Beam-Beam Simulation for PEP-II

Transcription

1 Beam-Beam Simulation for PEP-II Yunhai Cai Accelerator Research Department-A, SLAC e + e Factories 2003, SLAC October 14, 2003

2 Outline Method of simulation: - Boundary condition - Parallel computing - Longitudinal slicing Present parameters: - Dynamical β and emittance - Luminosity - Spectrum Crossing angle. - Symplectic treatment -Benchmark Future prospect - Improvements - Upgrades Conclusion and outlook. 2

3 Solving Poisson s Equation with Reduced Region a Beam Boundary: L Beam pipe Assign potential on the reduced boundary: Φ(x, y) = dx dy G(x x,y y )ρ c (x,y ) for (x, y) L. Solve Poisson s equation with inhomogeneous boundary condition. a Y. Cai, A. W. Chao, S. I. Tzenov, and T. Tajima, Phys. Rev. ST Accel. Beams (2001). 3

4 Strategy of Parallel Computing processor total ρ exchange Sum local ρ positron beam electron beam Macro particles are distributed on many processors. The processors are divided into two groups, one for positron beam and the other for electrons. The beam distribution on grid is summed within each group and then exchanged between the groups. 4

5 Scaling on Parallel Supercomputers 140 Number of completed turns in 30 minutes ALVAREZ SP T3E Number of processors SP(IBM), T3E(CRAY), ALVAREZ(LINUX PC) are the super computers at NERSC. We gain a factor of 24 in speed with 32 processors on the SP. 5

6 Luminosity Convergence with Equal-Spacing Slices at the Peak Beam Intensities L b (10 28 cm 2 s 1 ) slices 11 slices 15 slices 21 slices 31 slices turns 1400 L b (10 28 cm 2 s 1 ) Number of slices 961(31 31) two-dimensional collisions are required for each three-dimensional collision. It takes a week to finish a simulation on 32 processors at NERSC. It is still not very well convergent. 6

7 Horizontal Size Convergence with Equal-Spacing Slices σ x + (µm) slices 11 slices 15 slices 21 slices 31 slices σ x (µm) slices 11 slices 15 slices 21 slices 31 slices turns 7

8 Vertical Size Convergence with Equal-Spacing Slices σ y + (µm) slices 11 slices 15 slices 21 slices 31 slices σ y (µm) slices 11 slices 15 slices 21 slices 31 slices turns 8

9 Convergence with Equal-Area Slices b at the Peak Intensities L b (10 28 cm 2 s 1 ) slices 5 slices 7 slices 9 slices 11 slices 13 slices 15 slices turns 1400 L b (10 28 cm 2 s 1 ) equal spacing slices equal area slices Number of slices 225(15 15) two-dimensional collisions are required for each three-dimensional collision. It takes two days to complete a run. b K. Hirata, H. Moshammer, and F. Ruggiero, Particle Accelerator, (1993). 9

10 Convergence with Equal-Area Slices and Linear Interpolation between the Slices L b (10 28 cm 2 s 1 ) slices 5 slices 7 slices 9 slices 11 slices 13 slices turns L b (10 28 cm 2 s 1 ) equal spacing slices equal area slices equal area slices with interpolation Number of slices 25(5 5) two-dimensional collisions are required for each three-dimensional collision. It takes eight hours to complete a job. 10

11 Dynamic Beta and Emittance in Horizontal Plane 1. Beam-beam parameter ξ 0 = r e Nβ 0 2πγσ x (σ x + σ y ) 2. Beam-beam tune shift: where µ =2πν. 3. Dynamic beta: cos µ =cosµ 0 2πξ 0 sin µ 0 β = β 0 sin µ 0 sin µ 4. Dynamic emittance: c 1+2πξ 0 cot µ 0 ɛ = ɛ 0 1+4πξ0 cot µ 0 4π 2 ξ0 2 c K. Hirata and F. Ruggiero, Particle Accelerator, (1990). 11

12 Dynamic Beta and Emittance for the HER 1 β x * (m) ε x (nm rad) σ x * (µm) LEGO calculation Hirata & Ruggiero formula ν x Positron Bunch Population(10 10 ) The method introduced by Hirata and Ruggiero is used in our beam-beam simulation. 12

13 Dynamic Beta and Emittance for the LER 1 β x * (m) ε x (nm rad) LEGO calculation Hirata & Ruggiero formula σ x * (µm) ν x Electron Bunch Population(10 10 ) The method introduced by Hirata and Ruggiero is used in our beam-beam simulation. 13

14 PEP-II Parameters, June 10, 2003 Parameter LER(e+) HER(e-) E (Gev) N βx (cm) βy (cm) ɛ x (nm-rad) ɛ y (nm-rad) ν x ν y ν s σ z (cm) σ p τ t (turn) τ s (turn)

15 Simulation Parameters Parameter Description Value N m # macro particles/bunch n (turn) tracking time mesh cell (µm 2 ) 15 2 n s number of slices 5 15

16 Long-Term Convergence in Slicing at High Intensities? 18 5 slices 7 slices L b (10 30 cm 2 s 1 ) turns At peak intensities(1.6/1.22ma), simulated luminosity with 5 slices is cm 2 s 1. With 7 slices, we have cm 2 s 1. The difference is less than 3%. 16

17 Reach Equilibrium at High Intensities? L b (10 30 cm 2 s 1 ) turns x 10 4 With interpolated slices, beams reach their equilibrium after three damping time. 17

18 Simulation at Low Intensities(0.16/0.12mA) L b (10 28 cm 2 s 1 ) e + e σ x (µm) e + e σ y (µm) Σ x = 141µm, Σ y =6.9µm, L sp = cm 2 s 1 ma 2 should be compared to the measured values: Σ max = 160µm, Σ min =7.01µm, L sp = cm 2 s 1 ma 2 on June 3,

19 Specific luminosity vs. LER X-tune Lsp (sim.) Lsp_geom

20 Y spot sizes vs LER X-tune s*ly s*hy

21 Beam-Beam Effects of PEP-II, June 10, 2003 (ν + x =0.5125) L b (10 30 cm 2 s 1 ) ν x = ν x = observation i + *i (ma 2 ) 5 L sp (10 30 cm 2 s 1 ma 2 ) ν x = ν = x observation i + *i (ma 2 ) Compared with the observed value cm 2 s 1 at the peak currents. 21

22 Beam-Beam Effects of PEP-II, June 10, 2003 (ν + x =0.5125) 20 L b (10 30 cm 2 s 1 ) L sp (10 30 cm 2 s 1 ma 2 ) σ x (µm) e + e σ y (µm) e + e Loss(%) e + e i + *i (ma 2 ) There are still room to increase the beam currents. Beams are matched at high currents in the vertical plane. There are more beam loss for electrons than positrons especially at higher currents. That can be a problem if the dynamic aperture is to large enough. 22

23 Beam-Beam Effects of PEP-II, June 10, 2003 (ν + x =0.5125) e + e ξ x i + *i (ma 2 ) e + e ξ y i + *i (ma 2 ) The beam-beam parameters are: ξ + x =0.2239, ξ + y =0.0776, ξ x =0.0714, and ξ y =

24 Beam-Beam Effects of PEP-II, June 10, 2003 (ν + x =0.5125) Σ x (µm) Luminosity scan Luminous region i + *i (ma 2 ) 10 8 Σ y (µm) 6 Luminosity scan Luminous region i + *i (ma 2 ) Based on the beam-beam scan at June 3, 2003, we have Σ max = 161µm, Σ min =7.01µm, and L sp = cm 2 s 1 ma 2. The luminous region seen by BaBar is Σ x =77µm. The larger horizontal beam sizes in the experimental observations may be attributable to the huge beta beating in the High Energy Ring. 24

25 Beam-Beam Effects of PEP-II, June 10, 2003 (ν + x =0.5125) 1 HER x ν x HER y ν y LER x ν x LER y ν y Compared to the measured spectrums of collision at peak currents on June 5, 2003, we have excellent agreements in the horizontal plane. It is not clear why the spectrums differ in the vertical plane. 25

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27 Symplectic Treatment of a Finite Crossing Angle X + -2 s + sinφ Z + IP φ X - Z - The whole process can be summarized and written as T ± (s ±,φ) O BB (x±,y ±,φ) T ± (s ±,φ) 1, (1) where T ± (s ±,φ)=d z ± (s ± ) R ± y (φ) D x ± (s ±,φ), (2) 27

28 Sympletic Rotation around Y Axis x = x cos φ(1 P x tan φ/p s ), P x = cosφp x +sinφp s, ỹ = P y x tan φ y + (P s P x tan φ), P y = P y, z = z δ = δ, (1 + δ)x tan φ (P s P x tan φ), where P s = (1 + δ) 2 P 2 x P 2 y. It can be expressed as a Lie operator exp(: xp s : φ). Itisalsoexactfor ultra-relativistic particles. 28

29 Geometric Effect from a Finite Crossing Angle d L/L φ(mrad) R L = L L 0 = 2 π aeb K 0 (b), a = σ y 2σ z σ p y,b= a 2 [1 + ( σ z σ x tan φ) 2 ], d K. Hirata, Phys. Rev. Lett (1995). 29

30 KEKB Parameters, May 13, 2003 Parameter LER(e+) HER(e-) E (Gev) N βx (cm) βy (cm) ɛ x (nm-rad) ɛ y (nm-rad) ν x ν y ν s σ z (cm) σ p τ t (turn) τ s (turn)

31 KEKB Parameters(φ = ±11mrad), 5/13/ Yunhai Ohmi & Tawada Observation L b (10 30 cm 2 s 1 ) Turn 31

32 KEKB Parameters(φ = ±11mrad), 5/13/2003 σ x + (µm) Yunhai Ohmi & Tawada σ x (µm) σ y (µm) σ y + (µm) Yunhai Ohmi & Tawada Yunhai Ohmi & Tawada Yunhai Ohmi & Tawada Turn 32

33 KEKB with a Different Crossing Angle, 20 L b (10 30 cm 2 s 1 ) σ x (µm) e + e σ y (µm) e + e Crossing angle(mrad) Similar results have been obtained by Ohmi and Tawada. 33

34 Comparison of KEKB and PEP-II Parameters Parameter KEKB(e+) PEP-II(e+) E (Gev) N β x (cm) β y (cm) ɛ x (nm-rad) ɛ y (nm-rad) ν x ν y ν s σ z (cm) σ p τ t (turn) τ s (turn)

35 Break Down of Luminosity Between KEKB and PEP-II Description L b L sp Gain present PEP-II % aspect ratio % aspect ratio % aspect ratio % β y/2, σ z / % τ/ % ν x,ν y,kekb(φ = 0) % L b is the bunch luminosity in unit of cm 2 s 1 and L sp is the specific luminosity in unit of cm 2 s 1 ma 2. 35

36 PEP-II with a Crossing Angle 8 L b (10 30 cm 2 s 1 ) σ x (µm) e + e σ y (µm) e + e Crossing angle(mrad) For φ = ±3mrad, we see a degradation of luminosity by 43%. Similar results have been obtained by Ohmi and Tawada. 36

37 Horizontal Beam Spectrums with a Crossing Angle 1 φ = ν x 1 φ = 1 mrad ν x 1 φ = ν x Tail-tail motion is seen in the spectrums. 37

38 Conclusion Over the past few years, we have completed the work of implementing a three-dimensional, fully self-consistent, and parallel PIC code. Now, we can complete a single simulation within 8 hours on parallel supercomputers. The code has been benchmarked against a similar code developed by Ohmi at KEK. The simulated luminosity is in a agreement with the experimental observation. We have interpolated achievable luminosities based on the simulations and observations at KEKB and PEP-II. It showed that it is very important to reduce the vertical beam size to gain higher luminosity. We found a large degradation of luminosity even with a small crossing angle at current working point. This effects should be verified experimentally at PEP-II. 38

39 Acknowledgments Witold Kozanecki (PEP-II & BaBar) Ilya Narsky (BaBar) Frank Porter (BaBar) John Seeman (PEP-II) Kazuhito Ohmi (KEKB) Masafumi Tawada (KEKB) 39