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Introduction to Accelerator Physics Beam Dynamics for „Summer Students“ Bernhard Holzer, CERN-LHC

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Introduction to Accelerator Physics Beam Dynamics for „Summer Students“ Bernhard Holzer, CERN-LHC Introduction to Accelerator Physics Beam Dynamics for „Summer Students“ Bernhard Holzer, CERN-LHC IP5The Ideal World I.) Magnetic Fields and Particle Trajectories IP8 IP2 IP1 * Luminosity Run of a typical storage ring: LHC Storage Ring: Protons accelerated and stored for 12 hours distance of particles travelling at about v ≈ c L = 1010-1011 km ... several times Sun - Pluto and back intensity (1011) 3 - 2 - 1 - guide the particles on a well defined orbit („design orbit“) focus the particles to keep each single particle trajectory within the vacuum chamber of the storage ring, i.e. close to the design orbit. 1.) Introduction and Basic Ideas „ ... in the end and after all it should be a kind of circular machine“ need transverse deflecting force Lorentz force F q*() E v B 8 m typical velocity in high energy machines: vc3 *10 s Example: m Vs B 1 T F q 3 10 8 1 2 s m technical limit for el. field: MV F q 300 MV m E 1 m equivalent el. field ... E old greek dictum of wisdom: if you are clever, you use magnetic fields in an accelerator wherever it is possible. y The ideal circular orbit ρ θ ● s circular coordinate system condition for circular orbit: Lorentz force F L e v B p m v 2 B centrifugal force F 0 centr e B ρ = "beam rigidity" m v 2 0 e v B 2.) The Magnetic Guide Field Dipole Magnets: n I define the ideal orbit B 0 homogeneous field created h by two flat pole shoes Normalise magnetic field to momentum: convenient units: p 1 e B Vs GeV B B T p e p m 2 c Example LHC: 8.3 Vs 8 m 1 2 8.3 s * 3 *10 e m s B 8.3 T 9 2 7000 *10 9 eV 7000 *10 m c GeV p 7000 1 8.3 c 0.333 1 7000 m The Magnetic Guide Field ρ α ds field map of a storage ring dipole magnet 2.53 km 2πρ = 17.6 km B 1 ... 8 T ≈ 66% rule of thumb: 1 B T „normalised bending strength“ 0.3 p GeV / c 2.) Focusing Properties – Transverse Beam Optics classical mechanics: there is a restoring force, proportional pendulum to the elongation x: dx2 m** c x dt2 general solution: free harmonic oszillation x( t ) A * cos( t ) Storage Ring: we need a Lorentz force that rises as a function of the distance to ........ ? ................... the design orbit F()**() x q v B x Quadrupole Magnets: required: focusing forces to keep trajectories in vicinity of the ideal orbit linear increasing Lorentz force linear increasing magnetic field B y g x B x g y normalised quadrupole field: g k p / e g (T / m ) simple rule: k 0.3 p(GeV / c ) LHC main quadrupole magnet g 25 ... 220 T / m B what about the vertical plane: E y B x B =j 0 g ... Maxwell t x y Focusing forces and particle trajectories: normalise magnet fields to momentum (remember: B*ρ = p / q ) Dipole Magnet Quadrupole Magnet BB1 g k : p/ q B pq/ 3.) The Equation of Motion: B ( x ) 1 1 1 k x m x 2 n x 3 ... p / e 2! 3! only terms linear in x, y taken into account dipole fields quadrupole fields Separate Function Machines: Split the magnets and optimise them according to their job: bending, focusing etc Example: heavy ion storage ring TSR * man sieht nur dipole und quads linear yˆ The Equation of Motion: ● Equation for the horizontal motion: * ρ y θ ● 1 x x ( k ) 0 x 2 s * Equation for the vertical motion: 1 y 0 no dipoles … in general … 2 k k quadrupole field changes sign x y k y 0 4.) Solution of Trajectory Equations Define … hor. plane: Kk1 2 x K x 0 … vert. Plane: Kk Differential Equation of harmonic oscillator … with spring constant K Ansatz: Hor. Focusing Quadrupole K > 0: 1 x() s x00 cos( K s ) x sin( K s ) K x() s x00 K sin( K s ) x cos( K s ) For convenience expressed in matrix formalism: 1 x x cos K l sin K l M K M foc * foc x x s1 s 0 K sin K l cos K l s = 0 s = s1 hor. defocusing quadrupole: x K x 0 Ansatz: Remember from school 1 coshK l sinh K l K x(s) a1 cosh( s) a 2 sinh( s) M defoc Ksinh K l cosh K l drift space: K = 0 x(s) x 0 * s 1 l M drift 01 ! with the assumptions made, the motion in the horizontal and vertical planes are independent „ ... the particle motion in x & y is uncoupled“ Transformation through a system of lattice elements combine the single element solutions by multiplication of the matrices focusing lens MMMMMM**** total QF D QD Bnde D *..... dipole magnet defocusing lens x x M (s , s ) * x ' 2 1 x ' s2 s1 court. K. Wille in each accelerator element the particle trajectory corresponds to the movement of a harmonic oscillator „ x(s) 0 typical values in a strong foc. machine: s x ≈ mm, x´ ≤ mrad 5.) Orbit & Tune: Tune: number of oscillations per turn 64.31 59.32 Relevant for beam stability: non integer part LHC revolution frequency: 11.3 kHz 0.31 *11 .3 3.5kHz LHC Operation: Beam Commissioning First turn steering "by sector:" One beam at the time Beam through 1 sector (1/8 ring), correct trajectory, open collimator and move on. Question: what will happen, if the particle performs a second turn ? ... or a third one or ... 1010 turns x 0 s II.) The Ideal World: Particle Trajectories, Beams & Bunches Bunch in a Storage Ring (Z X Y) Astronomer Hill: differential equation for motions with periodic focusing properties „Hill„s equation“ Example: particle motion with periodic coefficient equation of motion: x( s ) k ( s ) x ( s ) 0 restoring force ≠ const, we expect a kind of quasi harmonic k(s) = depending on the position s oscillation: amplitude & phase will depend k(s+L) = k(s), periodic function on the position s in the ring. 6.) The Beta Function „it is convenient to see“ ... after some beer ... general solution of Mr Hill can be written in the form: Ansatz: ε, Φ = integration constants x() s * ()*cos(() s s ) determined by initial conditions β(s) periodic function given by focusing properties of the lattice ↔ quadrupoles ()()s L s ε beam emittance = woozilycity of the particle ensemble, intrinsic beam parameter, cannot be changed by the foc. properties. scientifiquely spoken: area covered in transverse x, x´ phase space … and it is constant !!! Ψ(s) = „phase advance“ of the oscillation between point „0“ and „s“ in the lattice. For one complete revolution: number of oscillations per turn „Tune“ 1 ds Q y 2 (s ) 7.) Beam Emittance and Phase Space Ellipse ()*sxs22 () 2()()() sxsxs ()() sxs x´ Liouville: in reasonable storage rings area in phase space is constant. ● A = π*ε=const ● ● ● x ● ● x(s) s ε beam emittance = woozilycity of the particle ensemble, intrinsic beam parameter, cannot be changed by the foc. properties. Scientifiquely spoken: area covered in transverse x, x´ phase space … and it is constant !!! Particle Tracking in a Storage Ring Calculate x, x´ for each linear accelerator element according to matrix formalism plot x, x´as a function of „s“ ● … and now the ellipse: note for each turn x, x´at a given position „s1“ and plot in the phase space diagram Emittance of the Particle Ensemble: (Z X Y) Emittance of the Particle Ensemble: x(s) (s) cos( (s) ) xˆ(s) (s) 1 x 2 Gauß N e 2 2 x Particle Distribution: ( x ) e 2 x particle at distance 1 σ from centre ↔ 68.3 % of all beam particles single particle trajectories, N ≈ 10 11 per bunch LHC: 180 m 5 * 10 10 m rad * 5 *10 10 m *180 m 0.3 mm aperture requirements: r 0 = 12 * σ III.) The „not so ideal“ World Lattice Design in Particle Accelerators x , y D 1952: Courant, Livingston, Snyder: Theory of strong focusing in particle beams Recapitulation: ...the story with the matrices !!! Equation of Motion: Kk1 2 Solution of Trajectory Equations Kk x K x 0 … hor. plane: x x M * x x … vert. Plane: s1 s 0 1 l M drift 0 1 1 cos( K l ) sin( K l ) K M foc K sin( K l ) cos( K l ) 1 cosh( K l ) sinh( K l ) K M defoc K sinh( K l ) cosh( K l ) M total M QF * M D * M B * M D * M QD * M D * ... 8.) Lattice Design: „… how to build a storage ring“ Geometry of the ring: B*/ p e p = momentum of the particle, ρ = curvature radius Bρ= beam rigidity Circular Orbit: bending angle of one dipole ds dl Bdl B The angle run out in one revolution must be 2π, so for a full circle Bdl 2 B p Bdl 2 … defines the integrated dipole field around the machine. q Example LHC: 7000 GeV Proton storage ring dipole magnets N = 1232 B dl N l B 2 p / e l = 15 m q = +1 e 2 7000 10 9 eV B 8.3 Tesla m 1232 15 m 3 10 8 e s FoDo-Lattice A magnet structure consisting of focusing and defocusing quadrupole lenses in alternating order with nothing in between. (Nothing = elements that can be neglected on first sight: drift, bending magnets, RF structures ... and especially experiments...) Starting point for the calculation: in the middle of a focusing quadrupole Phase advance per cell μ = 45°, calculate the twiss parameters for a periodic solution 9.) Insertions x , y D β-Function in a Drift: 2 l l () 0 0 * β0 At the end of a long symmetric drift space the beta function reaches its maximum value in the complete lattice.
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