Impedance, Damper, Radial Modes, Coupled Bunches, Beam- Beam and more - PowerPoint PPT Presentation

Nested Head Tail Vlasov Solver: Impedance, Damper, Radial Modes, Coupled Bunches, Beam- Beam and more… A. A. Burov ov Fermilab milab special thanks - to V.Danilov, E.Metral, N.Mounet, S.White, X.Buffat, and T.Pieloni JAI, Oxford, 10/10/13 1

ects Intensity ensity Effects • Beams in accelerators consist of charged particles which interact with each other: – By means of direct Coulomb fields (space charge). By itself, this cannot drive an instability. – By means of image charges and currents. By itself, this cannot drive an instability. – By means of fields left behind (wake fields due to radiation). This may drive collective instability. • When a beam is intense enough, wake fields make it unstable. • We have to distinguish collective motion of beam particles from their mutual Coulomb scattering (intra-beam scattering). This is possible due to huge number of particles inside the beam, like ~1E11 /bunch. • Here, only collective instabilities are discussed. 2 AB

Liuville ville/Collisionless /Collisionless Boltzmann/ tzmann/Jeans/Vlasov Jeans/Vlasov ation Equation • Collective motion of beam particles can be described as a flow of a medium in the phase space:      f f         [ , ] 0 0 r f p f H f     t r p t       ; ; f f f H H H H H    0 0 ; r p    p r [ , ] 0 H f 0 0        f H f H f f    0   [ , ] [ , ] 0 H f H f      t p r r p  0 0 t , H f - steady state values 0 0          ( ) (z z ) ( ) H W f d , - perturbations H f As a result, we have a linear integro-differential equation to solve. See details in e.g. A.Chao , “ Physics of Collective Beam Instabilities ” 3 AB

is Nested ted Head-Tail Tail Basis My basis functions for transverse oscillations p of bunched beams: z         exp( cos ) ; il i i t   l b   Q r    0 ; z   c n r I am using equally populated rings which radii  r are chosen to reflect the phase space density. 4 AB

ch Starting rting Equation, ation, single gle bunch • In the air-bag single bunch approximation, beam equations of motion can be presented as in Ref [A. Chao, Eq. 6.183]: ˆ ˆ ˆ       X S X Z X D X where is a vector of the HT mode amplitudes, X   ˆ ˆ                 l m ( ) ( ) ( ) ( ) S Z il i d Z J J       lm lm l m n  r d ˆ      m l ( ) ( ) D i J J    lm l m n r N r R   0 0 d b is the damper gain in units of the damping rate,   2 8 Q Q b s time is in units of the angular synchrotron frequency. 5 AB

utions Analysis lysis of solutions 1. For every given gain and chromaticity, the eigensystem is found for the LHC impedance table (N. Mounet).     l 2. The complex tune shifts are found from the eigenvalues   l l 3. The stabilizing octupole current is found from the stability diagram for every mode, then max is taken. Stability diagram at +200 A of octupoles Impedances Z, Ohm cm Im Q Qs Im / dQ Q s Gaussian, 105 5 transverse only 105 2 0.03 105 1 0.02 104 5 104 0.01 2 104 1 6 Re Re Q Qs / dQ Q AB s f, Hz 0.4 0.2 0.2 0.4 108 109 1010 1011

ches Coupled pled Equidistant idistant Bunches Main idea: For LHC, wake field of preceding bunches can be taken as flat within the bunch length. The only difference between the bunches is CB mode phase advance, otherwise they are all identical. Thus, the CB kick felt by any bunch is proportional to its own offset, so the ˆ ˆ CB matrix has the same structure as the damper matrix : D C ˆ ˆ ˆ ˆ         ; X S X Z X D X C X d ˆ ˆ ˆ           m l ( ) ( ); 2 ( ) / ; D i J J C i W D d      lm l m n r     2 ( { }) Q             ( ) ( )exp( ) ; x ; 0 1. W W ks ik M    0 b M  k 1 b Wake and impedance are determined according to A. Chao book. 7 AB

per gain Old damper g  ( ) Old narrow-band ADT gain profile (W. Hofle, D. Valuch) . At 10 MHz it drops 10 times. The new damper is bbb for 50ns beam. Below gain is measured in omega_s units, max gain=1.4 is equivalent to 50 turns of the damping time. 8 AB

e CB Mode e Damping ping Rate g  ( ) With as the frequency response function of the previous plot, the time- domain damper’s “wake” is           ( ) ( )cos / , G g d 0 assuming this response to be even function of time (no causality for the damper!). From here (equidistant bunches!):      (0) 2 ( )cos( ) G G k k  0   1 k ; d d      (0) 2 ( ) G G k 0  k 1 d where is the rate provided for low-frequency CB zero-head-tail modes at zero chromaticity. 9 AB

tors for the Old ADT CB Wake e and Gain n Factors 10 AB

2  (S teau (SB B and CB), , flat t ADT, , Tunes es at the Plateau tunes tune shifts • All unstables -0.1<Re[dQ/Qs]<0. all tunes • Weak head-tail is justified at the plateau. • Mode with max rate (MUM) has ~max tune shift as well. • For unstables -Re[dQ]/Im[dQ]~20-30. 11 AB

te) NHT vs BeamBeam3D mBeam3D tracking cking (S. White) Highest growth rates for single bunch, gain=1.4 and nominal impedance 12

2  (S , MUM (SB B and CB), , flat t ADT, d d Q ’ Q ’ Growth rate and -tune shift of the most unstable mode (MUM) vs chroma and gain. Both are in units of Qs. Note that at the plateau the rate (Im[dQ_c]) is ~20-30 times smaller than the shift (Re[dQ_c]). 13 AB

2  (S pling (SB B and CB), , flat t ADT, , MUM CM and Coupling d d Q ’ Q ’     l ( ) / ; . A i J n x X A   l l r n   r   2 2 2 | | | | ; | | 1; X X X  l l l   1 l    2 2 2 :| | max | | ; HTC 1 | | l X X X m l l l lm m Center of mass (CM) and head-tail coupling parameters for MUM. Note strong suppression of CM at the plateau by the damper. 14 AB Note that at plateau the weak head-tail approximation is well-justified.

AB I? Intensity ensity scan, n, flat t ADT, , MUM: : where re is TMCI? Gain=0 Gain=1.4 Q ’ Q ’ Q' 0, 5, 15, 20 Q' 0, 5, 15, 20 Rate ImpFactor Rate ImpFactor Rate/ImpFactor Rate/ImpFactor 0.04 0.08 0.03 0.06 0.02 0.04 0.01 0.02 15 0.00 10 ImpFactor 0.00 ImpFactor 0 2 4 6 8 0 2 4 6 8 10

Beam Coherent erent Beam-Beam Main assumption: bunch length << beta-function. For transversely dipolar modes, CBB is a cross-talk of bunch CM – thus, intra-bunch matrix structure is similar to the ADT and CB: ˆ ˆ ˆ ˆ ˆ           ; X S X Z X D X C X b B X 1 1 1 1 1 12 2 ˆ ˆ ˆ ˆ ˆ           ; X S X Z X D X C X b B X 2 2 2 2 2 2 1 1   K K   ˆ ˆ      / ( / ) k cos( ) k ; B i D d k     bb 2 2   k K k K k k      * 1 exp( ). b b i 1 2 21 Here 2 identical opposite IRs are assumed (IR1 and IR5 for LHC) with 2K+1 LR   collisions for each, every one with its beta-function and separation . , k k  Alternating x/y collision for IR1/IR5 is assumed with as a difference between   the two phase advances, while is the incoherent beam-beam tune shift bb per IR. 16 AB

AB 0% Cohere erent nt BB at Platea teau: u: effect ect ~30% 1 beam        3 2.5 10 , / 2, Q bb  7 K 1 beam 17

ation Dispersion persion Equation Let’s consider a small fraction of the beam described by an NHT amplitude vector x p : ˆ ˆ ˆ ˆ ˆ              ( ) x i x S x Z X Z X i I iS X p p p p p c    Due to the frequency spread eigenvalues are slightly changed, but c eigenvector at the first approximation are the same (similar to QM). From here  1 ˆ ˆ  ˆ   ˆ          ( ) x I iS I iS X     p p p c  1 ˆ ˆ ˆ ˆ             ( ) X I iS I iS X     p p c  1 ˆ ˆ ˆ ˆ              † 1 ( ) X I iS I iS X     p p c  2 | ( ) | X J J F         1 ( ) l s x l d         c s l i J l s x x       2 | ( ) | 1 X J Fd Fd 18 l s AB