The Physics of Accelerators : The Physics of Accelerators C.R. Prior
Rutherford Appleton Laboratory
and Trinity College, Oxford
Contents : Contents Basic concepts in the study of Particle Accelerators (including relativistic effects)
Methods of acceleration
Linacs and rings
Controlling the beam
Confinement, acceleration, focusing
Animations
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 Pre-requisites
Applications of Accelerators : Based on
possibility of directing beams to hit specific targets
production of thin beams of synchrotron light
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. Applications of Accelerators
Basic concepts : Energy of a relativistic particle
For v<
Basic concepts : Basic concepts Momentum
Connection between E and p:
so for ultra-relativistic particles
Units: 1eV = 1.6021 x 10-19 joule.
E[eV] = where e = electron charge
Rest energies: Electron E0 = 511 keV
Proton E0 = 938 MeV
Velocity v. Kinetic energy : Velocity v. Kinetic energy At low energies, Newtonian mechanics may be used; relativistic formulae necessary at high energies
Motion in electric and magnetic fields : Motion in electric and magnetic fields Governed by Lorentz force
Acceleration along a uniform electric field
Constant energy with spiralling along a uniform magnetic field
Methods of Acceleration : Methods of Acceleration Linear accelerator (linac)
Vacuum chamber with one or more DC accelerating structures with E-field aligned in direction of motion.
Avoids expensive magnets
no loss of energy from synchrotron radiation
but many structures, limited energy gain/metre
large energy increase requires long accelerator Part of the Fermilab linear accelerator
Methods of Acceleration : Methods of Acceleration Circular machines
Use magnetic fields to force particles to pass through accelerating fields at regular intervals.
Cyclotrons
Constant B field
Constant accelerating frequency f
Spiral trajectories
For synchronism f=nw, which is possible only at low energies, ~1.
Use for heavy particles (protons, deuterons, -particles).
Methods of Acceleration : Methods of Acceleration Isochronous Cyclotron
Higher energies => relativistic effects => w no longer constant.
Particles get out of phase with accelerating fields; eventually no overall acceleration.
Solution: vary B to compensate and keep f constant.
Thomas (1938): need both radial (because varies) and azimuthal B-field variation for stable orbits.
Construction difficulties.
Methods of Acceleration : Methods of Acceleration Synchro-cyclotron
Modulate frequency f of accelerating structure instead.
McMillan & Veksler (1945): oscillations are stable.
Betatron (Kerst 1941)
Particles accelerated by rotational electric field generated by time varying B :
Theory of “betatron oscillations”
Overtaken by development of the synchrotron
Methods of Acceleration : Methods of Acceleration Synchrotron
Principle of frequency modulation but in addition variation in time of B-field to match increase in energy and keep revolution radius constant.
Magnetic field produced by several dipoles, increases linearly with momentum. At high energies: Br = p/q E/qc
E[GeV]=0.29979 B[T] r [m] per unit charge.
Limitations of magnetic fields => high energies only at large radius
e.g. LHC E = 8 TeV, B = 10 T, r = 2.7 km
Types of Synchrotron : 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.
Example of variation of parameters with time in a synchrotron
Confinement, Acceleration and Focusing of Particles : Confinement, Acceleration and Focusing of Particles By increasing E (hence p) and B together, possible to maintain cyclotron at constant radius and accelerate a beam of particles.
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, experimental areas, vacuum pumps. ISIS dipole
Ring Layout : Ring Layout Mean radius of ring R>
e.g. CERN SPS
R=1100m, =225m
Can also have large machines with a large number of dipoles each of small bending angle.
e.g. CERN SPS
744 magnets, 6.26m long, angle =0.48o
Acceleration : Acceleration A positive charge is accelerated when crossing uniform E-fields. Simple model of an RF cavity: uniform field between parallel plates of a condenser containing small holes to allow for the passage of the beam
Acceleration : Acceleration Important concepts in rings:
Revolution period
Revolution frequency
If several bunches in machine, introduce RF cavities in straight sections with oscillating fields
h is the harmonic number.
Energy increase E when particles pass RF cavities can increase energy only so far as can increase B-field in dipoles to keep constant .
Effect on Particles of an RF Cavity : Effect on Particles of an RF Cavity Cavity set up so that centre of bunch, called the synchronous particle, acquires just the right amount of energy.
Particles see voltage V0sin2rft=V0sin (t)
In case of no acceleration, synchronous particle has s = 0
Particles arriving early see <s
Particles arriving late see >s
energy of those in advance is decreased wrt synchronous particle and vice versa.
To accelerate, make 0 < s< so that synchronous particle gains energy E=qV0sins
Bunching Effect
Limit of Stability : Not all particles are stable. There is a limit to the stable region (the separatrix or “bucket”) and, at high intensity, it is important to design the machine so that all particles are confined within this region and are “trapped”. Limit of Stability
Weak (Transverse) Focusing : Weak (Transverse) Focusing Particles injected horizontally into a uniform, vertical, magnetic field follow a circular orbit
Misalignment errors, difficulty in perfect injection cause particles to drift vertically and radially and to hit walls.
severe limitations to a machine
Require some kind of stability mechanism
Vertical stability requires negative field gradient. i.e. horizontal restoring force is towards the design orbit. Overall stability condition: Weak focusing
Strong (Transverse)Focusing - Alternating Gradient Principle : Strong (Transverse)Focusing - Alternating Gradient Principle A sequence of focusing-defocusing fields provides a stronger net focusing force.
Quadrupoles focus horizontally, defocus vertically or vice versa. Forces are proportional to displacement from axis.
A succession of opposed elements enable particles to follow stable trajectories, making small oscillations about the design orbit.
Technological limits on magnets are high.
Focusing Elements : Sextupoles are used to correct longitudinal momentum errors. Focusing Elements
Thin Lens Analogy of AG Focusing : Thin Lens Analogy of AG Focusing
Examples of Transverse Focusing : Examples of Transverse Focusing
Ring Studies : Ring Studies Typical example of ring design
basic lattice
beam envelopes
phase advances
Importance of matching
phase space non-linearities
Poincaré maps
Electrons and Synchrotron Radiation : Electrons and Synchrotron Radiation Particles radiate when they are accelerated, so charged particles crossing the magnetic dipoles of a lattice in a ring (centrifugal acceleration) emit radiation in a direction tangential to their trajectory.
After one turn of a circular accelerator, total energy loss by synchrotron radiation is
Proton mass : electron mass = 1836. For the same energy and radius,
Synchrotron Radiation : Synchrotron Radiation In electron machines, strong dependence of radiated energy on energy E.
Losses must be compensated by cavities
Technological limit on maximum energy a cavity can deliver upper band for electron energy in an accelerator:
Better to have larger accelerator for same power from RF cavities at high energies.
To reach twice LEP energy with same cavities would require a machine 16 times as large.
e.g. LEP with 50 GeV electrons, =3.1 km, circumference =27 km:
Energy loss per turn is 0.18 GeV per particle
energy is halved after 650 revolutions, in a time of 59 ms.
Synchrotron Radiation : Synchrotron Radiation Radiation is produced within a light cone of angle
For electrons in the range 90 MeV to 1 GeV, is in the range 10-4 - 10-5 degs.
Such collimated beams can be directed with high precision to a target - many applications, for example, in industry.
Luminosity : Luminosity Measures interaction rate per unit cross section - an important concept for colliders.
Simple model: Two cylindrical bunches of area A. Any particle in one bunch sees a fraction N/A of the other bunch. (=interaction cross section). Number of interactions between the two bunches is N2 /A.
Interaction rate is R=f N2 /A, and
Luminosity
CERN and Fermilab p-pbar colliders have L ~ 1030 cm-2s-1. SSC was aiming for L ~ 1033 cm-2s-1
Reading : Reading E.J.N. Wilson: Introduction to Accelerators
S.Y. Lee: Accelerator Physics
M. Reiser: Theory and Design of Charged Particle Beams
D. Edwards & M. Syphers: An Introduction to the Physics of High Energy Accelerators
M. Conte & W. MacKay: An Introduction to the Physics of Particle Accelerators
R. Dilao & R. Alves-Pires: Nonlinear Dynamics in Particle Accelerators
M. Livingston & J. Blewett: Particle Accelerators