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) ru rqu zu r q z dv (r rq 2 )u (2rq rq)u zu dt r q z v B rqBzur (zBr rBz )uq rqBr u z rq (B0 gx)ur (zgz 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) x0S(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 L1 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 p Pseudo-harmonic solution x(s) (s) cos((s) ) (s) 0 p s ds 1 p 2(s) Betatron Dispersion