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

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

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 pp Quadrupole : the Lorenz force focuses the trajectories  bending radius B v L F

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≤

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 ] CERS Tevatron Examples quadrupole strength

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

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

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

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

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

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=-        

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

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

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

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

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

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

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)

Acceleration mechanism

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

LHC luminosity

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