An Introduction to Particle Accelerators

Erik Adli,, University of Oslo/CERN November, 2007 Erik.Adli@cern.ch

v1.32 References

• Bibliograp hy: – CAS 1992, Fifth General Accelerator Physics Course, Proceedings, 7-18 September 1992 – LHC Design Report [online] – K. Wille, The Physics of Particle Accelerators, 2000

• Other references – USPAS resource site, A. Chao, USPAS January 2007 – CAS 2005, Proceedings (in-print), J. Le Duff, B, Holzer et al. – O. Brüning: CERN student summer lectures – N. Pichoff: Transverse Beam Dynamics in Accelerators, JUAS January 2004 – U. Am aldi, presentation on Hadron therapy at CERN 2006 – Various CLIC and ILC presentations – Several figures in this presentation have been borrowed from the above references, thanks to all! Part 1

Introduction Particle accelerators for HEP

•LHC:theworld: the world biggest accelerator, both in energy and size ((gas big as LEP) •Under construction at CERN today •End of magnet installation in 2007 •First collisions expected summer 2008 Particle accelerators for HEP The next big thing. After LHC, a Linear Collider of over 30 km lengg,th, will probably be needed (why?) Others accelerators

• Histor ica lly: t he ma in dr iv ing force o f acce lerator deve lopment was collision of particles for high-energy physics experiments

• However, today there are estimated to be around 17 000 particle accelerators in the world, and only a fraction is used in HEP

• Over half of them used in medicine

• Accelerator physics: a disipline in itself, growing field

• Some examples: Medical applications

• Therapy – The last decades: electron accelerators (converted to X-ray via a target) are used very successfully for cancer therapy)

– Todayyp's research: proton accelerators instead (hadron therapy): energy deposition can be controlled better, but huge technical challenges

• Imaging – Isotope production for PET scanners Advantages of proton / ion-therapy

(() Slide borrowed from U. Am aldi ) Proton therapy accelerator centre

HIBAC in Chiba

What is all this? Follow the ( Slide borrowed from U. Am aldi ) lectures... :) Synchrotron Light Sources

• the las t two d eca des, enormous increase in the use o f sync hrony ra dia tion, emitted from particle accelerators • Can produce very intense light (radiation), at a wide range of frequencies (visible or not) • Useful in a wide range of scientific applications Outline of presentation

Part 1: Intro + Main parameters + Basic Concepts Part 2: Longgyitudinal Dynamics Part 3: Transverse Dynamics Case: LHC Part 4: Intro to synchrotron radiation Part 5: The road from LEP via LHC to CLIC Case: CLIC Main Parameters Main parameters: particle type

• Hadron collisions: compoundild particles – Mix of quarks, anti-quarks and gluons: variety of processes – Parton energy s pread – Hadron collisions ⇒ large discovery range

• Lepton collisions: elementary particles – Collision process known – Well defined energy – Lepton colli si ons ⇒ preciiision measurement

“If you know what to look for, collide leptons, if not collide hadrons” Main parameters: particle type

Discovery Precision

SS/LHCSppS / LHC LEP / LC Main parameters: particle energy

• NhibfdtlNew physics can be found at larger unpro bdibed energies

• Energy for particle creation: centre-of-mass energy, ECM

• Assume particles in beams with parameters m, E, E >> mc2

= – Particle beam on fixed target: ECM mE = – Colliding particle beams: ECM 2E

• ⇒ Colliding beams much more efficient Main parameters: luminosity

• Hig h energy is no t enoug h !

• Cross-sections for interestinggp processes are ver y small (~ pb = 10−36 cm² ) ! σ 2 2, 2 – (gg → H) = 23 pb [ at s pp = (14 TeV) mH = 150 GeV/c ] R = Lσ – We need L >> 1030 cm-2s-1 in order to observe a significant amount of interesting processes!

• L [cm-2s-1] for “bunched colliding beams” depends on

– number of particles per bunch (n1, n2) σ σ – bunch transverse size at the interaction point ( x, y ) – bunch collision rate ( f) n n L = f 1 2 πσ σ 4 x y Main parameters: LEP and LHC

LEP LHC

Particle type(s) e+ and e- p, ions (Pb, Au)

(~ Collision energy (Ecm) 209 GeV (max) p: 14 TeV at p 2-3 TeV mass reach, depending on physics) Pb: 1150 TeV

Luminosity (L) Peak: 1032 cm-2s-1 Peak: 1034 cm-2s-1 Daily avg last years: (IP1 / IP5) 1031 cm-2s-1 Integrated: ~ 1000 pb-1 (per experiment) Capabilities of particle accelerators

• A mod ern HEPtilHEP particle acce ltlerator can acce ltlerate par tilkticles, keep ithing them within millimeters of a defined reference trajectory, and transport them over a distance of several times the size of the solar system

HOW? HOW?

• In thi s presen tati on we t ry t o expl ai n thi s by s tu dy ing: – the basic components of an accelerator – the physical mechanisms that determines the particle motion – how particles (more or less) follow a specified path, even if our accelerator is not designed perfectly

• At the end, we use what we have learned in a case-study: the LHC Part 2

Basic concepts An accelerator

• Structures in which the particles will move • Structures to accelerate the particles • Structures to steer the particles • Structures to measure the particles Lorentz equation

• The t wo mai n t ask s of an accel erat or – Increase the particle energy – Change the particle direction (follow a given trajectory, focusing)

• Lorentz equation: r r r r r r r r r F = q(E + v × B) = qE + qv × B = FE + FB

⊥ • FB v ⇒ FB does no wor k on the par tic le

– Only FE can increase the particle energy

≈ •FE or FB for deflection? v c ⇒ Magnetic field of 1 T (feasible) same bending power as en electric field of 3⋅108 V/m (NOT feasible)

– FB is by far the most effective in order to change the particle direction Acceleration techniques: DC field

• The s imp les t acce lera tion me tho d: DC vo ltage

• Energy kick: ΔE=qV

• Can accelerate particles over many gaps: electrostatic accelerator

• Problem: breakdown voltage at ~10MV

• DC field still used at start of injector chain Acceleration techniques: RF field

• Oscill a ting RF (ra dio-f)fildfrequency) field

• “Widerøe accelerator”, after the pioneering work of the Norwegian Rolf Widerøe (b roth er of th e avi ator Vi ggo Wid erøe )

• Particle must sees the field onlyyg when the field is in the accelerating direction

– Requires the synchronism condition to hold: Tparticle =½TRF L = (1/ 2)vT

• Problem: high power loss due to radiation Principle of phase focusing

...what happens to particles with energies slightly off the nominal values...? Acceleration techniques: RF cavities

• Electromagnetic power is stored in a resonant volume instead of being radiated

• RF power feed into cavity, originating from RF power generators, like Klys trons

• RF power oscillating (from magnetic to electric energy), at the didfdesired frequency

• RF cavities requires bunched beams (as opposed to coasting beams ) – particles located in bunches separated in space Acceleration techniques: Pill-Box cavity

• Ideal cyli nd ri cal cavit y:

• Solution for E and H are oscillating modes (of increasing frequency)

• The fundamental mode normally used for acceleration is named TM010 with the following features:

–Ez is constant in space along the axis of acceleration, z, at any instant – λ010 = 2.6a, l < 2 a

• Acceleration efficiency ωof cavity depends on the transit-time factor – Ratio of “actual energy gain”, versus “energy gain if the field was constant in time“ eEˆ cos( t)dz ΔE ∫ z eVˆT sin(θ / 2) θ ωl T = var = = = , = Δ ˆ θ Econst eEˆ dz eV / 2 v ∫ z – Example: l = λ / 2 gives θ=π and T=0.64 From pill -box to real cavities

(from A. Chao)

LHC cavity module ILC cavity Why circular accelerators?

• Technological limit on the electrical field in an RF cavity (breakdown)

• Gives a limited ΔE per distance

• ⇒ Circular accelerators, in order to re-use the same RF cavity

• This requires a bending field FB in order to follow a circular trajectory (later slide) The synchrotron

• Acce lera tion i s perf ormed b y RF cav ities

• (Piecewise) circular motion is ensured by a guide field FB

•FB : Bending magnets with a homogenous field ρ 2 ρ v 1 qB 1 − B[T ] • In the arc section: F = m ⇒ = ⇔ [m 1] ≈ 0.3 B p ρ p[GeV / c]

•RF freqqyuency must sta y locked to the revolution fre qypquency of a particle (later slide)

• Almost all ppypresent day particle accelerators are s ynchrotrons Digression: other accelerator types

• CltCyclotron: – constant B field – constant RF field in the gap increases energy – radius increases proportionally to energy – limit: relativistic energy, RF phase out of synch – In some respects simpler than the synchrotron, and often used as medical accelerators

•Synchro-cyclotron – Cyclotron with varying RF phase

• Betatron – Acceleration induced by time-varying magnetic field

• The synchrotron will be the only type discussed in this course Particle motion

• We separate the particle motion into: – longitudinal motion: motion tangential to the reference trajectory along the

accelerator structure , u s

– transverse motion: degrees of freedom orthogonal to the reference trajectory,

ux, uy

•us, ux, uy are unit vector in a moving coordinate system, following the particle ? Part 3

LidildiLongitudinal dynamics and acceleration

LitdilDidLongitudinal Dynamics: degrees o ffdtf freedom tangen tiltthftial to the reference tjttrajectory

us: tangential to the reference trajectory RF acceleration

= ˆ ω • We assume a cavity with an oscillating RF-field: Ez Ez sin( RF t)

• In this section we neglect the transit-transit factor – we assume a field constant in time while the particle passes the cavity

• Work done on a particle inside cavity: W = Fdz = q E dz = q Eˆ sin(ω t)dz = qVˆ sin(ω t) ∫ ∫ z ∫ z RF RF Synchrotron with one cavity

• The energy kick of a particle, ΔE, depends on the RF phase seen, φ Δ = = ˆ ω = ˆ φ E W qV sin( RF t) qV sin • We define a “synchronous particle”, s, which always sees the same phase φ s passihing the cav ity ω ω ⇒ RF =h rs ( h: “harmonic number” )

• E.g. at constant speed, a synchronous particle circulating in the φ synchrotron, assuming no losses in accelerator, will always see s=0 Non-synchronous particles

φ Δ • AhA synchronous par tilPticle P1 sees a phase s and ge t a energy ki c k Es

φ= φ −δ • A particle N1 arriving early with s will get a lower energy kick

φ= φ +δ • A particle M1 arriving late with s will get a higher energy kick

• RbihthbhithhbfRemember: in a synchrotron we have bunches with a huge number of particles, which will always have a certain energy spread! Frequency dependence on energy

• IdtIn order to see th thfftftl/hihe effect of a too low/high ΔEdttdthE, we need to study the relation between the change in energy and the change in the revolution frequency (η: "slip factor")

η = dfr / fr dp / p

• Two effects: 1. Higher energy ⇒ higher speed (except ultra-relativistic) βc f = r 2πR

2. Higher energy ⇒ larger orbit “Momentum compaction” Momentum compaction

• Increase i n energy /mass will lea d to a larger or bit

dR / R • We define the “momentum compaction factor” as: α = dp / p

α α=< • is a function of the transverse focusing in the accelerator, Dx> / R – ⇒ α is a well defined quantity for a given accelerator Calculating η

• LithidifftitiiLogarithmic differentiation gives: β β dfr = dβ − dR fr β R β dp d 2 dβ = (1+ β ) = γ 2 p 1− 2 β df / f 1 ⇒η = r r = −α dp / p γ 2

• For a momentum increase dp/p: η – >0: velocity increase dominates ( fr increases ) η – <0: circumference increase dominates ( fr decreases ) Phase stability

η • >0: velocity increase dominates, fr increases

φ • Synchronous particle stable for 0º< s<90º φ= φ −δ – A particle N1 arrivinggy early with s will gggy,et a lower energy kick, and arrive relatively later next pass φ= φ +δ – A particle M1 arriving late with s will get a higher energy kick, and arrive relatively earlier next pass

η< φ • 0: stability for 90º< s<180º

• η=0 is called transition. When the syyggy,nchrotron reaching this energy, the φ 180−φ RF phase needs to be switched rapidly from s to s Longitudinal phase-space

(from A. Chao)

δ = Δp/p

Phase-space for a harmonic oscillator: H = p2 / 2 + x2 / 2

Longitudinal phase-space showing the synchrotron motion

Synchrotron motion: "particles rotate in the phase-space" Synchrotron oscillations

• Ana lys is o f sma ll amp litu de osc illa tions g ives:

• As result of the ppygyphase stability, the energy and phase will oscillate, resultin g in longitudinal synchrotron oscillations:

φ + Ω 2 ηωφ −φ = && s ( s ) 0, ˆ φ Ω 2 = h rseV cos s S π 2 Rs ps

•Equation of an Harmonic Oscillator

• For derivations, please refer to [Wille2000] Synchrotron: energy ramping

• We now wi sh ht to ramp up th e energy of fth the parti tilcle. Wh Whtdat do we do ?

• Each time the synchronous particle passes the cavity it will receive a momentum kick: Δ = Δ = ˆ φ π p ( E) / v qV (sin s )(2 Rfs ) • We want the synchronous particle to on the same trajectory (reference trajectory) regardless of particle energy. Therefore, we require the field and the particle momentum to increase proportionally: p& = qρB& Δ = ˆ φ π = ρ • Combining gives: pffs p& ⇒ qV sin s (1/ 2 R) q B&

• Thus, for a given dB/dt profile the synchronous phase is given as: 2πRqρB& φ = arcsin s qVˆ φ • For a given B’ , the synchronous particles will, by definition, see the phase s Summary: longitudinal dynamics for a synchrotron

• StSummary: to ramp up th thihte energy in a synchrotron

– Simply ramp up the magnetic field

– With the (automatic) RF frequency modulation the synchronous particle will stay on the reference orbit

– Due to the phase-stability, the particles in the phase-space vicinity of the synchronous particle will be captured by the RF and will also be accelerated at the same ra te, un dergo ing sync hro tron osc illa tions ? Part 4

Transverse dynamics

Transverse dyygnamics: degrees of freedom ortho gonal to the reference tra jyjectory

ux: the horizontal plane

uy: the vertical plane Bending field

• Circu lar acce lera tors: de flec ting forces are nee de d r r r r r r F = q(E + v × B) = FE + FB

• Circular accelerators: piecewise circular orbits with a defined bending radius ρ – Straight sections are needed for e.g. particle detectors – In circular arc sections the magnetic field must provide the desired bending radius: 1 = eB ρ p • For a constant particle energy we need a constant B field ⇒ dipole magnets with homogenous field

• In a synchrotron, the bending radius,1/ρ=eB/p, is kept constant during acceleration (last section) The reference trajectory

– We need t o st eer and f ocus th e b eam, k eepi ng all par tic les c lose to the re ference orbit

Reference trajectory

ρ

Dipole magnets to steer Focus?

homogenous field or cosθ distribution Focusing field: quadrupoles • Quadrupole magnets gives linear field in x and y:

Bx = -gy

By = -gx

• However, forces are focusing in one plane and defocusing in the orthogonal

plane: Fx = -qvgx (focusing)

Fy = qvgy (defocusing)

Alternating gradient scheme, leading to betatron oscillations The Lattice

• AltiAn accelerator is compose dfbditfid of bending magnets, focusing magne tdts and non-linear magnets (later)

• The ensemble of magnets in the accelerator constitutes the “accelerator lattice” Stability of a FODO structure

• There are li m itions to the achi eva ble focus ing e ffec t; too s hor t focal leng th will give overfocusing, and an unstable trajectory:

stability ⇒ f > l / 4 Example: lattice components Equations of motion: coordinates

• CditCoordinate sys tem:

• x, y are small deviations from the reference trajectory – x: deviations in the horizontal plane – y: deviations in the vertical plane

•r = ρ + x Linear equations of motion I

• FlilhitthttiftiFrom classical mechanics we get the exact equaθ tions of motion : 2 m(&x&− r & ) = −erθ&(B − gx) m&y& = −erθ&gy • Preferred: x(s) instead of x(t), x’(s) is the slope

2 θ = ≈ = d x ds 2 ≈ 2 ′′ •Approximation: r & vθ v ⇒ &x&(t) ( ) v x (s) ds2 dt 1 e • Equations of motions become: x′′ = − (B − gx) r mv eg y′′ = − y mv Linear equations of motion II

ρ 1 1 1 x 1 1 1 Δp • AitiApproximations: = ≈ ρ (1− ), = ≈ (1− ), neglecting 2nd o. t. + ρ + Δ r x p p0 p p0 p0

• We use the quadrupole strength: k = eg / p

• Basic linear trajectory equations: 1 1 Δp x′′ − (k − )x = x ρ 2 ρ p0 y''+ky = 0

– k: normalized quadrupole strength –1/ρ: normalized dipole strength Δ – p0 is the reference momentum, p is the momentum deviation Mathematical description

• The linearized deviations from the reference orbit can be described by Hill's equation x′′ + K(s)x = 0 • no field: K(s)=0 • inside a dipole K(s) = 1/ ρ2 • inside a quadrupole K(s)=+\-k

x′′ + Kx = 0 εβ x′′ + K(s)x = 0 ⇓ φ⇓ φ s = + φ = dt x(s) = Asin( K s +φ ) x(s) (s) sin( (s) 0 ), (s) 0 ∫s 0 β Particle motion: general solutions of Hill' s equation

• WhWe have cal cu ltdlated par tilticle mo tiftion for a silingle partic le by s tu dy ing the transfer matrices M(s). We now want to find characteristics of the general aspects of the motion

• Solution of Hill’s equation with K(s) =K → harmonic oscillator x′′ + Kx = 0 ⇓ = +φ x(s) Asin( K s 0 ) • The general solution of εβ Hill ’ s equation with is: x′′ +φK(s)x = 0 φ ⇓ φ β(s): the "beta s function" = + = dt x(s) (s) sin( (s) 0 ), (s) ∫s 0 β • Oscillating solution, but with amplitude and phase-advance dependent on s ! – a “quasi-harmonic” oscillator The transverse beam size

• A very important parameter – Vacuum chamber – Interaction point and luminosity

• The transverse beam size is given by the envelope of the particles: E(s) = εβ (s)

La ttice Beam quality The beta function, β

• Twiss parameter β(s), ”the beta function ”, defines the envelope for the solutions of Hill’s, and thus envelope for the particle motion (Δp=0)

• IFODOIn a FODO structure th hbfiie beta function is at maxi mum i ihiddlfhFn the middle of the F quadrupole and at minimum in the middle of the D quadrupole

• NB: Even if beta function is periodic, the particle motion itself is in general not periodic (after one revolution the initial condition φ0 is altered)

• The beta function should be kept at minimum, β∗, at interaction points to maximize the luminosity Example: motion in a FODO lattice

(From CAS 1992) Transverse phase-space

• The phase-space of the horizontal β plane is spanned by the two coordinates [x, x’] φ = + φ x(s) A (s) sin( (s) 0 ) ⇓ β A φ β ' φ x'(s) = (cos( (s) + φ ) + sin( (s) + φ )) 0 2 0 • ⇒ For a fixed pp,,oint, s, on the accelerator [ [,]x, x’] is a p arametric representation of an ellipse

– envelope: E(s) = ε β ε – divergence: A(s) = γ

• For each turn the particle moves around the ellipse according to its tune (non-integer part)

• The shape of the ellipse depends β(s) and β'(s), and thus the position along the ring εβ Emittance φ = +φ x(s) (s) sin( (s) 0 ), • The sol uti on of Hill' s for a g iven par tic le is de term ine d by in itia l con ditions

[x0, x'0]. An ideal particle will have [x0, x'0] = [0, 0] and x(s) = 0

• A given beam consists of particles of various amplitude and angle. At a certain point s, the particles will fill the phase-space ellipse

• As long as a particle is inside the phase-space ellipse, it will remain there, and the area of the ellipse is constant (Liouville's theorem) area = πε = constant • ε is called the beam emittance (horizontal / vertical) – very important parameter for beam quality

• Small emittance strongly desired: – Keep beam envelope and beam divergence small – Keep luminosity high RMS transverse beam size

• In reality: what is often quoted is the RMS emittance, and the RMS beam size

ε • RMS em ittance rms: resulting p hase- space ellipse contains one σ of particles

σ = ε β • RMS beam siz e: (s) rms (s)

Beam quality Lattice Conclusion: transverse dynamics

• WhWe have now st tdidthtudied the transverse op tics o f a c ircu lar acce lera tor an d we have had a look at the optics elements, – the dipole for bending – the quadrupole for focusing – (sextupole for chromaticity correction – not discussed here)

• All optic elements (+ more) are needed in a high performance accelerator, like the LHC ? Intermezzo Norske storheter innen akseleratorfysikk

Rolf Wideröe Bjørn Wiik Professor og direktør ved Pioneer både for Europas nest største betatronprinsippet og for Odd Dahl akseleratorsenter (DESY i lineære akseleratorer Hamburg) Leder av CERN PS prosjektet Kjell Johnsen (en viktig del av LHC- komplekset den dag i dag) Involvert i en rekke CERN- prosjekter, leder av ISR og CERN's gruppe for akseleratorforskning Case: LHC LHC LHC: wrt . to earlier slides

• proton-proton co llisi ons ⇒ two vacuum chambers, with opposite bending field

• RF cavities ⇒ bunched beams

• Synchrotron with alternating-gradient focusing

• Supercond ucti ng l atti ce magnet s and supercondtiRFitiducting RF cavities

• Regular FODO arc-section with sextupoles for chromaticity correction

• Proton chosen as particle type due to low synchrotron radiation

• Magnetic field-strength limiting factor for particle energy LHC injector system

• LHC is respons ible for acce lera ting protons from 450 GeV up to 7000 GeV

• 450 GeV protons injected into LHC from the SPS

• PS injects into the SPS

• LINACS injects into the PS

• The protons are generated by a Duoplasmatron Proton Source LHC layout

• circumference = 26658. 9 m

• 8 interaction points, 4 of which contains detectors where the beams intersect

• 8 straight sections, containing the IPs, around 530 m long

• 8 arcs with a regular lattice structure, containing 23 arc cells

• Each arc cell has a FODO structure, 106.9 m long LHC beam transverse size

σ = εβ ≈ arc typ 0.3mm

σ = εβ * ≈ μ IP 17 m

β β beta in drift space: ≈ 180m, * = 0.55m,ε ≈ 0.5nm × rad typ β(s) = β* + (s-s*)2 / β∗ LHC cavities

• Superconducting RF cavities (standing wave, 400 MHz) • Each beam: one cryostats with 4+4 cavities each • Located at LHC point 4 LHC main parameters at collision energy

Particle type p, Pb

Proton energy Ep at collision 7000 GeV Peak luminosity (ATLAS, 10 x 1034 cm-2s-1 CMS) Circumference C 26 658.9 m BdidiBending radius ρ 2804. 0 m

RF frequency fRF 400.8 MHz 11 # particles per bunch np 1.15 x 10

# bunches nb 2808 ? Part 5

Synchrotron radiation 1) Synchrotron radiation

• Charged particles undergoing acceleration emit electromagnetic radiation

• Main limitation for circular electron machines – RF power consumption becomes too high

• The main limitation factor for LEP... – ...the main reason for building LHC !

• However, synchrotron radiations is also useful (see later slides) Show RAD2D here

(anim) Characteristic of SR: power Characteristics of SR: distribution

• Elec tron res t-fditiditibtd"Htframe: radiation distributed as a "Hertz-dipo le "

dP S ∝ sin2ψ dΩ

• Relativist electron: Hertz-dipole distribution in the electron rest-frame, but transformed into the laboratory frame the radiation form a very sharply peaked light-cone Characteristics of SR: spectrum

• BdBroad spec t(dthtltra (due to short pulses as seen by an observer) • But, 50% of power contained within a well defined "critical frequency"

Summary: advantages of Synchrotron Radiation 1. Very high intensity 2. Spectrum that cannot be covered easy with other sources 3. Critical frequency easily controlled Typical SR centre

Accelerator + Users Some applications of Synchrotron Radiation: •material/molecule analysis (UV, X-ray) •crystallography •Archaeology ? References

• Bibliograp hy: – CAS 1992, Fifth General Accelerator Physics Course, Proceedings, 7-18 September 1992 – LHC Design Report [online] – K. Wille, The Physics of Particle Accelerators, 2000

• Other references – USPAS resource site, A. Chao, USPAS January 2007 – CAS 2005, Proceedings (in-print), J. Le Duff, B, Holzer et al. – O. Brüning: CERN student summer lectures – N. Pichoff: Transverse Beam Dynamics in Accelerators, JUAS January 2004 – U. Am aldi, presentation on Hadron therapy at CERN 2006 – Various CLIC and ILC presentations – Several figures in this presentation have been borrowed from the above references, thanks to all!