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Introduction to particle accelerators Walter Scandale CERN - AT department Roma, marzo 2006

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Lecture II - single particle dynamics topics Guiding fields and transverse motion Weak versus strong focusing Equation of motion Unperturbed case Orbit errors Quadrupole errors Chromaticity Resonances and dynamic aperture Low-ß insertion Longitudinal stability

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Synchrotron: guiding field Synchrotron ring: particle trajectories at fixed radius to keep constant B should increase as p increases during acceleration, RF frequency synchronized to the particle revolution bending and focusing fields Dipole : the Lorenz force provides the centripetal acceleration magnetic rigidity bending angle f L f x º º p s L º ByBy X0X0 f pp Quadrupole : the Lorenz force focuses the trajectories bending radius B v L F

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Weak focusing Weak focusing of the transverse particle motion: to get vertical stability, the bending field should decrease with , as in cyclotrons, to get horizontal stability the the decrease of B with should be moderate, so that, for > 0, the Lorenz force exceeds the centripetal force. BrBr FyFy BrBr FyFy n > 0 vertical stability r N S y y r N S BrBr FyFy BrBr FyFy n < 0 no vertical stability horizontal stability: weak focusing centripetal forceLorenz force≤

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Strong focusing d f1f1 f2f2 Horizontal and vertical focusing for a large range of f 1 f 2 and d separated functions: the alternate gradient is made with quadrupoles of opposite focusing strength combined functions: the alternate gradient is made with dipoles with radial shape of opposite sign normalized quadrupole gradient Ring p [GeV/c] B 0 [T] [m] B [Tm] 1/ 2 [m -2 ] weak focus L quad [m] B ’ [T/m] K [m -2 ] CERS5.20.1896.417.310 -4 0.550.298 Tevatron10004.475833351.7 10 -6 1.7760.0228 Examples quadrupole strength

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Particle equation of motion transport matrix approach B-field expansionMaxwell equations and quadrupolar gradients Equation of motion g g skew

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Weak versus strong focusing Equation of motion FODO transfer matrices Strong focusing Weak focusing cosmotronAGS Strong focusing u Smaller pipe u Smaller magnet u Reduced cost

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Hill equation solution : Floquet theorem phase advance between s 0 and s Unperturbed equation of motion periodicity condition -function (periodic) phase advance variation envelop equation and depend on the lattice arrangement and 0 depend on the initial conditions of the trajectory envelop cos-like orbit sin-like orbit several orbits Here ß is NOT the relative speed v/c

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Courant-Snyder parameters Courant & Snyder invariant and more Courant-Snyder invariant z z’ Slope=- emittance is the particle emittance is the area of the ellipse mapped turn by turn in the phase plane (z,z’) beam is the (1·rms) beam emittance if the area beam encloses 39 % of the circulating particles N z z’ Poincaré section s S0S0 C beam /N is a constant of the motion (Liouville theorem) u The Liouville theorem holds in absence of acceleration, losses, scattering effects and radiation emission Liouville theorem

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Adiabatic invariant normalized emittance The Courant-Snyder invariant emittance ε decreases if we the accelerate the particle. This is called “adiabatic damping” (a pure cinematic effect, since there is no damping process involved). The slope of the trajectory is z’ = p z /p s. Accelerate the particle: p s increases to p s +∆p s, but p z doesn’t change => slope changes. p psps pzpz z’ p+ p ps+psps+ps pzpz z’+ z’ Invariant of the motion In a stationary Poincaré section -> In an accelerating Poincaré section -> = the relative speed = the relativistic factor -> Courant Snyder parameter

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Stability of the motion Transfer matrix from s 0 to s One turn transfer matrix Condition for the stability of the motion tune 4D resonance condition --> order Condition for the invariance of z z’ Slope=-

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Exact & approximate solution Exact solution of the Hill equation Exact solution in compact form with u This is a pseudo-harmonic oscillation modulated both in amplitude and in frequency u Q is the total number of oscillation per turn The phase advances faster in the sections with a smaller Approximate solution (smooth approximation) with Ten cell lattice n Cell length L = 1 m n Ring length C = 10 m n Focal length f = 0.45 m

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Perturbed equation of motion chromaticitysextupoleoctupole dispersion Solution with dipoles, quadrupoles sextupoles and octupoles betatron motionclosed orbit dispersive orbit nonlinear terms Uncoupled motion (x-plane) orbit distortion dispersive orbit with betatron oscillation natural chromaticity aberration chromaticity correction by sextupoles geometric aberration with

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Dipolar and quadrupolar field errors B localized in s k over the length L (kick approximation) kick Q = 1/2 integer kick Q = integer Periodicity of the closed orbit At every turn the perturbation is compensatedAt every turn the perturbation is enhanced Dipole error Quadrupole error B’ localized in s k over the length L (thin lens approximation) first order second order Avoid tune close to 1/2 integer - the range of forbidden tunes is called stop-band Avoid tune close to integer

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Design orbit On-momentum particle trajectory Off-momentum particle trajectory Chromatic close orbit Momentum dispersion and chromaticity First order solution Dispersion function Divergent for Q=integer Gradient error induced by momentum dispersion in a FODO lattice (thin lens approximation) Chromaticity correction with sextupoles Sextupole strength

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QxQx QyQy Int(Q x )Int(Q x )+1 Int(Q y ) Int(Q y )+1 High energy particles Low energy particles The beam rigidity increases with p K decreases with p the tune decreases with p Q’ is negative Why chromaticity should be corrected Q’ non zero produces a tune shift with p In a beam Q’ produces a tune spread Be aware of resonance crossing

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Phase space with only linear fields Distortion due to sextupoles Distortion due to octupoles Dynamic aperture Nonlinear fields imply multiple traversal of resonances Emittance distortion and growth Tune shift and spread with the amplitude Coupling of the degrees of freedom Chaotic motion Particle loss -> dynamic aperture

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since L*>> * Low ß insertion A low-ß insertion is used to focalize the beams at the collision point This is achieved with triplets or doublets of quadrupoles In the drift space where the experimental devices sit ß growth with the square of the length The chromaticity induced by the triplet can be large (local correction scheme may be needed)

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Acceleration mechanism

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Longitudinal stability Momentum compaction Slip factor measures how closely packed orbits with different momenta are measures how how much off- momentum particles slip in time relative to on-momentum ones Phase stability principle < tr B is late respect to A B will receive a larger voltage and will increase its speed B will be closer to A one turn later > tr B is late respect to A B will receive a smaller voltage and will see a shorter circumferential path one turn later B will be closer to A small oscillations

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LHC luminosity

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Lecture II - single particle dynamics reminder In a circular accelerator the guiding fields provide the required forces to keep the particles in a closed orbit along the magnet axis Strong focusing allows building much smaller magnets and is a fundamental progress respect to weak focusing The particle trajectory is a pseudo-harmonic function modulated both in amplitude and phase rather well approximated by a sinusoidal function oscillating at the betatron frequency (tunerevolution frequency) with an amplitude proportional to the square root of the emittance The imperfections of the guiding field and of the momentum particles produce resonances and eventually chaotic motion The low ß insertions are basic devices to focus the beam size at the collision point of a collider ring The phase stability principle guarantees the stability of the longitudinal motion

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