Characteristics of beam-electron cloud interaction

Transcription

1 Charatriti of bam-ltron loud intration Tun hift and intabilit K. Ohmi KEK Int. Workhop on Two-tram Intabiliti in Partil Alrator and Storag KEK Tukuba Japan

2 Bam-ltron intration Bam partil ar loalid in th rgion of Eltron ditribut all ovr th ring mab. Not that th long rang Coulomb for ~1/r. Eltron mov during th intration with th bam. How do ltron ontribut ingl bunh intabiliti tun hift and lod orbit?

3 Tun hift Naiv ida. L r ρ β γ ν ε ρ m. lindrial r E r L r E ρ β γ ν ε ρ m. planar

4 Modl of th bam-ltron loud tm Bam i aumd to b an uniform flat top ditribution along <<. W onidr onl dipol momnt in th tranvr motion. Tranvr bam ditribution i haratrid b a ontant Gauian r.m.. i. Th bam i rprntd b th dipol momnt or it of a ri of rigid tranvr Gauian miro-bunh j j ditributd along in numrial approah.

5 I. Linar and rigid Gauian loud modl Eltron loud i rprntd b a tranvr rigid Gauian ditribution. Linar for btwn bam and loud. Analti approah in whih bam and loud ar rprntd b th dipol momnt and t rptivl. 1 t k r d d b b b γ λ β t t k r dt t d b b λ λ. λ b : lin dnit of ltron loud and bam k: linar oupling offiint. k for qual bam-loud i

6 What i k λ b If w aum a rigid Gauian loud with r.m.. i Σ A and Σ A apart from th rliabilit A A A A A k ρ π λ for A A k ρ π λ

7 A>>1 for th rigid Gauian modl Eltron loud do not mov during / if A A A A k A A L r k r ρ π λ ρ β γ γ λ ν β Cl. 1 >> 1. 1 >> << k k r b λ Tun hift i dtrmind b onl th quation for bam. Stm i tabl. No intabilit! k r γ λ β β

8 A A 1 Eltron mov obing Eq.. λbr 1 KEKB t. 1. W olv th oupld quation Eq.1 and.. Eltron loud ditribut in th rgion of <<L. 3. Intration bgin from tt. 4. Cntr of ltron loud i ro amplitud and ro vloit bfor intration. 5. Bam ha dipol momnt along. b : finit for <<. k. t'in t t t b t' dt'

9 Eq.1 b b b d W r t d t d β γ λ ' ' ' ' in L W b λ λ Wak fild Th tm i untabl dpnding on nhrotron motion and hromatiit. r γ λ β β

10 I th ohrnt tun hift dtrmind b β. No. W hav to inlud a orrtiv fft of ltron loud: I.. W. Cohrnt Tun hift A. Chao ttbook uniform modl oh γ λ β β β g Z d T r i g π in 1 b L Z λ λ 1 1 Im

11 Cohrnt tun hift for A A 1 λr γ β oh β β in Stati loud / ν Th tun hift i rdud b th orrtiv fft. Intration of bam and an ltron ltron bam

12 Anali for th rigid Gauian modl A>>1 Tun hift i dtrmind b tati loud ditribution. No intabilit. A A 1 Tun hift i rdud b orrtiv fft Intabilit. Numrial anali with a oft ltron loud.

13 ; 1 j j a n j G a a r K d d δ γ F n j a a j G j t t r n N dt d 1 ; δ F r i F i F 1 Bam i rprntd b a ri of rigid tranvr Gauian miro-bunh ditributd along. Numrial approah II. Inluding nonlinarit and oft ltron loud ditribution

14 Paramtr KEKB n : Numbr of maro-ltron >1 n : Numbr of poitron miro-bunh ~1 CloudiA A 11.4mm.6mm ~ 15 4mm 3mm.

15 Tun hift Calulat kik whih miro-bunh prin from ltron loud whn all th miro-bunh with a rtain mall diplamnt < or < pa through th loud.

16 Vrtial dp m For 11 tun hift i onitnt with th analtial timat for 11. Inraing th loud i kik bom a ontant along. Th kik i lo to that from th tati loud.

17 Horiontal -7 dp m For 11 tun hift i onitnt with th analtial timat for 11. Inraing th loud i kik bom a ontant along. Th kik i lo to that from th tati loud.

18 H-kik for 11 and 55 ar th am. H-kik for 15 i largr than that for 55. Tun hift i dtrmind b maroopi trutur mmtr of th ltron loud. Eltron loud do not m to mov b th bam. Intabilit?

19 Wak for Calulat kik whih miro-bunh prin from ltron loud whn th firt miro-bunh with a rtain mall diplamnt < or < pa through th loud.

20 Vrtial wak fild W 1 m m

21 Horiontal wak fild W 1 m m

22 Horiontal wak fild trngth and loud i 3 16 R S /Q m Σ

23 Vrtial and horiontal wak fild for vr larg loud [15 and 15] Vrtial wak for 15 and 15 Vrtial wak for 15 and 15 -W1 m^ W1 m^ m - m

24 Charatriti of Wak for Vr. wak i about twi largr than th analtial timat. ρ th 1~1 1 m -3 Hor. wak i muh largr than th analti timat. ρ th 1~1 1 m -3 Analti timat A1 W /W / 3/ Numrial imul. A>>1 W /W / 1/ Th wak for i aturatd for inraing th loud i largr than 15. Eltron nar th bam r vral ontribut th wak for not. Th wak for do not dpnd on th loud apt ratio if th i i largr than vral.

25 Summar Th naiv formula for tun hift i right. ν r γ β ρ L Th tun hift i dtrmind b th mmtr of ltron loud. Eltron far from th bam i important. Eltron nar th bam r vral ontribut th wak fild. Eltron far from th bam ar not movd b bam for.

26 Rmaind ompl Ar th wak and kik onitnt? dp W1 m^- Vrtial wak for 15 and m - m Pinh fft. Suprpoition rul of th wak for.

27 Simulation for had-tail intabilit aud b ltron loud Rigid Gauian modl Partil-in-ll imulation trong-trong modl

28 Rigid Gauian modl a Turn V. i mm Turn 1 b Turn Chromati had-tail fft i n. Work with F.Zimmrmann.

29 Partil-in-ll mthod m 1 4 <> b <> b ig m b 11^ ^11 4 Q / Turn m <> b <> b / ig m a 11^ Turn 4 11^11 Q.15 ig m Q Q 4 Q 8 Q Turn Q 1 Thrhold ~51 11 m -3 Conitnt with R.Giovanni t.al. Chromatiit ur th intabilit. No hromati had-tail. PAC1 KEK-prprint 1-49

30 Aknowldgmnt Thi work i mainl don at a viit to SLAC in thi wintr and ummr 1. Th author thank fruitful diuion with A. Chao H. Fukuma S. Hift T. Iiri K.Oid E. Prvdntv G.Rumolo G. Stupakov F.Zimmrmann and mmbr of KEKB ommiioning.

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