Cllective effects in Beam Dynamics Givanni Ruml, Hannes Bartsik and Kevin Li USPAS, ne-week curse, 19-24 January, 2015 http://uspas.fnal.gv/index.shtml January 2015 USPAS lectures 2
Particle beam Nminal clsed rbit in the acceleratr (usually nt a circle!) Mtin f particles is described in reference frame c-mving with synchrnus particle (i.e. with nminal mmentum n nminal clsed rbit, mtin determined by external fields) Transverse à x (plane f clsed rbit) r y (cnjugate variables are divergences x and y ) Lngitudinal à z (cnjugate variable is mmentum spread δ) Vlume ccupied by beam in phase space (x, x, y, y, z, δ) is called beam emittance 3
A general definitin f cllective effects Class f phenmena in beam dynamics, in which the evlutin f a particle in a beam depends n bth the external EM fields and the extra EM fields created by the presence f ther particles. Hw ther particles can affect a single particle s mtin: Self-induced EM fields Space charge frm beam particles EM interactin f whle beam with surrunding envirnment EM interactin f whle beam with its wn synchrtrn radiatin Culmb cllisins Lng range and multiple tw beam particle encunters à Intra-beam scattering Shrt range and single events tw beam particle encunters à Tuschek effect Elastic and inelastic scattering against residual gas EM fields frm anther charge distributin (generated r nt by the beam itself), like a secnd beam Beam-beam in clliders In trapping fr electrn beams Electrn cluds fr psitrn/hadrn beams Interactins with electrn lens r electrn cling system 4
Cllective effects start playing a rle when the beam density is very high They are als referred t as high current, high intensity, high brightness effects and exhibit a threshld behaviur They result int a measurable respnse f the beam t the cllective interactin, which can be detrimental and lead t beam degradatin and lss Transverse cllective effects Due t self-induced EM fields Cherent: The beam centrid is affected, resulting in betatrn tune shift and pssibly in expnential grwth (single r multi-bunch instabilities, strng head-tail) à Cherent beam instability Can be seen with standard Beam Psitin Mnitrs Incherent: Beam centrid nt affected Result int emittance grwth but als hal/tail frmatin and slw particle lss (pr beam lifetime) 5
Hw d we recgnize a cherent beam instability? A beam becmes unstable when a mment f its distributin exhibits an expnential grwth (e.g. mean psitins <x>, <y>, <z>, standard deviatins σ x, σ y, σ z, etc.) resulting int beam lss r quality degradatin! y x (x, y, z, x 0,y 0, ) z 2 x = 1 N N = hxi = 1 N Z 1 1 Z 1 1 Z 1 1 (x, y, z, x 0,y 0, )dxdx 0 dydy 0 dzd x (x, y, z, x 0,y 0, )dxdx 0 dydy 0 dzd (x hxi) 2 (x, y, z, x 0,y 0, )dxdx 0 dydy 0 dzd And similar definitins fr hyi, y, hzi, z 6
An example f transverse cherent beam instability A beam becmes unstable when a mment f its distributin exhibits an expnential grwth (e.g. mean psitins <x>, <y>, <z>, standard deviatins σ x, σ y, σ z, etc.) resulting int beam lss r quality degradatin! Beam Psitin Mnitr 0.4 Stable beam Bunch 10 Hr. psitin. [a.u.] 0.3 0.2 0.1 0-0.1-0.2-0.3-0.4 Thusands f turns, i.e. millisecnds 0 5000 10000 15000 Turn # 7
An example f transverse cherent beam instability A beam becmes unstable when a mment f its distributin exhibits an expnential grwth (e.g. mean psitins <x>, <y>, <z>, standard deviatins σ x, σ y, σ z, etc.) resulting int beam lss r quality degradatin! Beam Psitin Mnitr Hr. psitin. [a.u.] 0.4 0.3 0.2 0.1 0-0.1-0.2-0.3 Unstable beam Beam Current Transfrmer ΔI -0.4 0 5000 10000 15000 Turn # 8
An example f transverse cherent beam instability A beam becmes unstable when a mment f its distributin exhibits an expnential grwth (e.g. mean psitins <x>, <y>, <z>, standard deviatins σ x, σ y, σ z, etc.) resulting int beam lss r quality degradatin! Beam Prfile Mnitr Beam Prfile Mnitr Beam Psitin Mnitr Hr. psitin. [a.u.] 0.4 0.3 0.2 0.1 0-0.1-0.2-0.3 Unstable beam Beam Current Transfrmer ΔI -0.4 0 5000 10000 15000 Turn # 9
Cllective effects start playing a rle when the beam density is very high They are als referred t as high current, high intensity, high brightness effects and exhibit a threshld behaviur They result int a measurable respnse f the beam t the cllective interactin, which can be detrimental and lead t beam degradatin and lss Lngitudinal cllective effects Due t self-induced EM fields Lngitudinal space charge, energy lss, ptential well distrtin (synchrnus phase shift, bunch lengthening) Instabilities (negative mass instability, single r cupled bunch instabilities, micrwave instability) 10
Example f lngitudinal cherent mtin The beam prfile, measured at a Wall Current Mnitr, shws a bunch scillating in its buckets (plt 2) r executing quadruple scillatins (plt 3) 11
Why it is imprtant t study beam cherent mtin The beam cherent mtin becmes unstable fr a certain beam intensity, which is the maximum that a machine can stre/accelerate (perfrmance limitatin) y x z hxi (mm) Typical situatin A beam centrid instability appears when the intensity is raised abve a certain threshld The threshld can be ptimized by an accurate chice f the machine settings 12
Why it is imprtant t study beam cherent mtin The beam cherent mtin becmes unstable fr a certain beam intensity, which is the maximum that a machine can stre/accelerate (perfrmance limitatin) Understanding the type f instability limiting the perfrmance, and its underlying mechanism, is essential because it: Allws identificatin f the surce and the pssible measures t mitigate/suppress the effect Allws specificatin f an active feedback system t prevent the instability 13
Types f cherent instabilities Beam instabilities ccur in bth linear and circular machines Lngitudinal phase plane (z,δ) Transverse phase plane (x,y,x,y ) Beam instabilities can affect the beam n different scales Crss-talk between bunches The unstable mtin f subsequent bunches is cupled The instability is cnsequence f anther mechanism that builds up alng the bunch train Single bunch effect Casting beam instabilities 14
Instabilities can affect the beam n different scales Crss-talk between bunches The unstable mtin f subsequent bunches is cupled The instability is cnsequence f anther mechanism that builds up alng the bunch train Single bunch effect 15
Instabilities can affect the beam n different scales Crss-talk between bunches The unstable mtin f subsequent bunches is cupled The instability is cnsequence f anther mechanism that builds up alng the bunch train Single bunch effect 16
Instabilities can affect the beam n different scales Crss-talk between bunches The unstable mtin f subsequent bunches is cupled The instability is cnsequence f anther mechanism that builds up alng the bunch train Single bunch effect 17
Cllective effects start playing a rle when the beam density is very high They are als referred t as high current, high intensity, high brightness effects and exhibit a threshld behaviur They result int a measurable respnse f the beam t the cllective interactin, which can be detrimental and lead t beam degradatin and lss Cllisinal effects (transverse and lngitudinal) Due t scattering events (cllisins) between individual particles Shrt range encunters causing large angle deviatins (single events, mdeled as Tuschek effect) Lng range encunters causing small angle deviatins (cumulative effect, mdeled as Intra Beam Scattering) Tend t deppulate the denser beam cre and degrade emittance (i.e. vlume ccupied by the beam in the phase space) and lifetime, similar t what is caused by incherent cllective effects, like direct space charge. 18
Transverse incherent effects r cllisinal effects A beam exhibits slw lsses (n the time scale f the cycle r stre) and emittance grwth visible frm a beam prfile measurement device, pssibly assciated t develpment f hal r tails Beam Prfile Mnitr Beam Prfile Mnitr Beam Psitin Mnitr Stable beam Beam Current Transfrmer I(t) Several secnds r several hurs 19
Cllective effects start playing a rle when the beam density is very high They are als referred t as high current, high intensity, high brightness effects and exhibit a threshld behaviur They result int a measurable respnse f the beam t the cllective interactin, which can be detrimental and lead t beam degradatin and lss Tw-stream effects (transverse and lngitudinal) Due t the interactin with anther set f charged particles (e.g. electrn clud) Can cause cherent mtin as well as incherent emittance grwth and lsses, as previusly described 20
Electrn clud instability A cherent instability is visible fr the last bunches f a train (BPM signal and beam lsses), because an electrn clud has frmed alng the train and can nly make these bunches unstable 48b injectin test in LHC (26/08/11) 21
Mdeling f cllective effects due t self-induced EM fields Single particle mtin under the verall effect f externally applied fields (RF, magnets) and thse created by the beam itself with the prper bundary cnditins. Single particle dynamics nt sufficient, need t describe a system f many particles ü Thery: kinetic mdels based n distributin functins (Vlasv-Maxwell) y x (x, y, z, x 0,y 0,,t) z d dt =0! ~E = ~ Eext + ~ E( ) ~B = ~ B ext + ~ B( ) in Maxwell s equatins 22
Mdeling f cllective effects due t self-induced EM fields Single particle mtin under the verall effect f externally applied fields (RF, magnets) and thse created by the beam itself with the prper bundary cnditins. Single particle dynamics nt sufficient, need t describe a system f many particles ü Thery: kinetic mdels based n distributin functins (Vlasv-Maxwell) ü Simulatin: slve numerically the equatins f mtin f a set f macrparticles and use the EM fields f the macrparticle distributin y x 10 8 10 11 particles! 10 4 10 6 macrparticles z d~p mp = q ~E + ~vmp B dt ~ ~E = Eext ~ + E( ~ mp ) ~B = B ~ ext + B( ~ mp ) 23
Mdeling f cllective effects due t self-induced EM fields Single particle mtin under the verall effect f externally applied fields (RF, magnets) and thse created by the beam itself with the prper bundary cnditins. Single particle dynamics nt sufficient, need t describe a system f many particles ü Thery: kinetic mdels based n distributin functins (Vlasv-Maxwell) ü Simulatin: slve numerically the equatins f mtin f a set f macrparticles and use the EM fields f the macrparticle distributin q Direct space charge refers t the EM fields created by the beam as if it was mving in pen space, q Impedances are used t describe EM interactin f beam with bundaries y x z 24
Mdeling f cllective effects due t self-induced EM fields + Culmb cllisins Single particle mtin under the verall effect f externally applied fields (RF, magnets) and thse created by the beam itself with the prper bundary cnditins. Single particle dynamics nt sufficient, need t describe a system f many particles ü Thery: kinetic mdels based n distributin functins (Vlasv-Maxwell) ü Simulatin: slve numerically the equatins f mtin f a set f macrparticles ª Prbability f clse encunters can be included thrugh the apprpriate mdels y x d dt = @ @t cll! ~E = ~ Eext + ~ E( ) ~B = ~ B ext + ~ B( ) z Vlasv-Fkker-Planck frmalism 25
Mdeling f cllective effects due t EM fields frm anther charge distributin Single particle mtin under the verall effect f externally applied fields (RF, magnets) and thse created by the secnd beam. Single particle dynamics nt sufficient, need t describe evlutin (and smetimes generatin) f the ther system f particles t derive its EM fields ü Thery: simplified mdels t include the effect f the secnd beam ü Simulatin: describe numerically the secnd beam and calculate its fields as driving terms in the equatins f mtin f the set f macrparticles representing the beam y x z d~p mp1,mp2 = q ~E + ~vmp1,mp2 B ~ dt ~E = Eext ~ + E( ~ mp1, mp2 ) ~B = B ~ ext + B( ~ mp1, mp2 ) Ex. Beam ging thrugh an electrn clud 26
Mdeling f cllective effects due t EM fields frm anther charge distributin Single particle mtin under the verall effect f externally applied fields (RF, magnets) and thse created by the secnd beam. z Single particle dynamics nt sufficient, need t describe evlutin (and smetimes generatin) f the ther system f particles t derive its EM fields ü Thery: simplified mdels t include the effect f the secnd beam ü Simulatin: describe numerically the secnd beam and calculate its fields as driving terms in the equatins f mtin f the set f macrparticles representing the beam y x y z x d~p mp1,mp2 = q ~E + ~vmp1,mp2 B ~ dt ~E = Eext ~ + E( ~ mp1, mp2 ) ~B = B ~ ext + B( ~ mp1, mp2 ) Ex. Beam-beam effects in clliders 27
Outline f the curse 1. Intrductry cncepts 2. Space charge 3. Wake fields and impedance 4. Instabilities 2-particle mdel 5. Instabilities kinetic thery 6. Tw stream effects + Numerical methds + Experimental examples + Mitigatin techniques Additinal tpics interleaved within the abve list January 2015 USPAS lectures 28
Outline 1. Intrductry cncepts Cllective effects Transverse single particle dynamics including systems f many nn-interacting particles Lngitudinal single particle dynamics including systems f many nn-interacting particles 2. Space charge Direct space charge (transverse) Indirect space charge (transverse) Lngitudinal space charge January 2015 USPAS lectures 29
Outline 3. Wake fields and impedance Wake fields and wake functin Definitin f beam cupling impedance Examples resnatrs and resistive wall Energy lss Impedance mdel f a machine 4. Instabilities few-particle mdel Equatins f mtin Lngitudinal plane: Rbinsn instability Transverse plane: rigid bunch instability, strng head-tail instability, head-tail instability January 2015 USPAS lectures 30
Outline 5. Instabilities kinetic thery Intrductin t Vlasv equatin and perturbatin apprach Vlasv equatin in the lngitudinal plane Vlasv equatin in the transverse plane Oscillatin mdes, shift with intensity, instability 6. Tw stream effects Electrn clud build up Electrn clud induced beam instability In trapping and fast in instability January 2015 USPAS lectures 31
Outline + Numerical methds Basic mdels fr macrparticle simulatins: appraches, apprximatins, assumptins Beam tracking with cllective effects (PyHEADTAIL) + Experimental examples Real life instabilities, bservatins, implicatins Beam-based measurements t determine impedances Electrn clud and fast in bservatins in machines + Mitigatin techniques Impedance reductin, electrn clud suppressin Beam parameters, machine settings, Landau damping Feedback systems January 2015 USPAS lectures 32