X. Ding, UCLA AAG June 27, 2012

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##### X. Ding, UCLA AAG June 27, 2012

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1. Optimized Target Parameters and Meson Production by IDS120h with Focused Gaussian Beam and Fixed Emittance(Update) X. Ding, UCLA AAG June 27, 2012

2. Focused Incident Proton Beam at 8 GeV(Beam radius is fixed at 0.12 cm at z=-37.5 cm) Relative normalized meson production is 0.84 of max at β* of 0.3 m for x = y = 5 m. For low β* (tight focus) the beam is large at the beginning and end of the interaction region, and becomes larger than the target there.

3. Focused Incident Proton Beam at 8 GeV (Cont’d)(Beam radius is fixed at 0.12 cm at z=-37.5 cm) Non-Linear Fit (Growth/sigmoidal, Hill) Y=N/(1+K2/beta-2) N=1.018 Sqrt(K2)=0.1368 Linear emittance is 5 μm with beam radius of 0.1212 cm and β* of 0.3 m.

4. Gaussian distribution(Probability density) • In two dimensional phase space (u,v): where u-transverse coordinate (either x or y), v = α u+β u’ α, β are the Courant-Snyder parameters at the given point along the reference trajectory. In polar coordinates (r, θ): u = r cosθ v = r sinθ u’= (v-α u)/β=(r sinθ-α u)/β

5. Distribution function method Random number generator: Θ = 2π*rndm(-1) R = sqrt(-2*log(rndm(-1))*σ

6. Gaussian distribution(Fraction of particles) • The fraction of particles that have their motion contained in a circle of radius “a” (emittance ε = π a2/β) is

7. Fraction of particles Normalized emittance: (βγ)εKσ

8. Focused beam • Intersection point (z=-37.5 cm): α*= 0, β*, σ* • Launching point (z=-200 cm): L = 200-37.5 = 162.5 cm α = L/β* β = β*+ L2/β* σ = σ*sqrt(1+L2/β*2) These relations strictly true only for zero magnetic field.

9. Setting of simple Gaussian distribution • INIT card in MARS.INP (MARS code) INIT XINI YINI ZINI DXIN DYIN DZIN WINIT XINI = x0 DXIN = dcx0 YINI = y0 DYIN = dcy0 ZINI = z0 DZIN = dcz0 = sqrt(1-dcx02-dcy02) (Initial starting point and direction cosines of the incident beam)

10. Setting with focused beam trajectories • Modeled by the user subroutine BEG1 in m1510.f of MARS code xv or xh (transverse coordinate: u); x'v or x'h (deflection angle: u’) XINI = x0 + xh DXIN = dcx0 + x’h YINI = y0 + xv DYIN = dcy0 + x’v ZINI = z0 DZIN = sqrt(1-DXIN2-DYIN2)

11. Optimization of target parameters • Fixed beam emittance (εKσ) to π (σ)2/β • Optimization method in each cycle (Vary beam radius or beam radius σ*, while vary the β* at the same time to fix the beam emittance; Vary beam/jet crossing angle; Rotate beam and jet at the same time) We also optimized the beam radius and target radius separately (not fixed to each other).

12. Optimized Target Parameters and Meson Productions at 8 GeV(Linear emittance is fixed to be 4.9 μm )

13. Optimize beam radius and target radius separately(Linear emittance is fixed to be 4.9 μm ) We found almost no improvement in optimized meson production if the beam radius is not fixed at 30% of target radius and optimized separately!

14. Optimized Meson Productions at 8 GeV(Linear emittance is fixed to be 5 μm )

15. Optimized Target Parameters and Meson Productions at 8 GeV(Linear emittance is fixed to be 10 μm )

16. Procedure of Simulation without Angular Momentum 1). Create a beam (simple Gaussian or focused) with target radius of 0.404 cm, beam angle of 117 mrad and jet angle of 137.6 mrad at z=-37.5 cm. 2). Track back the beam to z=-200 cm under the SC field and no HG Jet. 3). Based on the backward beam at z=-200 cm, create the forward beam at z=-200 cm. 4). Read the forward beam and run MARS15.

17. Procedure of Simulation without Angular Momentum (Cont’d) • Meson production with emittance of 5 μm is 1.30141*10^5. (Old Method with single centroid particle track back and using Twiss formulae at z=-200 cm w/t SOL field, Meson production is 1.09957*10^5.)

18. Meson Productions without Angular Momentum

19. Formulae Related with Angular Momentum  q=1.610-19 (C), c = 3108 (m/s) 1 (GeV) = (1091.610-19) (J) = (109q) (J) 1 (J) =1/(109q) (GeV) Px= qBcy/2 (J) Py= -qBcx/2 (J) Px=Bcy/(2109) (GeV)~0.15By (GeV) Py=-Bcx/(2109) (GeV)~-0.15Bx (GeV)

20. Estimation of Angular Momentum Effect • Proton Beam: E0=0.938272 GeV KE=8 GeV ET=8.9382 GeV P=(ET2-E02)1/2=8.88888 GeV  • (x0,y0,Px,Py), without angular momentum (x0,y0,Px+Px,Py+Py), with angular momentum

21. Estimation of Angular Momentum Effect (Cont’d) • Z = -37.5 cm, B ~ 20.19 T • Z = -50 cm, B ~ 20.09 T • Z = -100 cm, B ~ 17.89 T • Z = -150 cm, B ~ 14.03 T • Z = -200 cm, B ~ 11.29 T  Pz = (P2-(Px+Px)2-(Py+Py)2)1/2

22. Procedure of Simulation with Angular Momentum 1). Create simple Gaussian distribution at -37.5 cm. 2). Transform each particle according to px -> px + qBy/2, py -> py> - qBx/2, using the x and y of the individual particle. (Use the B at the point we generate the particles, so -37.5 cm) 3). Track the resulting distribution backwards to z=-50 cm,-100 cm, -150 cm and -200 cm under the SC field and no HG Jet. Based on this, create the forward beam at different position. 4). Read the forward beam and run MARS15.

23. Procedure of Simulation with Angular Momentum (Cont’d) Meson production (simple Gaussian distribution) without angular momentum for launching point at z=-200 cm is 1.29918*10^5.

24. BackUp

25. Courant-Snyder Invariant

26. Emittance (rms) and Twiss Parameters

27. Effect of Solenoid Field[Backtrack particles from z = -37.5 cm to z = - 200 cm.](Could then do calculation of α, β, σ at z = -200 cm, but didn’t)

28. Effect of Solenoid Field