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Optimal Beamlines for Beams with Space Charge Effect S.V.Miginsky Budker Institute of Nuclear Physics, Novosibirsk, Russia.

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Presentation on theme: "Optimal Beamlines for Beams with Space Charge Effect S.V.Miginsky Budker Institute of Nuclear Physics, Novosibirsk, Russia."— Presentation transcript:

1 Optimal Beamlines for Beams with Space Charge Effect S.V.Miginsky Budker Institute of Nuclear Physics, Novosibirsk, Russia

2 Basics and previously published results An optimizer for high-current beamlines. ICAP’04 -> NIM A 558 (2006). Minimization of Space Charge Effect. SR'06 -> NIM ??? Kapchinsky – Vladimirsky : rms values Space charge can be neglected only if: AND! If space charge is significant and charge density is not homogeneous, emittance dilution occurs due to nonlinearity of the effect.

3 Basic space charge effects Effect of longitudinal inhomogeneity : each slice is uniformly charged and independent from others. Slices have different currents and states. Effect of transverse inhomogeneity : nonlinear radial force

4 Basic assumptions Longitudinal circular symmetric beam; slices are charged uniformly; no thermal emittance; slices move independently; independent on transverse inhomogeneity. Transverse circular symmetric beam; no thermal emittance; laminar motion : trajectories don't cross each other; independent on longitudinal inhomogeneity.

5 Emittance compensation Longitudinal Transverse Small (linear!) vibration around main phase trajectory, x + δx = x·(1+δ): If the initial beam emittance is zero and vibration is perfectly linear, phase portraits of all the slices coincide twice per a period of vibration and the emittance is zeroed at these points.

6 Basic results Emittance dilution in an optimized space charge affected beamline is many times less than in arbitrary one due to emittance compensation effect. Effect of longitudinal inhomogeneity is typically the strongest. Effect of transverse inhomogeneity is usually weaker, but comparable to the previous one. Combined effect causes greater emittance dilution. Parameters of the optimal beamline in this case are compromise between the two above cases. The best emittance compensation occurs when the charge vibration phase is about 2π. The emittance is significantly better in 2nπ minima, than in (2n+1)π ones. The obtained analytical formulae improved by numerical simulation provide good estimations of emittance dilution in optimized beamlines in the mentioned cases. They also gives approximate parameters of optimized beamlines. The differences between uniformly focusing beamlines and non- uniform ones are moderate, so the obtained estimates can be applied well to arbitrary beamlines.

7 Basic scaling I : peak current; r : rms beam size; n : φ = 2nπ; ξ : a coefficient depending on the considered effect, the type of the beamline, etc. 8·10 -3 …5·10 -2 Valid for longitudinal, transverse and combined effect; for uniform and nonuniform beamlines.Valid for longitudinal, transverse and combined effect; for uniform and nonuniform beamlines. For optimal beamlines only. An optimal beamline is to provide matched focusing and φ = 2nπ.For optimal beamlines only. An optimal beamline is to provide matched focusing and φ = 2nπ. Parameters of optimal beamlines were estimated.Parameters of optimal beamlines were estimated. All these parameters are considered as a good initial approximation for a designed beamline.All these parameters are considered as a good initial approximation for a designed beamline.

8 Bunching : basic equations Hamiltonian Basic equation Longitudinal effect.Longitudinal effect. Persistent energy.Persistent energy. Matched continuous focusing : x = const.Matched continuous focusing : x = const. Adiabatic damping Analytical estimate Charge vibration phase taper Valid only if υ:υ:υ:υ:

9 Bunching : simulation Effect of longitudinal inhomogeneity.Effect of longitudinal inhomogeneity. 2π-minimum emittance (triangles; red solid line – 0.0215·υ 1/3 ).2π-minimum emittance (triangles; red solid line – 0.0215·υ 1/3 ). Optimal focusing (blue dashed line).Optimal focusing (blue dashed line). Optimal length (green dash-dot line), ≈ 15.7·υ -1/3.Optimal length (green dash-dot line), ≈ 15.7·υ -1/3.

10 Bunching : simulation Effect of longitudinal inhomogeneity.Effect of longitudinal inhomogeneity. No bunching, υ = 1.No bunching, υ = 1. Emittance vs. length L and focusing g.Emittance vs. length L and focusing g.

11 Bunching : simulation Effect of longitudinal inhomogeneity.Effect of longitudinal inhomogeneity. υ = 2 bunching.υ = 2 bunching. Emittance vs. length L and focusing g.Emittance vs. length L and focusing g.

12 Bunching : simulation Effect of longitudinal inhomogeneity.Effect of longitudinal inhomogeneity. υ = 5 bunching.υ = 5 bunching. Emittance vs. length L and focusing g.Emittance vs. length L and focusing g.

13 Bunching : simulation Effect of longitudinal inhomogeneity.Effect of longitudinal inhomogeneity. υ = 10 bunching.υ = 10 bunching. Emittance vs. length L and focusing g.Emittance vs. length L and focusing g.

14 Bunching : simulation Effect of longitudinal inhomogeneity.Effect of longitudinal inhomogeneity. υ = 20 bunching.υ = 20 bunching. Emittance vs. length L and focusing g.Emittance vs. length L and focusing g.

15 Bunching : simulation Effect of combined inhomogeneity.Effect of combined inhomogeneity. 2π-minimum emittance (triangles; red solid line – 0.0349·υ 0.28 ).2π-minimum emittance (triangles; red solid line – 0.0349·υ 0.28 ). Optimal focusing (blue dashed line).Optimal focusing (blue dashed line). Optimal length (green dash-dot line), ≈ 12.7·υ -0.28.Optimal length (green dash-dot line), ≈ 12.7·υ -0.28.

16 Bunching : simulation Effect of combined inhomogeneity.Effect of combined inhomogeneity. No bunching, υ = 1.No bunching, υ = 1. Emittance vs. length L and focusing g.Emittance vs. length L and focusing g.

17 Bunching : simulation Effect of combined inhomogeneity.Effect of combined inhomogeneity. υ = 2 bunching.υ = 2 bunching. Emittance vs. length L and focusing g.Emittance vs. length L and focusing g.

18 Bunching : simulation Effect of combined inhomogeneity.Effect of combined inhomogeneity. υ = 5 bunching.υ = 5 bunching. Emittance vs. length L and focusing g.Emittance vs. length L and focusing g.

19 Bunching : simulation Effect of combined inhomogeneity.Effect of combined inhomogeneity. υ = 10 bunching.υ = 10 bunching. Emittance vs. length L and focusing g.Emittance vs. length L and focusing g.

20 Bunching : simulation Effect of combined inhomogeneity.Effect of combined inhomogeneity. υ = 20 bunching.υ = 20 bunching. Emittance vs. length L and focusing g.Emittance vs. length L and focusing g.

21 Accelerating : basic equations Hamiltonian Basic equation Longitudinal effect.Longitudinal effect. Persistent current.Persistent current. Matched continuous focusing : x = const.Matched continuous focusing : x = const. Adiabatic damping Analytical estimate Charge vibration phase taper Valid only if α:

22 Accelerating : simulation Effect of longitudinal inhomogeneity.Effect of longitudinal inhomogeneity. 2π-minimum emittance (triangles; red solid line – 0.0220·α -0.136 ).2π-minimum emittance (triangles; red solid line – 0.0220·α -0.136 ). Optimal focusing (blue dashed line).Optimal focusing (blue dashed line). Optimal length (green dash-dot line), ≈ 11.96 + 6.05·α.Optimal length (green dash-dot line), ≈ 11.96 + 6.05·α.

23 Accelerating : simulation Effect of longitudinal inhomogeneity.Effect of longitudinal inhomogeneity. No accelerating, α = 1.No accelerating, α = 1. Emittance vs. length L and focusing g.Emittance vs. length L and focusing g.

24 Accelerating : simulation Effect of longitudinal inhomogeneity.Effect of longitudinal inhomogeneity. α = 2 accelerating.α = 2 accelerating. Emittance vs. length L and focusing g.Emittance vs. length L and focusing g.

25 Accelerating : simulation Effect of longitudinal inhomogeneity.Effect of longitudinal inhomogeneity. α = 5 accelerating.α = 5 accelerating. Emittance vs. length L and focusing g.Emittance vs. length L and focusing g.

26 Accelerating : simulation Effect of longitudinal inhomogeneity.Effect of longitudinal inhomogeneity. α = 10 accelerating.α = 10 accelerating. Emittance vs. length L and focusing g.Emittance vs. length L and focusing g.

27 Accelerating : simulation Effect of longitudinal inhomogeneity.Effect of longitudinal inhomogeneity. α = 20 accelerating.α = 20 accelerating. Emittance vs. length L and focusing g.Emittance vs. length L and focusing g.

28 Accelerating : simulation Effect of combined inhomogeneity.Effect of combined inhomogeneity. 2π-minimum emittance (red solid line).2π-minimum emittance (red solid line). Optimal focusing (blue dashed line), ≈0.115·α 0.227.Optimal focusing (blue dashed line), ≈ 0.115·α 0.227. Optimal length (green dash-dot line), ≈ 10.89 + 5.03·α.Optimal length (green dash-dot line), ≈ 10.89 + 5.03·α.

29 Accelerating : simulation Effect of combined inhomogeneity.Effect of combined inhomogeneity. No accelerating, α = 1.No accelerating, α = 1. Emittance vs. length L and focusing g.Emittance vs. length L and focusing g.

30 Accelerating : simulation Effect of combined inhomogeneity.Effect of combined inhomogeneity. α = 2 accelerating.α = 2 accelerating. Emittance vs. length L and focusing g.Emittance vs. length L and focusing g.

31 Accelerating : simulation Effect of combined inhomogeneity.Effect of combined inhomogeneity. α = 5 accelerating.α = 5 accelerating. Emittance vs. length L and focusing g.Emittance vs. length L and focusing g.

32 Accelerating : simulation Effect of combined inhomogeneity.Effect of combined inhomogeneity. α = 10 accelerating.α = 10 accelerating. Emittance vs. length L and focusing g.Emittance vs. length L and focusing g.

33 Accelerating : simulation Effect of combined inhomogeneity.Effect of combined inhomogeneity. α = 20 accelerating.α = 20 accelerating. Emittance vs. length L and focusing g.Emittance vs. length L and focusing g.

34 Summary formula & coeffs BeamlineBeamξ Bunching, υ = I end /I start Transverse : uniform; longitudinal : Gaussian 0.0215·υ 1/3 Totally Gaussian 0.0349·υ 0.28 Accelerating, α = (βγ) 1 /(βγ) 0 Transverse : uniform; longitudinal : Gaussian 0.0220·α -0.136 Totally Gaussian 0.0350

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