Accelerators Defining the Basic Parameters of a SR based Light Source

hc 2 2 E1  p  1 2  1 Ku   2  Photon energy range + Energy Insertion device Technology + Top-up Injection

2  g  g   B [Tesla]  3.694exp5.068 1.52   0        p  p  

Diameter Emittance (brightness) requirements  2q 3  0  Cq F June 2017 CERN Accelerator School – Vacuum for Accelerators12 15J x Electrostatic SR

Stores 25 keV ions. S.Moller, EPAC98

June 2017 CERN Accelerator School – Vacuum for Accelerators Beam Guiding

Why magnetic fields ?

    Lorentz Force F  e0 (E  v  B)

at 3.0 GeV, B= 1.0 T E = 500 MV/m !!

June 2017 CERN Accelerator School – Vacuum for Accelerators Transverse Beam Dynamics

Zeroth order: guide fields (dipoles) First order : Focusing – linear oscillations (quadrupoles). Alternating Gradient. Second order: Chromatic Aberrations and corrections (sextupoles) Effects of perturbations, non-linearities Dynamic Aperture.

June 2017 CERN Accelerator School – Vacuum for Accelerators Damping/Excitation of Longitudinal Oscillations Photon emission depends on particle energy (larger energy, more emission). This adds a dissipative term to the eqs. of motion. However, emission happens in the form of discrete events (photons). At each emission, there is a sudden change in particle energy (but no sudden change in particle position. Both effects together lead to an equilibrium state that defines the bunch dimensions in longitudinal phase space Energy spread Bunch length

1 2 3 c 2   s      C l p p q s J s 1 2  s Depends on lattice June 2017 CERN Accelerator School – Vacuum for Accelerators Damping/Excitation of Transverse Oscillations Oscillations

Discrete photon emission changes momentum along the direction of propagation If this happens in a dispersive region of the magnet lattice, a transverse (betatron) oscillation will be excited. Momentum is regained at the RF cavity only along the longitudinal direction. This causes a reduction of the particle angles (damping). Both effects together lead to an equilibrium state that define the transverse beam dimension and angular spread, i.e., the emittance.

3 Number of dipoles  2 1 IsomagneticH   C s 0 q 2 3 J x 1  q   2C F 0 sq 12 15J H (s)   (s)(s)2  2(s)(s)x(s)   (s)(s)2

Lattice

June 2017 CERN Accelerator School – Vacuum for Accelerators The Challenge of High Brightness Source Source Design: a beam dynamics perspective

Low Emittance (High Brightness) Reduction of Lifetime/Injection Efficiency Strong Focusing: high natural chromaticity

Reduction of Strong Sextupoles lead to Dynamic Large Non-linearities Aperture

June 2017 CERN Accelerator School – Vacuum for Accelerators