The Physics of Accelerators C.R
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The Physics of Accelerators C.R. Prior Rutherford Appleton Laboratory and Trinity College, Oxford Contents Basic concepts in the study of Particle Accelerators Methods of acceleration Linacs and rings Controlling the beam Confinement, acceleration, focusing Electrons and protons Synchrotron radiation Luminosity Pre-requisites Basic knowledge for the study of particle beams: Applications of relativistic particle dynamics Classical theory of electromagnetism (Maxwell’s equations) More advanced studies require Hamiltonian mechanics Optical concepts Quantum scattering theory, radiation by charged particles Computing ability Applications of Accelerators Based on directing beams to hit specific targets or colliding beams onto each other production of thin beams of synchrotron light Particle physics structure of the atom, standard model, quarks, neutrinos, CP violation Bombardment of targets used to obtain new materials with different chemical, physical and mechanical properties Synchrotron radiation covers spectroscopy, X-ray diffraction, x-ray microscopy, crystallography of proteins. Techniques used to manufacture products for aeronautics, medicine, pharmacology, steel production, chemical, car, oil and space industries. In medicine, beams are used for Positron Emission Tomography (PET), therapy of tumours, and for surgery. Nuclear waste transmutation – convert long lived nucleides into short-lived waste Generation of energy (Rubbia’s energy amplifier, heavy ion driver for fusion) Basic Concepts I Speed of light c = 2.99792458×108 msec−1 2 2 Relativistic energy E = mc = m0γ c Relativistic momentum p = mv = m0γ β c − 1 v ⎛ v2 ⎞ 2 1 β = γ = ⎜1− ⎟ = ⎜ 2 ⎟ c ⎝ c ⎠ 1− β 2 E 2 = p2 + m 2c2 E-p relationship c2 0 ultra − relativistic particles β ≈ 1, E ≈ pc 2 2 Kinetic energy T = E − m0c = m0c (γ −1) Equation of motion under Lorentz force r dp r d r r r r = f ⇒ m ()γ v = q(E + v ∧ B) dt 0 dt Basic Concepts II Electron charge e =1.6021×10−19 Coulombs Electron volts 1eV = 1.6021×10-19 joule mc2 m γ c2 Energy in eV E[]eV = = 0 e e Energy and rest mass 1eV c2 =1.78×10−36 kg 2 -31 Electron m0 = 511.0keV c = 9.109×10 kg 2 -27 Proton m0 = 938.3MeV c =1.673×10 kg 2 -27 Neutron m0 = 939.6MeV c =1.675×10 kg Motion in Electric and Magnetic Fields r dp r r r Governed by Lorentz force = q[]E + v × B dt 2 r 2 2 2 4 E = p c + m0 c A magnetic field does not dE dpr alter a particle’s energy. ⇒ E =c2pr ⋅ Only an electric field can dt dt do this. 2 2 dE qc r r r r qc r r ⇒ = p ⋅()E + v ∧ B = p ⋅ E dt E E Acceleration along a uniform electric field (B=0) z ≈ vt ⎫ ⎪ eE parabolic path for v << c x ≈ t 2 ⎬ ⎪ 2mγ 0 ⎭ Behaviour under constant B-field, E=0 Motion in a uniform, constant magnetic field Constant energy with spiralling along a uniform magnetic field m γ v2 0 = qvB ⇒ ρ m γ v (a) ρ = 0 qB v qB (b) ω = = ρ m0γ p qBc 2 v ρ = ω = = qB E ρ Method of Acceleration: Linear Simplest example is a vacuum chamber with one or more DC accelerating structures with the E- field aligned in the direction of motion. Limited to a few MeV To achieve energies higher than the highest voltage in the system, the E- fields are alternating at RF cavities. Avoids expensive magnets No loss of energy from synchrotron SLAC linear accelerator radiation (q.v.) But requires many structures, limited energy gain/metre Large energy increase requires a long accelerator SNS Linac, Oak Ridge Structure 1: Travelling wave structure: particles keep in phase with the accelerating waveform. Phase velocity in the waveguide is greater than c and needs to be reduced to the particle velocity with a series of irises inside the tube whose polarity changes with time. In order to match the phase of the particles with the polarity of the irises, the distance between the irises increases farther down the structure where the particle is moving faster. But note that electrons at 3 MeV are Structure 2: already at 0.99c. A series of drift tubes alternately connected to high frequency oscillator. Particles accelerated in gaps, drift inside tubes . For constant frequency generator, drift tubes increase in length as velocity increases. Beam has pulsed structure. Methods of Acceleration: Circular George Use magnetic fields to force Lawrence particles to pass through and cyclotron accelerating fields at regular intervals Cyclotrons Constant B field Constant accelerating frequency f Spiral trajectories For synchronism f = nω, which is possible only at low energies, γ~1. Use for heavy particles (protons, deuterons, α-particles). p qBc 2 v ρ = ω = = qB E ρ Methods of Acceleration: Circular Higher energies => relativistic effects => ω no longer constant. p ρ = Particles get out of phase with accelerating fields; qB eventually no overall acceleration. Isochronous cyclotron f = nω Vary B to compensate and keep f constant. For stable orbits need both radial (because ρ varies) and azimuthal B-field variation qBc 2 v Leads to construction difficulties. ω = = E ρ Synchro-cyclotron Modulate frequency f of accelerating structure instead. In this case, oscillations are stable (McMillan & Veksler, 1945) Betatron Particles accelerated by the rotational electric field generated by a time varying magnetic field. r Magnetic r ∂B flux, Φ ∇ ∧ E = − ∂t r r d r r ⇔ E ⋅ dl = − B ⋅ dS r ∫ dt ∫∫ dΦ Generated ⇒ 2π r E = − E-field dt In order that particles circulate at constant radius: p Oscillations about B = − the design orbit qr B-field on orbit is one half of the average B over the circle. This p& E 1 dΦ are called betatron ⇒ B&()r,t = − = − = imposes a limit on the energy that qr r 2π r 2 dt oscillations can be achieved. Nevertheless the 1 1 constant radius principle is attractive ⇒ B()r,t = 2 B dS 2π r ∫∫ for high energy circular accelerators. Methods of Acceleration: Circular Synchrotron Principle of frequency modulation but in addition variation in time of B-field to match increase in energy p ρ = and keep revolution radius constant. qB Magnetic field produced by several bending magnets f = nω (dipoles), increases linearly with momentum. For q=e and high energies: p E Bρ = ≈ so E[GeV] ≈ 0.3 B[T] ρ [m] per unit charge qBc 2 v . ω = = e ce E ρ Practical limitations for magnetic fields => high energies only at large radius e.g. LHC E = 8 TeV, B = 10 T, ρ = 2.7 km Types of Synchrotron Storage rings: accumulate particles and keep circulating for long periods; used for high intensity beams to inject into more powerful machines or synchrotron radiation factories. Colliders: two beams circulating in opposite directions, made to intersect; maximises energy in centre of mass frame. Variation of parameters with time in the ISIS synchrotron: B=B0-B1 cos(2πft) B E ω Fixed Field Alternating Gradient Circular Machines (FFAG) An old idea, dating from 1950’s, given a new lease of life with the development of new magnetic alloy cavities. Field constant in time, varies with radius according to a strict mathematical formula. Wide aperture magnets and stable orbits. High gradient accelerating cavities combine with fixed field for rapid acceleration. Good for particles with short Prototype FFAG, accelerating protons from half-lives (e.g. muons). 50 keV to 500 keV, was successfully built and tested at the KEK laboratory in Japan, 2000. Summary of Circular Machines Machine RF frequency Magnetic Orbit Radius Comment f Field B ρ Cyclotron constant constant increases with Particles out of synch energy with RF; low energy beam or heavy ions Isochronous constant varies increases with Particles in synch, Cyclotron energy but difficult to create stable orbits Synchro-cyclotron varies constant increases with Stable oscillations energy Synchrotron varies varies constant Flexible machine, high energies possible FFAG varies constant in time, increases with Increasingly varies with radius energy attraction option for 21st century designs p qBc 2 v ρ = ω = = qB E ρ Confinement, Acceleration and Focusing of Particles By increasing E (hence p) and B together, possible to maintain a p constant radius and accelerate a ρ = beam of particles. qB In a synchrotron, the confining magnetic field comes from a system of several magnetic dipoles forming a closed arc. Dipoles are mounted apart, separated by straight sections/vacuum chambers including equipment for focusing, acceleration, injection, extraction, collimation, experimental areas, vacuum pumps. ISIS dipole Ring Layout Mean radius of ring Dipoles R > ρ Injection e.g. CERN SPS R = 1100 m, ρ = 225 m R.F. Can also have large machines with a large Collimation number of dipoles each of small bending angle. e.g. CERN SPS Focusing elements 744 dipole magnets, 6.26 m long, angle θ = 0.48o Extraction Dipoles Ring Concepts 2π R L Important concepts in rings: τ = ≈ v c Revolution period τ Revolution frequency ω ω 1 c = ≈ If several bunches in a machine, introduce RF cavities 2π τ L in straight sections with fields oscillating at a multiple of hc the revolution frequency. ωrf = hω ≈ h is the harmonic number. L Energy increase ∆E when particles pass RF cavities ⇒ p can increase energy only so far as can increase B-field ρ = qB in dipoles to keep constant ρ. p Bρ = q Effect on Particles of an RF Cavity Cavity set up so that particle at the centre of bunch, called the synchronous particle, acquires just the right amount of energy. Particles see voltage V0 sin 2πωrf t = V0 sinϕ(t) In case of no acceleration, synchronous particle has ϕs = 0 Particles arriving early see ϕ < ϕs Particles arriving late see ϕ > ϕs energy of those in advance is decreased relative to the synchronous particle and vice versa.