Report

Beam dynamics in RF linacs Step 2 : Beam transport Nicolas PICHOFF France CEA-DSM/IRFU/SACM/LEDA 1 Outlines • Beam representation • Distribution • Sigma matrix • Emittance • Matching • Mechanism of emittance growth 2 Beam representation Beam definition A beam could be defined as a set of particles whose maximum average momentum in one direction (z) is higher than its dispersion : pz x 2 px2 p y2 pz pz 2 p z y 3 Beam representation Particle representation Each particle is represented by a 6D vector : p1 p2 p3 P p4 p 5 p 6 x x y y W or x px y py z p z or … 4 Beam phase-space representation Beam representation It is represented by a particle distribution in the 6D phase-space (P). It can be plotted in 2D sub-phase-spaces : 5 Beam representation Beam modelisation Beam : Set of billions (N) of particles evolving as a function of an independent variable s Macroparticle model: set of n macroparticles (n<N) macroparticle: statistic sample of particle Distribution function model: of 6 coordinates Number ofparticles f P, s dP Between P and P dP f P , s dP N 1.0 0.8 0.6 0.4 0.2 -4 4 -3 3 -2 2 -1 1 0 0 1 -1 2 -2 3 -3 4 -4 6 Statistics Beam representation Average of a function A on beam : 1 N A P A Pi N i 1 1 n A Pj n j 1 1 f P A P dP N 7 Beam representation First order momentum: beam Centre of Gravity (CoG) AP P P0 p1 p2 p3 p4 p5 p6 Average : position, phase, Angle, Energy … 8 Beam representation Second order momentum : Sigma matrix The beam can be represented by a 6×6 matrix containing the second order momentum in the 6D phase-space : the sigma matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i , j pi pi p j p j x det12,12 y det 34,34 z det 56,56 are the beam 2D rms emittances 9 2D RMS Emittance 2 2 2 ~ x x x x x x x x x Beam representation (for example) The statistic surface in 2D sub-phase-space occupied by the beam Indicator of confinement 10 Beam Twiss parameters Beam representation The goal is to model the beam shape in 2D sub-phase-space with ellipses. The beam Twiss parameters are : ~2 ~2 w w w ~ w ~ w w w w w w w ~ w w w2 2 w ww w w2 5 w 5 : uniform elliptic distribution with same rms size. 11 Linear transport 6D transport matrix The transverse force is generally close to linear. The longitudinal force can be linearized when << s. The particle transport can be represented by a 6×6 transfer matrix : p1 . p2 . p . 3 M s1 / s0 p4 . p . 5 . p 6 s1 . . . . . . . . . . . . . . . . . . . . . . . . . p1 . p2 . p3 . p4 . p5 . p6 s 0 12 Sigma matrix transport in linear forces Linear transport The transport of sigma matrix can be obtained from particle transfer matrix : s M s1 / s0 s M s1 / s0 T 1 0 13 Linear transport RMS emittance evolution in linear, uncoupled forces ~x2 x 2 x2 x x 2 x x x x x x d~x2 2 x x x 2 x x x x ds If linear force : d~x 0 ds x k x The emittance is constant. dz x If acceleration : x k x z ds 1 d~x 1 dz ~ x ds z ds The emittance is damped. ~xn z ~x , the normalized rms emittance, is conserved. 14 Linear matching Linear transport The linear matching is the association of 2 notions : - Slide 36 : In uncoupled, linear & periodic forces, particles are turning around periodically oscillating ellipses Their shapes are given by Courant-Snyder parameters wm s w2 2 wm s ww wm s w2 0 - Slide 57 : A beam can be represented by 2D-ellipses Their shape are given by Twiss parameters w s w2 2 w s ww w s w2 15 Beam dynamics Beam linear matched The beam is linear-matched : Beam Twiss parameters (S57) Phase-space trajectory Phase-space periodic looks Channel Courant= Snyder parameters (S36) w wm w wm w wm Matched beam Bigger input beam Smaller input beam Phase-space scanned by the mismatched beams 50% mismatched beam The beam second order (or envelope) motion is periodic 16 Linear (rms) matching Beam dynamics The matching is done between sections by changing focusing force with quadrupoles (transverse) and cavities (longitudinal). Calculations are made with « envelope codes » where the beam is modelled by its sigma matrix. This type of code calculates automatically the focusing strength in elements that matches the beam. Not matched beam Matched beam 17 Beam dynamics Space-charge forces Electromagnetic interaction between particles. It is linear if beam is uniform, non-linear otherwise (generally). Example : axi-symmetric beam 2 Firled Density 1.5 1 0 1 0.5 0 1 2 3 4 Radial position Distributions : uniform 5 0 0 1 2 3 Radial position parabolic 4 5 gaussian Equivalent beams: Same current, same sigma matrix 18 Introduction Mechanisms of emittance growth and particle losses 19 Introduction The main source of emittance growth is the beam mismatching in non linear forces acting through 3 mechanisms : - The distribution intern mismatching, - The beam filamentation, - Resonant interactions between particle and beam motions Other mechanisms play a (small) role : - Coupling between directions (x, y, phi), - Interaction with residual gas, - Intra-beam scattering. 20 1 – Intern mismatching Beam dynamics In an intern matched beam, beam distribution in phase-space is constant on particle trajectory in phase-space. p H = Cste r f u, u f H u, u If not, the beam distribution “re-organise” itself. 21 2 - Filamentation Beam dynamics When the confinement force is non linear (multipole, longitudinal, space-charge), the particle oscillation period depends on its amplitude : d 2w k w s , w w 0 2 ds Particle do not rotate at the same speed in the phase-space : possible filamentation Linear force Non linear force 22 Beam dynamics 3 – Space-charge resonance - In non-linear forces, the particle oscillation period depends on its amplitude - The space-charge force acting on a particle depends on beam average size - If the beam is mismatched, its average size oscillates with 3 “mismatched” modes Quadrupolar mode High-freq breathing mode Low-freq breathing mode - Some particles can have oscillation period being a multiple of these modes - The amplitude of these particles will resonantly growth and decreased 23 4 - Coupling Beam dynamics The preceding developments assumed that the force along each direction was depending only on the particle coordinates in this direction (even non-linear). When the force also depends on other coordinates 2 particles with the same sub-phase-space position can feel different forces and get separated in the sub-phase-space. 2D emittance growth Sources of coupling : - Transverse defocusing in cavities depending on phase, - Transverse focusing in quadrupoles depending on energy, - Energy gain in cavities depending on transverse position and slope, - Phase-delay due to transverse trajectory increase, - Skew quadrupoles, - Space charge-force, - … 24 5 – Residual gas interaction Beam dynamics Atom nucleus Charge : +Z.e Mass : M bmin b Particle Charge : z.e Mass : m Energy : E Momentum: p Cross section : (Rutherford) Probability : b bmax Electrons d cm2 d min 13 z 2 Z 2 1 26 10 2 EMeV 3 dPm1 ( ) d max 0.31 Z 2 PhPa 1 3 2 EMeV 25 Beam dynamics 5 – Residual gas interaction(2) -30 -25 -20 -15 -10 -5 0 1 (a) "high" N2 pressure 10-1 Beam profile (a) Simulation result Beam core (b) "low" N2 pressure (b) Simulation result 10-2 10-3 Good agreement Mismatching (a) 10-4 10-5 (b) x (mm) 10-6 26 Beam dynamics 6 – Intra-beam scattering Transfer of energy between 2 directions in a two-body collision. Very efficient if different longitudinaltransverse emittances 1.E+00 =1 = 1.5 =2 =3 I = 98 mA ; f = 352 MHz x0 = y0 = 2.5 mm ; z0 = 6.75/ mm x'0 = y'0 = 3.57.6MeV/ mrad z'0 = x'0 1.E-01 1.E-02 1.E-03 : Ratio between longitudinal and transverse energy 1.E-04 n(x) 1.E-05 1.E-06 1.E-07 1.E-08 1.E-09 1.E-10 1.E-11 1 1.E-12 0 0.5 1 1.5 1.5 x 2 2 3 2.5 3 Tails induced by 2 body collision in a uniform proton beam 27 Summary conclusion • Beam is a set of particles • Beam can be modeled with: • macroparticles, • distribution function, • statistic properties • The simplest is the 2nd order momentum : the sigma matrix, including • rms emittance (confinement), • Twiss parameters and 2D ellipses, • Emittance is conserved and damped in linear motion • Sigma matrix can be transport with matrix when the force is linear(ised) • (some) Source of emittance growth and halo are : • In mismatched beam in non-linear forces : filamentation & resonances • forces coupled between directions • scattering (intra or with residual gas) 28