Accelerator Physics Statistical and Beam-Beam Effects. G. A. Krafft Old Dominion University Jefferson Lab Lecture 14

Transcription

1 Accelerator Phsics Statistical and Beam-Beam Effects G. A. Krafft Old Dominion Universit Jefferson Lab Lecture 4 Graduate Accelerator Phsics Fall 7

2 Waterbag Distribution Lemons and Thode were first to point out SC field is solved as Bessel Functions for a certain equation of state. Later, others, including m advisor and I showed the equation of state was eact for the waterbag distribution. p HT e m A H H T m m SC nr ddn H / e H H d d m H p d d m SC v m SC Graduate Accelerator Phsics Fall 7

3 Integrals m m SC dda dd p H m SC H m m m SC A rdrd A H p p H m SC H m 3 H r drd p dd HA p v m dd m m m SC A H p H H m H m m SC SC SC m H H m H Graduate Accelerator Phsics Fall 7

4 Self-consistent potential solves SC en SC D H D m H H v p e n m e n m / BK r / Debe Length Analtic solutions in terms of Modified Bessel Functions e r m AI r B b boundar condition A chosen so that solution without I D solution to inhomogeneous eqn. D Graduate Accelerator Phsics Fall 7

5 Now Equation for Beam Radius r r r r r D p Ir p p b D A m At r r b the densit vanishes / p p p Irb / D p b n r n In figure nˆ b / D / D I r / I r I r b n b D Graduate Accelerator Phsics Fall 7

6 Debe Length Picture* *Davidson and Qin Graduate Accelerator Phsics Fall 7

7 Collisionless (Landau) Damping Other important effect of thermal spreads in accelerator phsics Longitudinal Plasma Oscillations ( D) n vn t d e E dt m E en Graduate Accelerator Phsics Fall 7

8 Graduate Accelerator Phsics Fall 7 Linearied p i t p v n n t e E t m E e n en e n E n n t m m en n e m In fluid limit plasma oscillations are undamped

9 Vlasov Analsis of Problem,,, F p t p dp dp e v e Fe t p e th order solution e F F p linearied e F dp e i, F v Fe e t p F dp e n Graduate Accelerator Phsics Fall 7

10 Initial Value Problem Laplace in t and Fourier in it Fˆ dte F t Im large enough to converge it ˆ, t ˆ, F t d e Fˆ il/ L / / i Fe l, p, t e l F / p F,, ˆ e l p l, v l L L v l L ˆ l, C it d dte F t i F F t dt l L l l t e e lf p / p ˆ dp l, L v l / L ei Fe l, p, t dp v l / L Graduate Accelerator Phsics Fall 7

11 Dielectric function Landau (self-consistent) dielectric function, ˆ,, D l l N l e L F p / p Dl, dp l v l / L Solution for normal modes are Dl, D l, p dp e dp m v l / L v F p / n l / L F p Graduate Accelerator Phsics Fall 7

12 Collisionless Damping For Lorentian distribution F n p p i dp p l / Lm p p l / Lm Landau damping rate l p i L Graduate Accelerator Phsics Fall 7

13 Luminosit and Beam-Beam Effect Luminosit Defined Beam-Beam Tune Shift Luminosit Tune-shift Relationship (Krafft-Ziemann Thm.) Beam-Beam Effect Graduate Accelerator Phsics Fall 7

14 Events per Beam Crossing In a nuclear phsics eperiment with a beam crossing through a thin fied target Target Number densit n Beam Probabilit of single event, per beam particle passage is P σ is the cross section for the process (area units) l n l Graduate Accelerator Phsics Fall 7

15 Collision Geometr Beam Beam Probabilit an event is generated b a single particle of Beam crossing Beam bunch with Gaussian densit* N ep / ep / P d 3/ ep / N ep / ep / * This epression still correct when relativit done properl Graduate Accelerator Phsics Fall 7

16 Collider Luminosit Probabilit an event is generated b a Beam bunch with Gaussian densit crossing a Beam bunch with Gaussian densit Event rate with equal transverse beam sies Luminosit P L NN dn dt fnn 4 fn N 4 L 33 ~ sec cm, for f MH, N N, microns Graduate Accelerator Phsics Fall 7

17 Beam-Beam Tune Shift As we ve seen previousl, in a ring accelerator the number of transverse oscillations a particle makes in one circuit is called the betatron tune Q. An deviation from the design values of the tune (in either the horiontal or vertical directions), is called a tune shift. For long term stabilit of the beam in a ring accelerator, the tune must be highl controlled. M tot * cos sin * / f sin / cos * cos sin * * cos / f sin / cos / f sin Graduate Accelerator Phsics Fall 7

18 * Tr M cos tot cos sin f 4 f * * Q f Graduate Accelerator Phsics Fall 7

19 Bessetti-Erskine Solution -D potential of Bi-Gaussian transverse distribution Q, ep ep Potential Theor gives solution to Poisson Equation, Bassetti and Erskine manipulate this to, ep ep Q q q dq 4 q q Graduate Accelerator Phsics Fall 7

20 i Q i E Im w ep w i Q i E Re w ep w w Comple error function We need -D linear field for small displacements E, dq q Q 3/ q Graduate Accelerator Phsics Fall 7

21 Can do the integral analticall dq q q q q q q dq 3/ / / q q q q dq Similarl for the -direction E, Q Graduate Accelerator Phsics Fall 7

22 Linear Beam-Beam Kick Linear kick received after interaction with bunch mc q E v B t, t dt b relativit, for oppositel moving beams mc q E t, t dt Following linear Bassetti-Erskine model E,,, t q ep q moves with t,, ct ct Graduate Accelerator Phsics Fall 7

23 / f Linear Beam-Beam Tune Shift mc r 4 mc / f Both beams relativistic q q c N r e Nr From linear Bassetti-Erskine model, and replacing the beam sie N r i / / / Argument entirel smmetric wrt choice of bunch and i N r Niri i N iri i i i / / / i Graduate Accelerator Phsics Fall 7

24 Luminosit Beam-Beam tune-shift relationship Epress Luminosit in terms of the (larger!) vertical tune shift (i either or ) L fn r e r i i i i I i * * ii ii i / / Necessar, but not sufficient, for self-consistent design Epressed in this wa, and given a known limit to the beam-beam tune shift, the onl variables to manipulate to increase luminosit are the stored current, the aspect ratio, and the β* (beta function value at the interaction point) Applies to ERL-ring colliders, stored beam (ions) onl Graduate Accelerator Phsics Fall 7

25 Luminosit-Deflection Theorem Luminosit-tune shift formula is linearied version of a much more general formula discovered b Krafft and generalied b V. Ziemann. Relates eas calculation (luminosit) to a hard calculation (beam-beam force), and contains all the standard results in beam-beam interaction theor. Based on the fact that the relativistic beam-beam force is almost entirel transverse, i. e., -D electrostatics applies. Graduate Accelerator Phsics Fall 7

26 E -D Electrostatics Theorem Q 4 F F d d on n / Q n b / Q ero centerred Q d b d / Q i F i F Q Q b n n d d b Graduate Accelerator Phsics Fall 7

27 b b b b b b F b d Generalies c E take b Transverse interaction in the beam-beam problem p q q Graduate Accelerator Phsics Fall 7

28 D b m / m q q b n n mc d d b b 4 e e 4 mc D b N r n b n d r L b N N n b n d L b Lb N 4 r e N 4 r e b b D b e Graduate Accelerator Phsics Fall 7

29 Graduate Accelerator Phsics Fall 7 * / / as before Maimum when, e D b D b f N L r D D b b b b

30 Luminosit-Deflection Pairs Round Beam Fast Model D b Gaussian Macroparticles Nreb NN L b Bassetti _ Erskine ; ; D b D b b b Lb Db Lb NN b ep ep Smith-Laslett Model bˆ 4 bˆ 4 4 ˆ ˆ bˆ b ˆ bˆ bˆ ˆ ˆ4 ˆ ˆ4 3 sinh sinh 5/ b 4 ˆ ˆ3 ˆ ˆ Nreb 4b b 3b b sinh sinh ˆ 4 3/ b AB bˆ bˆ 4 b b NN 4 4 ˆ 3 ˆ ˆ b b b AB 4b b 4b b bˆ b b A B Graduate Accelerator Phsics Fall 7