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Transcript
Introduction to Collective Effects in Particle Accelerators Part 1: Space Charge and Scattering Effects Andy Wolski University of Liverpool Outline • Space charge Principles of space-charge forces The envelope equations Space-charge tune shifts Longitudinal space-charge effects • Beam-beam effects Beam-beam tune shifts • Scattering effects Touschek effect and Touschek lifetime Intrabeam scattering Collective Effects in Accelerators: Part 1 2 Learning Objectives By the end of this lecture, you should be able to: • describe some of the main effects of space-charge on transverse and longitudinal beam dynamics, including betatron tune shifts, emittance growth, and longitudinal instability; • explain what is meant by the “perveance” of a beam, and determine, with reference to the envelope equation, whether the transport of a beam with given parameters is dominated by emittance or space-charge effects; • explain the significance of the beam-beam interaction in colliders, and estimate the beam-beam tune shift for the collision of beams with a given set of parameters; • describe some of the effects of scattering between particles within a bunch, including emittance growth (intrabeam scattering), and particle loss (Touschek effect), and explain how these effects depend on parameters such as beam energy, beam size, and bunch charge. Collective Effects in Accelerators: Part 1 3 Space-charge forces • Charged particles in an accelerator generate electromagnetic fields that can be calculated from Maxwell’s equations: E 0 1 E B 2 0 J c t • The fields affect the motion of other particles in the accelerator, through the Lorentz force: F q E vB Collective Effects in Accelerators: Part 1 4 Fields around ultra-relativistic charges • Solving Maxwell’s equations for an ultra-relativistic charge moving in a straight line shows that the fields become “flattened” in a plane perpendicular to the direction of motion of the charge. • For two (like) charges moving parallel to each other: • the electric force pushes the charges apart; • the magnetic force pulls the charges together. • At ultra-relativistic energies ( >> 1), the electric and magnetic forces almost cancel. Collective Effects in Accelerators: Part 1 5 Space-charge forces in a uniform beam • The simplest realistic case in an accelerator is a continuous beam with uniform density within a circular cross-section: radius a velocity v nq particles (with charge q) per unit length • Each charge experiences a net radial force: q 2 nq r Fr 2 0 2 a 2 Collective Effects in Accelerators: Part 1 6 Equation of motion with space charge Consider motion in the horizontal transverse (x) direction. The equation of motion for a particle in the beam is: d 2x K 2x 2 ds a where s is the distance along the beamline, and K is the perveance of the beam: 2I K 3 3 b Ic b = v/c, I is the beam current, and Ic is the critical current: 4 0 mc3 Ic q For electrons, Ic is the Alfvén current, IA 17.045 kA. Collective Effects in Accelerators: Part 1 7 Equation of motion with space charge • Notice that the space-charge force varies linearly with transverse position in the beam. This applies only to the special case of a uniform transverse distribution. • For a uniform transverse distribution, space-charge forces act like forces from quadrupole magnets, except that the spacecharge forces are simultaneously defocusing in x and y. • It follows that important properties, in particular conservation of emittance, apply to beam transport in the presence of (linear) space-charge forces. Collective Effects in Accelerators: Part 1 8 The envelope equation • Since we know the equation of motion for individual particles, we can work out the equation of motion for the beam distribution. • The equation of motion for the rms beam size is known as the envelope equation: d 2 x x2 K k1 x 3 0 2 ds x 2 x y where k1 is the quadrupole focusing strength, and: x x 2 and x x 2 x xx 2 2 • Similar equations apply for the vertical (y) motion. Collective Effects in Accelerators: Part 1 9 The envelope equations • So far we have considered only the special case of a beam with uniform charge density within a circular cross-section. • In this case, the horizontal and vertical rms beam sizes evolve according to the envelope equations: d 2 x x2 K k 0 1 x 2 3 ds x 2 x y d 2 y ds 2 y2 K k1 y 3 0 y 2 x y where the emittances x and y are constant. • It turns out that the envelope equations apply to any continuous beam with elliptical symmetry – although in general, the emittances are not conserved. Collective Effects in Accelerators: Part 1 10 Emittance and space charge • Notice that the terms in the envelope equation have a clear physical origin: d 2 x x2 K k1 x 3 0 2 ds x 2 x y quadrupole focusing emittance space charge • If the emittance term is much larger than the space-charge term, then the beam transport is said to be emittance dominated. • If the space-charge term is much larger than the emittance term, then the beam transport is said to be space-charge dominated. Collective Effects in Accelerators: Part 1 11 Beam transport with space charge Example: transport of a beam with emittance 10 mm mrad through a drift space of length 1 m. Initial conditions chosen to minimise the maximum beam size. K = 5×10-4 K = 2×10-4 K=0 Collective Effects in Accelerators: Part 1 12 Space-charge tune shifts • The defocusing effects of space-charge forces lead to changes in the betatron oscillation frequencies. • For a continuous beam with uniform charge density, all particles experience the same frequency shift. • For a bunch in a storage ring, the situation is much more complicated: the space-charge forces are nonlinear the local charge density seen by a particle varies in a way that depends on the amplitude of the synchrotron oscillations of the particle. particles near the centre of a bunch experience a large tune depression particles at the head or tail of a bunch experience almost no tune depression Collective Effects in Accelerators: Part 1 13 Incoherent space-charge tune shifts • Although each particle within a bunch experiences a betatron frequency shift from space-charge forces, the net spacecharge force on a bunch is zero. • This kind of tune shift is sometimes called an incoherent tune shift: it cannot be measured by observing the coherent motion of a bunch of particles. • For particles in a bunch with non-uniform density, there is in addition a spread in the tune shifts for different particles. The vertical tune spread can be estimated from the formula: by K y ds 4 y x y where K is the peak perveance in the bunch, by is the vertical beta function, and the integral is over the full circumference. Collective Effects in Accelerators: Part 1 14 Laslett tune shifts • In addition to direct forces between particles in a bunch, particles experience forces due to image charges in the walls of the vacuum chamber. • The forces from image charges depend on the geometry of the vacuum chamber, and the position of the bunch within the chamber. • Tune shifts resulting from image charges are known as Laslett tune shifts. • In general, there will be coherent Laslett tune shifts as well as incoherent Laslett tune shifts. Collective Effects in Accelerators: Part 1 15 Laslett tune shifts Beam on the mid-plane between infinite plane parallel plates: incoherent tune shift. Collective Effects in Accelerators: Part 1 Beam displaced from the midplane between infinite plane parallel plates: incoherent and coherent tune shifts. 16 Longitudinal space-charge effects • In a continuous beam with uniform charge density (within a given cross-section) the longitudinal fields are zero, by symmetry. • Longitudinal components of the fields occur when there is some variation in the density of particles along the beamline. High particle density E radius a velocity v = bc Low particle density nq(s) particles (with charge q) per unit length Collective Effects in Accelerators: Part 1 17 Longitudinal space-charge fields • The longitudinal components of the fields are related to the transverse components through Maxwell’s equations: this provides one route to calculating the longitudinal fields. B E t C E dl t B dA • In a chamber (of radius b) with perfectly conducting walls, the longitudinal component of the electric field vanishes at the walls. Collective Effects in Accelerators: Part 1 18 Longitudinal space-charge effects The longitudinal electric field is conveniently expressed in terms of the Fourier components of the particle density. nq(s) s radius a velocity v = bc nq ( s, t ) n~q ( )ei ( ks t ) d Es r 0, s, t i qg k ~ ( ) ei ( ks t ) d n q 4 0 b 2c Collective Effects in Accelerators: Part 1 bc b g 1 2 ln a 19 Longitudinal space-charge effects • The main effect of the longitudinal space-charge fields is to change the energy of the particles. • The change in energy of a particle over the distance from s1 to s2 is found by integrating the force on the particle over that distance: s q 2 sz ds ( z ) Es s, t E0 s1 bc E E0 E0 velocity v = bc s1 z reference particle at position s at time t Collective Effects in Accelerators: Part 1 s2 20 Longitudinal space-charge impedance Combining the expressions for the longitudinal electric field and the energy change, we find that in a storage ring, the energy change over one revolution can be written: z q E0 ~ i z c I Z e d || sc ~ where I is the Fourier spectrum of the beam current observed at a fixed location in the ring, and Z|| sc is the longitudinal space-charge impedance, given by: gZ 0 Z|| sc i 2b 2 0 g is the geometry factor, and 0 is the ring revolution frequency. Collective Effects in Accelerators: Part 1 21 Longitudinal space-charge in a storage ring In a synchrotron storage ring, the longitudinal space-charge impedance leads to a change in the synchrotron frequency and the equilibrium longitudinal distribution. Storage ring below transition. equilibrium distribution without space-charge equilibrium distribution with space-charge Collective Effects in Accelerators: Part 1 22 Beam-beam effects • Particles in colliding beams experience electromagnetic forces from the particles in the opposing beam. • For particles moving in opposite directions, the electric and magnetic forces add up rather than tending to cancel out. • Beam-beam forces can be significant even at energies large enough for space-charge effects to be negligible. • For colliding beams of E opposite charge, one of e the principal effects is a F transverse focusing force e at the interaction point, B e+ moving leading to variation in into page; betatron tunes and e- moving out of page. beta functions. Collective Effects in Accelerators: Part 1 23 Beam-beam tune shifts • Beam-beam forces can be strongly nonlinear and result in complicated effects. • However, they are often characterised by the beam-beam parameter, which quantifies the linear kick experienced by a particle as a function of its transverse position: b y dp y y 4 dy y 0 • The transverse kick leads to a (coherent) tune shift of the same size as the beam-beam parameter, e.g. for a Gaussian beam: y y N b re b y 2 y x y Nb is the number of particles per bunch, re is the classical radius (of the electron), by* is the vertical beta function at the interaction point. Collective Effects in Accelerators: Part 1 24 Scattering effects • When discussing space-charge effects, we considered the impact of fields calculated with the approximation of a smooth charge distribution. • In reality, bunches consist of large numbers of “point-like” particles. • The fields close to a single particle can be very much larger than the fields calculated in the approximation of a smooth charge distribution. • Occasionally, two particles within a bunch can approach each other closely enough for the local peaks in the field strengths to have observable effects. Collective Effects in Accelerators: Part 1 25 Touschek effect • A “collision” between two electrons (or positrons) is described as Møller scattering. • In the rest frame of a bunch, the differential cross-section for the scattering process gives the distribution for the deflection angle. • When transformed into the lab frame, even a small deflection angle can lead to a large change in energy deviation. • If the energy deviation of a particle is outside the energy acceptance of the machine, the particle will be lost from the beam: this is the Touschek effect. Collective Effects in Accelerators: Part 1 26 Touschek effect • The rate at which particles are lost from a beam can be calculated from the density of particles in the beam, and the energy acceptance of the storage ring. • In the simplest case (neglecting dispersion and coupling, and assuming non-relativistic motion in the bunch rest frame), the loss rate from the Touschek effect is given by: 2 b 2 0 dN b N cr dt 8 2 x y z 3 b D max where max is the energy acceptance of the ring, and: 2 max b 2 2x b x Collective Effects in Accelerators: Part 1 D e u u 1 u 1 ln du u 2 2 3/ 2 27 Touschek effect Collective Effects in Accelerators: Part 1 28 Touschek lifetime • The loss rate depends on the square of the number of particles in the bunch. • Therefore, the bunch population does not (strictly speaking) decay exponentially; nevertheless, we define the Touschek lifetime: 3 2 b N b cr0 1 1 dN b D 2 T N b dt 8 x y z max • Other effects (in particular, gas scattering) can lead to the loss of particles from the beam in a storage ring. • But in low-emittance electron rings (e.g. in third-generation synchrotron light sources) the Touschek effect is usually the dominant lifetime limitation. • The Touschek lifetime has a strong dependence on the energy acceptance: usually, the goal is to achieve max > about 4%. Collective Effects in Accelerators: Part 1 29 Intrabeam scattering • Even if a Møller scattering event does not result in the loss of the particles involved, there can be a transfer of momentum from the transverse direction to the longitudinal. • If the scattering takes place at a location with non-zero dispersion, then the momentum transfer can lead to an increase in the transverse emittance. • Intrabeam scattering (IBS) leads to emittance increase by a mechanism very similar to quantum excitation by synchrotron radiation. • The main difference between IBS and quantum excitation is that IBS happens wherever the dispersion is non-zero: not just in dipoles. Collective Effects in Accelerators: Part 1 30 Intrabeam scattering • There are two widely-used formalisms for the analysis of intrabeam scattering: Piwinski (Proc. 9th Int. Conf. High Energy Accel., 1974) Bjorken and Mtingwa (Particle Accelerators 13, 115-143, 1983) • Both formalisms lead to expressions for the growth rates of the transverse and longitudinal emittances. Simplified formulae can be derived using various approximations. • In an electron (or positron) storage ring, the IBS growth rates are usually slow compared to the emittance damping rates from synchrotron radiation. • Increases in equilibrium emittance from IBS are usually not observable, except with: high bunch population (> few × 109 particles) low or moderate energy (< few GeV) very low emittance (vertical emittance < few pm) Collective Effects in Accelerators: Part 1 31 Observations of IBS at the KEK-ATF K.L.F. Bane et al, “Intrabeam scattering analysis of measurements at KEK’s Accelerator Test Facility damping ring,” PRST-AB 5, 084403 (2002). Collective Effects in Accelerators: Part 1 32 Summary • Space-charge forces have transverse defocusing effects. The size of the space-charge forces can be characterised by the perveance, which is a function of the beam current and energy. The evolution of the transverse beam size, including focusing, emittance and space-charge, is described by the envelope equations. For a uniform transverse distribution, the space-charge forces are linear, and the emittance is conserved. The Laslett tune shifts describe the effects of image charges in the walls of the vacuum chamber. • Longitudinal variations in particle density lead to changes in particle energy, which can be described by an impedance. • Touschek scattering leads to loss of particle from the beam. • Intrabeam scattering leads to an increase in beam emittance. Collective Effects in Accelerators: Part 1 33