Transverse Phase Space 1. Under linear forces, any x´ x´ particle moves on an ellipse in phase space (x,x´). 2. Ellipse rotates in magnets and shears between magnets, but x x its area is preserved: General equation of ellipse is Emittance β x′2 + 2α x x′ + γ x 2 = ε α, β, γ are functions of distance (Twiss parameters), and ε is a constant. Area = πε. For non-linear beams can use 95% emittance ellipse or RMS emittance ε rms = x′2 − xx ′ 2 x2 (statistical definition) R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Circular 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. p Magnetic field produced by several bending magnets ρ= qB (dipoles), increases linearly with momentum. For q=e and high energies: f = nω p E Bρ = ≈ so E [GeV] ≈ 0.3 B [T] ρ [m] per unit charge . e ce Practical limitations for magnetic fields => high energies qBc 2 v ω= = only at large radius E ρ e.g. LHC: B = 8.36 T, ρ = 2.7 km => E ~ 7 TeV 2 R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Synchrotrons 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 3 3 R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Stability Conditions Recall, the tune is the number of betatron oscillations per turn Provided integer values of the tune are avoided the machine operates reliably Synchrotron motion during acceleration cycle: p=0.3 B R Bending dipoles and focussing quadrupoles carry same (increasing) current The optics – (magnets effectively perform the functions of prisms and lenses) – looks the same Tune stays constant Setting the tune to something sensibly non-resonant means it stays there? Imperfections: Errors in position, current etc, in a magnet means a particle gets the wrong ‘kick’. Over many turns this smears out – if the particle is at different points on its betatron oscillation each time 4 R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Synchrotron Oscillations In principle, synchrotrons or linacs are designed such RF accelerating cavity that the synchronous particle is accelerated continuously. RF accelerating cavities In practice, non- synchronous particles will require acceleration too! Thus, the question of phase Series of accelerating cavities in: stability, or do particles (a) synchrotron and (b) a linac with different energies and phases remain close to the synchronous phase? In order to derive an equation for the passage of non-synchronous particle through the accelerator we consider the time interval τ between passages of two successive modules of accelerating cavities (or sometimes called accelerating stations): τ = L/v 5 where L is the distance between stations and v the particle velocity v: R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Synchrotron Oscillations -cont The fractional change from small changes in L and v is: Δτ ΔL Δv = − τ L v Thus, a particle moving faster than the synchronous (ideal) particle will take less time between accelerating stations and a small error in the circumference will increase the transit time (not appropriate to linacs) The second term is readily expressed as: Δv 1 ⎛ Δp ⎞ = 2⎜ ⎟ v γ ⎝ p ⎠ In a circular accelerator the orbit circumference is slightly larger for those particles having a slightly larger momentum and thus: ΔL 1 ⎛ Δp ⎞ = 2⎜ ⎟ L γt ⎝ p ⎠ Here γt is determined according to the device design (in linacs we expect 1/ γt = 0). We introduce a slip factor (which changes sign at the transition energy γ = γt): 1 1 η= 2 − 2 6 γt γ R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Synchrotron Oscillations –cont. We will investigate the equations Station n Station n+1 of the particle with arbitary phase and energy. En En+1 At the n+1 station the phase is: ψn ψn+1 ψ n +1 = ψ n + ωrf ( τ + Δτ )n +1 Passage of particle through n and n+1 ⎛ Δτ ⎞ accelerating stations (enters nth station = ψ n + ωrf τn +1 + ωrf τn +1 ⎜ ⎟ with energy En and phase ψn) ⎝ τ ⎠ n +1 As the synchronous particle always arrives at the nth station at the same phase, we switch to a variable which is relative to the rf phase (Tn being the time at the entrance to the nth station) φn ≡ ψ n − ωrf Tn Thus in terms of this variable: ⎛ Δτ ⎞ φn +1 + ωrf Tn +1 = φn + ωrf Tn + ωrf τn +1 + ωrf τn +1 ⎜ ⎟ , and as Tn +1 = Tn + τn +1 ⎝ τ ⎠n +1 Where we have used: Δτ/τ = ηΔp/p ⎛ Δτ ⎞ ⎛ Δp ⎞ φn +1 = φn + ωrf τn +1 ⎜ ⎟ = φn + ηωrf τn +1 ⎜ ⎟ We will drop the suffix on ⎝ τ ⎠n +1 ⎝ p ⎠n +1 (ωrfτ)n as is designed into the geometry of the accelerating structure. 7 R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Synchrotron Oscillations –cont. We now consider the deviation in the energy from that of the synchronous energy: ΔE n +1 = ΔE n + eV ( sin φn − sin φs ) , where ΔE n = E n − E s Also as E=γmc 2 and p=γmv it is straightforward to show: Δp c 2 ΔE = 2 p v E Recall, the phase equation is of the form: ⎛ Δp ⎞ φn +1 = φn + ηωrf τn +1 ⎜ ⎟ ⎝ p ⎠n +1 Thus the phase equation and energy equations are: ηωrf τc 2 φn +1 = φn + 2 ΔE n +1 v Es ΔE n +1 = ΔE n + eV ( sin φn − sin φs ) It remains to solve these difference equations numerically……….. 8 R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Phase Stability A few iterations of the difference equations, for initial phase equal to the synchronous phase (φ1 = φs) and for various initial value energy differences ΔE. Particles close to the ideal, synchronous particle, remain close to it. Solution of difference eqns for Particles with larger ΔE depart from the neighbourhood synchrotron motion, with φ1=φs of the ideal particle –and are lost from acceleration Solution of difference eqns for synchrotron motion with various initial φ1 conditions Similarly for a whole range of initial phase (φ1) conditions It is notable that there is a boundary between stable and unstable motion –the separatrix. Two qualitatively different points: See Mathematica notebook for details on how these and other plots are obtained and to explore other regimes 1. Stable fixed point at φ=φs with ΔE = 0 2. Outer edge of separatrix –unstable stable region 9 In general, for a circular accelerator, there are many stable fixed points (spaced by 2π) –three shown. R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Phase Stability –cont. The area in phase defined by the separatrix is known as a “bucket” It is not necessary to fully populate all buckets –particles occupying a bucket are known as bunches For example, when LEP, the electron-positron collider at CERN was in operation, 4 bucket were populated by each species of particle. As the circumference of LEP is 27 km and the bunch spacing is approx 1m, just over 0.1% of buckets are populated. Another interesting circumstance consists of phase space for φs = 0 or π. Here the ideal particle remains un-accelerated and the phase stable region is 2π in extent. These are known as stationary buckets. A particle will be un-accelerated and will remain in these buckets forever –unstable though, as it will continue to undulate in energy and wander away in phase from the ideal particle phase. C.f. the accelerated buckets, in which in this case particles outside the separatrix diverge in both energy and phase and eventually leave the accelerator (into the walls or similar obstacles) Stationary Buckets –for φs = 0 or π 10 R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Phase Stability –diff. eq. We now seek a closed form analytical solution and we replace the small difference changes by differentials –acceptable as the energy changes per station are usually small dφ ηωrf τc 2 dΔE = 2 ΔE, = eV ( sin φ − sin φs ) dn v Es dn Taking the derivative of the first equation allows them to be turned into one differential eq. d 2 φ ηωrf τeVc 2 - Horizontal 2 = 2 ( sin φ − sin φs ) lines denote T dn v Es -Curve is V -here we have assumed dEs/dn, dV/dn ~ 0 . -horizontal lines give T=0 Multiplying by dφ/dn and integrating over dn ⌠ d φ dφ dn − ηωrf τeVc ⌠ sin φ − sin φ dφ dn = 0 2 2 ⎮ 2 ⎮( s) ⌡ dn dn v2 Es ⌡ dn This is in the form: “U =T+U” 1 ⎛ dφ ⎞ ηωrf τeVc 2 2 -where the total “energy” is U ⇒ ⎜ ⎟ + ( cos φ + φ sin φs ) = const -the first term is the KE 2 ⎝ dn ⎠ 2 v Es U -the second is the potential energy T V 11 R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Phase Stability –diff. eq. Combining these two equations: 1 ⎛ dφ ⎞ ηωrf τeVc2 2 dφ ηωrf τc 2 ⎜ ⎟ + ( cos φ + φ sin φs ) = const. = 2 ΔE 2 ⎝ dn ⎠ 2 v Es dn v Es 2v 2 E s eV 2 ( ⇒ ΔE + 2 cos φ + φ sin φs ) = const . ηωrf τc Implicitly, in deriving this equation we have employed dEs/dn ~0 (not used in the difference equations) and so it remains to be seen that this is a faithful representation of the physics… 12 R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Phase Stability –contour sol. Contour of particle motion in longitudinal phase space for φs = π The form obtained earlier, namely, that obtained through solving the difference equations, is verified – here we have stationary buckets Contour of particle motion in longitudinal phase space for φs = 5π/6 Colours represent different contours of various values of the constant: 2v 2 E s eV ΔE + 2 ( cos φ + φ sin φs ) = const ηωrf τc2 13 R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Phase Stability –contour sol. Contour of particle motion in longitudinal phase space for φs = 2π/3 φs = 5π/4 14 R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Phase Stability –tune To investigate the tune we return to the phase equations and consider small excursions about the synchronous phase: φ = Δφ + φs d 2 φ ηωrf τeVc 2 2 = 2 ( sin φ − sin φs ) Here νs is number of synchrotron oscillations, or the dn v Es “tune”. For η<0 (i.e. γ<γt) the motion is stable provided ηωrf τeVc 2 cosφs>0 2 v Es ( sin ( φs + Δφ ) − sin φs ) For η>0 (i.e. γ>γt) the motion is stable provided cosφs<0 For circular accelerators that cross the transition d 2 Δφ ηωrf τeVc 2 energy during acceleration the rf system must ⇒ 2 − 2 cos φs Δφ = 0 perform a phase jump to maintain phase stability dn v Es At the transition (γ=γt) the synchrotron period becomes infinite and no phase focusing occurs. or d 2 Δφ ηωrf τc 2 eV cos φs + ( 2πν s ) Δφ = 0 where νs = − 2 dn 2 4π 2 v 2 E s 2πνs ηωrf c 2 eV cos φs and the angular frequency is = − τ τv 2 E s 15 R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Phase Stability –linac tune For linacs, we replace the slip factor η with -1/γ2 This requires cosφs>0 Hence, as the energy increases the synchrotron frequency approaches zero 1 ωrf τc 2 eV cos φs νs = γs 4π 2 v 2 E s Thus for particles initially γ>>1, the solution to dφ ηωrf τc 2 = 2 ΔE dn v Es is of the form Δφ ~ const –electrons start out at a particular phase and stay in phase. 16 R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Fixed Field Alternating Gradient 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 (scaling FFAG has a specific slowly varying, B-field B ~ Rk) Wide aperture magnets and stable orbits. High gradient accelerating cavities combine with fixed field for rapid acceleration. Prototype FFAG, accelerating protons from Good for particles with short half- 50 keV to 500 keV, was successfully built lives (e.g. muons). and tested at the KEK laboratory in Japan, 2000. 17 17 R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Fixed Field Alternating Gradient Machines (FFAG) nsFFAGs (non scaling FFAGS) Drop scaling requirement –ensures simplicty and more compact! Do not worry about going through an integer tune –just ensure you do not linger! Electron Machine with Many Applications (EMMA) 10-20 MeV electron accelerator 42 cells. 19 RF cavities. Accelerates in ~16 turns 18 R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Synchrotrons Principle of operation: This is a constant radius accelerator where both the frequency and the magnetic field vary with time in order to maintain the synchronism condition. qB Now ω= and it is straightforward to show that: qBR 2 2 2 m0 [1 + ]1 / 2 m0 c2 2 qB ⎛ ∂ω ⎞ q ω= and ⎜ ⎟ = m0γ 3 ⎝ ∂B ⎠ R m0γ 3 Therefore, angular frequency increases at B-dependent rate, initially fast but then more and more slowly, as g>>1. Energy range for p beams: Conventional magnets give fields of up to 2T. There is a min. field of about 10-2 T, hence Emin@5x10-3 Emax Hence, the beam has to be injected with Emin, which is provided by a linac or another synchrotron (‘booster’) or a linac-booster combination. Superconducting magnets provide Bmax@8T and Bmin=? 19 Synchrotrons 4. Energy range for (e-,e+) beams: • At medium energies, previous comments apply. • Above about 10GeV, SR loss is severe. Accelerator cost optimization requires that maximum machine radius Rmax~(Emax)2. Hence, Bmax ~1/Emax. Typically, LEP had Bmax @0.1T. 5. Synchrotrons do not need a continuous magnet around the orbit but operate with a ring of magnets. Therefore, they have achieved major savings in the magnet iron. 6. These are the dominant machine for high energy physics. Although to limit SR, linacs are envisaged for future e-e+ colliders –ILC (Low-loss SC) and CLIC (High gradient NC). 20 Synchrotron Radiation Accelerated charges emit radiation whose power (P) is given by: P= 2 e2 6 3c [ γ β .β − ( β × β ) 2 ] (Lienard, 1898) For linear motion, dβ 1 dp β × β = 0 and = , hence dt m0 cγ 3 dt 2 e 2 dp 2 2 e 2 dE 2 P= ( ) = ( ) 3 m0 c dt 2 3 3 m0 c dx 2 3 For a linear accelerator dE/dx must be of the order of 1014 MeV/m before this becomes significant compared to power supplied by external sources. For circular machines 2 e2 2 2 2 2 e2c 4 4 P= γ ω p = β γ , where ρ = radius. 3 m 02 c 3 3 ρ Note extra γ 2 term. 21 Synchrotron Radiation For b@1, the energy dE lost per turn has a numerical value given by: E4 δE ( MeV ) = 8.85 x10 −2 , ( E in GeV, ρ in m) ρ Some numbers: For LEP at 86GeV, dE@1.37GeV/electron. There are about 6x1012 electrons per beam and the hence power required to make up for this loss is @20MW. Power needed for RF is about 96MW. SR was originally perceived as a nuisance! Now it is a field with many important applications and has a large user community (both university and industry), with large, dedicated accelerators (ESRF at Grenoble, ALS at Brookhaven, USA and DIAMOND at RAL,UK and SSRL at SLAC, USA etc). 22 Synchrotron Radiation Radiation is produced within a light cone of angle 1 511 θ ≈ = for speeds close to c γ E [keV ] For electrons in the range 90 MeV to 1 GeV, θ 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. 23 R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Diamond Light Source, RAL: April 2006 R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Beam Transport and Delivery The term is meant to cover the optics of the charged-particle beam and the systems that focus and steer the beam (together with associated diagnostic devices). These devices are both internal and external to the accelerating structure. The optics of beams is a broad field covered by a number of specialist books. Key concept is that of ‘emittance’: each particle of the beam can be represented by a single point in the 6-dimensional phase space (x, y, z, px, py, pz). The volume occupied by these points is the 6-D emittance (e) of the beam. Emittance must be as small as possible and the density of points in phase space (‘brightness’) as high as possible. We normally use the 3 projections of the hyper-volume (x,px) (y,py) and (z,pz). For a beam drifting in a field-free space in the z-direction, pz= const. and px,py<<pz. Introduce p p x' = x y' = y pz pz 25 Beam Transport and Delivery The transverse emittances are usually plotted as (x,x’) and (y,y’) contours. For the third dimension (i.e. along the acceleration) we normally plot (Dz,Dpz), i.e. relative to a ‘reference’ particle. Equivalent phase spaces are (Dt,DE) or (f,DE). Emittances usually shown as ellipses in phase space, although they are frequently distorted. In reality, not all particles are contained within a single, elliptical contour. The usual convention is the quote the area of the ellipse occupied by 90% of the beam and, also, to include the factor p explicitly, e.g. 0.1 pmm.mrad. Normalized emittance en=bge, to take account of the fact that during acceleration pz increases and the measured emittance in, say, the (x,x’) plane decreases. 26 Liouville Liouville’s theorem: ‘In the absence of collisions or dissipative processes, the local density in phase space remains constant’. Alternatively: ‘For a system of non-interacting particles that are in a conservative system, the density in phase space along the trajectory is invariant’. There are a number of apparent violations of Liouville’s theorem, most of them leading to an increase of the beam emittance, e.g. Space charge forces between beam particles increase emittance. Beam-beam and beam-gas scattering do the same. Foil stripping of H- ions into H+, at the point of injection into a synchrotron, also increases emittance. Synchrotron radiation (see later) can lead to a reduction in phase- space volume (‘damping’). 27 Beam Cooling The principle: the emittance is a measure of ‘disorder’ in the beam, hence it is analogous to entropy. For a closed system DS>=0 but if system is not closed, it is possible to reduce S (or emittance). The beam ‘temperature’ is determined by transverse motion of particles relative to the central orbit. Methods available for cooling: 1. Electron cooling. 2. Stochastic cooling. Electron cooling relies on bringing a p beam in contact with a low- temperature e- beam, of the same b. Very effective for low energies, up to about 100MeV. 28 Beam Cooling Stochastic cooling relies on a pick-up electrode which detects large amplitude excursions of a particle from its central orbit (‘betatron oscillation’), sends a message to a ‘kicker’ downstream which applies a correction field to reduce this amplitude. If g is gain of detection system, W its bandwidth and N the number of circulating particles, then cooling time t is given by: 2N τ= W (2 g − g 2 ) Ionization cooling. All previous methods are ‘slow’. The neutrino factory will require fast cooling of the muons, before they decay. Ionization cooling relies on successive reduction of momenta and, then, addition of longitudinal momentum by acceleration. Net result is the reduction of transverse/longitudinal ratio, i.e. cooling. MICE experiment planned at RAL... 29 Targets and Kinematics Historically, fixed-target experiments came first. Still in use now for nuclear physics, radioactive beam facilities, neutron production, cancer therapy and neutrino factories. However, the really important quantity is the ‘centre-of-mass’ (or, better, the ‘centre-of-momentum’) energy T*, which depends on mass of target and its motion. In the relativistic case, m1 hitting stationary target m2, the CM energy of the system is: E * = c 2 m12 + m22 + 2γ 1m1m2 If m1=m2=m (e.g. proton-proton collision), then E* = mc 2γ + 2 and 2 1 the useful kinetic energy T* is only T * = E * −2m0 c 2 = m0 c 2 [ 2γ 1 + 2 − 2 ] Use head-on colliding beams E* ≅ 2 E1E2. For two colliding proton beams, each 28GeV: E* = 56GeV To get this from a stationary target, one would require ~1.8TeV. 30 Targets Fixed target experiments are not finished yet! A neutrino factory will be based on an intense p beam impinging on fixed target, capture of pions and then of muons, and then acceleration of the latter. Similar problems for the design of neutron spallation sources and of radioactive beam facilities. All the above areas will require considerable R/D in target design. 31 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 Nσ /A of the other bunch. (σ=interaction cross section). Number of interactions between the two bunches is Area, A N2σ /A. Interaction rate is R = f N2σ /A, and N2 Luminosity L= f A LHC and Fermilab p-pbar colliders have L ~ 1030 cm-2s-1. SSC was aiming for L ~ 1033 cm-2s-1 and CLIC aims for L>1034 32 R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Luminosity Exercise 3: The luminosity L is a measure of the probability rate of particle encounters in a collision process. Thus the total counting rate of a physics event is R = σphysL, where σphys is the cross- section of a physics process. For two beams colliding head-on the luminosity is: L = 2 fN1 N 2 ∫ ρ1 ( x,z,s1 ) ρ 2 ( x,z,s2 )dx ⋅ dz ⋅ ds ⋅ d(βct) Where s1=s+βct, s2=s-βct, f is the collision frequency N1 and N2 are the number of particles per bunch and ρ1 and ρ2 are the nomalised distribution functions 1. Using a Gaussian bunch distribution, 1 ⎛ x2 x2 x2 ⎞ ρ ( x,z,s ) = exp ⎜ − 2 − 2 − 2 ⎟ ( 2π ) σ xσ zσ s ⎝ 2σ x 2σ z 2σ s ⎠ 3/ 2 where σx , σz, and σ are the horizontal vertical rms bunch widths and rms bunch length, show that the luminosity for two bunches with identical distribution profiles is: fN1 N 2 L= 4πσ xσ z 2. Also show that when the two beams are offset by a horizontal distance b, the luminosity is reduced by a factor exp ( −b 2 / 4σ x ) 2 R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Accelerator Categories 34 R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Summary of Circular Accelerators Machine RF frequency Magnetic Field Orbit Radius ρ Comment ω/2π B 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- varies constant increases with Stable oscillations cyclotron 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 In General… Dipole magnets are used to bend the beam into a circular path p qBc 2 v ρ= ω= = Quadrupole to focus the beam and maintain stability qB E ρ (AG focusing) 35 R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Summary of Linacs Linacs Periodic Space Charge Wakefields Comments Low Energy No Yes No For intense beams, at low energy both space charge and wakefields are issues. Cavities must be modified in order to remain in phase with changing velocity of beam. At injection, or close to the particle source, space charge often dominates. Medium- ? ? Yes (1/γ2) Once the beam is sufficiently relativistic, cavities are periodic High Energy and wakefields are the main issue. Electron cloud is another serious concern. 36 R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010. Linacs Next week Linacs! 37 R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
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