A new Space-Charge Model for high-intensity Particle Beams - PowerPoint PPT Presentation

A new Space-Charge Model for high-intensity Particle Beams Introduction Basic Accelerator Physics Space Charge Michael Holz Model Beam Core Dynamics Spectator FREIA, Particle Dynamics Uppsala University Conclusion 14. September 2017 14/09/17 FREIA, Uppsala University - 1 - M. Holz

Outline Introduction 1 Introduction Basic Basic Accelerator Physics 2 Accelerator Physics Space Charge Model Space Charge Model 3 Beam Core Dynamics Spectator Beam Core Dynamics 4 Particle Dynamics Conclusion Spectator Particle Dynamics 5 Conclusion 6 14/09/17 FREIA, Uppsala University - 2 - M. Holz

Introduction to High Power Accelerators European Spallation Source (Lund) neutron source Introduction Basic High-intensity proton beam Accelerator Physics Space Charge Model Consequences of beam loss: Life- and Material Beam Core Dynamics Sciences heat load (cryogenics) Spectator radiation (maintanance) Particle Dynamics 5 MW rms beam power Conclusion loss threshold: 1 W m intensity-dependent effects 14/09/17 FREIA, Uppsala University - 3 - M. Holz

Space Charge same-charged particles Introduction ⇒ Space Charge Basic Accelerator Physics non-linear electric forces Space Charge Model (e.g. Gauss) Beam Core Dynamics Spectator particles pushed to higher Particle Dynamics amplitudes ⇒ halo Conclusion 150 100 halo is prone to be lost 50 Field / [a.u.] 0 -50 -100 -150 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 x / [m] 14/09/17 FREIA, Uppsala University - 4 - M. Holz

Research Questions What does the new space charge model address? Understand beam dynamics at high intensity Introduction Evolution of beam size Basic Accelerator Physics Mechanics of beam halo formation Space Charge Limits in beam current Model Beam Core Dynamics Spectator The new model works generally, but now: Rings. Particle Dynamics Conclusion 14/09/17 FREIA, Uppsala University - 5 - M. Holz

Propagation of Particles Particle Transport � ¯ � � x � � � x R 11 R 12 Linear optics (drift = ¯ ′ Introduction x ′ R 21 R 22 x space, dipoles, Basic quadrupoles) Accelerator Physics QF Space Charge R: lattice transfer Model matrix Beam Core Dynamics s L Spectator � 1 � Particle L R drift = Dynamics 0 1 Conclusion Distribution Transport � � x 2 � � T � R 11 � � σ 11 � � R 11 � R 12 σ 12 R 12 � xx ′ � σ = ; σ = � x ′ 2 � R 21 R 22 σ 21 σ 22 R 21 R 22 � xx ′ � σ = R σ R T 14/09/17 FREIA, Uppsala University - 6 - M. Holz

Visualize Particle Oscillations phase space Twiss Parameters Introduction α, β, γ angle vs. position Basic Accelerator Particle trajectory in Physics β : amplitude ring Space Charge J: action variable Model stable: follows ellipse Beam Core Dynamics Distribution Spectator ǫ = � J � ; emittance Particle σ x = √ ǫβ ; size Dynamics normalized phase Conclusion space particle action x ′ 2 x 2 + ˜ ˜ J = 2 has equivalent for distributions 14/09/17 FREIA, Uppsala University - 7 - M. Holz

Beam Focusing and Tune Quadrupole magnets focus the Harmonic Oscillator beam Introduction Tune Q Basic Tune: number of oscillations per Accelerator Physics turn for stability: fractional Space Charge equilibrium ⇒ stable motion Model tune Beam Core perturb equilibrium Dynamics ⇒ oscillations Spectator Particle Dynamics Effects of space-charge: Conclusion defocusing Space Charge Tune Shift reduces tune 14/09/17 FREIA, Uppsala University - 8 - M. Holz

Existing Simulation Methods 1 Envelope equations Eq. of Motion Introduction d 2 a a − ǫ 2 ds + a κ − F sc a 3 = 0 a(s): envelope Basic amplitude Accelerator Physics F sc : space charge Space Charge 2 Particle in Cell (PIC), multi-particle Model tracking ⇒ computationally expensive Beam Core density F Dynamics We complement these methods with a Spectator Particle space-charge model that x Dynamics Conclusion is valid for transverse, Gaussian beams density F includes non-linear space charge forces x is fully analytic 14/09/17 FREIA, Uppsala University - 9 - M. Holz

Basis of the Model Electric field of a Gaussian particle distribution: (Bassetti & Erskine, 1980) Introduction  � �  − x 2 + y 2 x σ y    σ x + iy σ x  Basic Q x + iy 2 σ 2 2 σ 2 σ y   E x − iE y = − i x x  w    − e w   Accelerator   �  �  �  2( σ 2 x − σ 2 2( σ 2 x − σ 2 � � y ) y )  σ 2 x − σ 2 Physics 2 ǫ 2 π y Space Charge w: complex error function Model Beam Core only valid for upright Gaussian beam Dynamics y Spectator Particle Dynamics Conclusion x ⇒ we use a covariant form F 0 = f 3 + if 1 to include tilt angles y ⇒ cross-plane coupling x 14/09/17 FREIA, Uppsala University - 10 - M. Holz

Self-Interaction of the Beam 2 σ − 1 det σ e − 1 1 mn x m x n 1 particle distribution → Ψ( � x ) = (2 π ) 2 √ Introduction 2 electric field → F 0 ( x 1 , x 3 , σ ) = f 3 + if 1 Basic Accelerator ⇒ beam matrix σ is identical for space charge Physics Space Charge Model Particle experiences a kick: Beam Core Dynamics � ′ � x 1 ¯ = x 1 ′ , y , y x , x = [ x 1 , x 2 , x 3 , x 4 ] Spectator Particle x 2 ¯ = x 2 + f 1 Dynamics x 3 ¯ = x 3 Conclusion x 4 ¯ = x 4 + f 3 What happens to a beam distribution? 14/09/17 FREIA, Uppsala University - 11 - M. Holz

Space Charge Map � � x 2 � � xx ′ � � Averaging over all particles σ = � x ′ 2 � � xx ′ � Introduction Basic Accelerator ¯ = � ¯ x 1 ¯ x 1 � = � x 1 x 1 � = σ 11 σ 11 Physics ¯ = � ¯ x 1 ¯ x 2 � = � x 1 ( x 2 + f 1 ) � = σ 12 + � x 1 f 1 � σ 12 Space Charge ¯ = � ¯ x 1 ¯ x 3 � = � x 1 x 3 � = σ 13 σ 13 Model ¯ = � ¯ x 1 ¯ x 4 � = � x 1 ( x 4 + f 3 ) � = σ 14 + � x 1 f 3 � σ 14 Beam Core x 2 � = � ( x 2 + f 1 )( x 2 + f 1 ) � = σ 22 + � 2 x 2 f 1 � + � f 2 ¯ = � ¯ x 2 ¯ 1 � Dynamics σ 22 ¯ = � ¯ x 2 ¯ x 3 � = � x 3 ( x 2 + f 1 � = σ 23 + � x 3 f 1 � Spectator σ 23 Particle ¯ = � ¯ x 2 ¯ x 4 � = � ( x 2 + f 1 )( x 4 + f 3 ) � = σ 24 + � x 2 f 3 � + � x 4 f 1 � + � f 1 f 3 � σ 24 Dynamics ¯ = � ¯ x 3 ¯ x 3 � = � x 3 x 3 � = σ 33 σ 33 Conclusion ¯ = � ¯ x 3 ¯ x 4 � = � x 3 ( x 4 + f 3 ) � = σ 34 + � x 3 f 3 � σ 34 x 4 � = � ( x 4 + f 3 )( x 4 + f 3 ) � = σ 44 + � 2 x 4 f 3 � + � f 2 ¯ = � ¯ x 4 ¯ 3 � σ 44 � 1 3 = ℜ{ F 0 } 2 = � 2 2 ( F 0 + ¯ linear terms � x k F 0 � f 2 F 0 ) � � 0 + ¯ 0 + 2 F 0 ¯ = 1 F 2 F 2 F 0 quadratic terms � f 2 1 � , � f 2 3 � , � f 1 f 3 � 4 14/09/17 FREIA, Uppsala University - 12 - M. Holz

Linear Terms ∞ N,K: scaling � d 4 x � x k F 0 � = NK x k Ψ( � x ) F 0 ( x 1 , x 3 , σ ); k = 1 , 2 , 3 , 4 Introduction −∞ Basic Complex error fct Accelerator Physics General Ansatz: integral form Space Charge ∞ Model 2 � d α e − α 2 +2 i α z w ( z ) = √ π Beam Core completing squares Dynamics 0 parametric Spectator � ∂ ∞ ∞ � n dx x n e − x 2 dx e Bx − x 2 � � � Particle 2 σ 2 = � differentiation 2 σ 2 Dynamics � ∂ B � B =0 −∞ −∞ Conclusion General solution to linear terms:     iNK  σ k 1 ( σ 33 − i σ 13 ) + i σ k 3 ( σ 11 + i σ 13 ) � x k F 0 � =  σ k 1 + i σ k 3 −     � 2( σ 11 − σ 33 + 2 i σ 13 )   σ 11 σ 33 − σ 2 13 14/09/17 FREIA, Uppsala University - 13 - M. Holz

Quadratic Terms ∞ � 3 = ℜ{ F 0 } 2 = � F 0 ¯ F 0 � = ( NK ) 2 d 4 x x ) F 0 ¯ f 2 Ψ( � F 0 � 1 � 2 = Introduction 2 ( F 0 + ¯ F 0 ) −∞ Basic � � 0 + ¯ 1 F 2 F 2 0 + 2 F 0 ¯ F 0 Accelerator . 4 . Physics . Space Charge fast-forward several weeks of work Model . w ( z ) = w ( ax 1 + bx 3 ) . . Beam Core Dynamics Spectator ( − NK ) 2 π S ( σ, a , b ) Particle � F 0 ¯ F 0 � = Dynamics � � 2( σ 11 − σ 33 + 2 i σ 13 ) 2( σ 11 − σ 33 − 2 i σ 13 ) Conclusion σ, 2 , a , ¯ σ, 3 , b , ¯ � � � � �� · P ( σ, 1 , a , ¯ a ) − P b − P ( σ, 2 , ¯ a , b ) + P b with � � �� P ( σ, n , a , b ) = 2 1 S 12 π 2 − arctan n π � S 11 S 22 − S 2 � S 11 S 22 − S 2 12 12 14/09/17 FREIA, Uppsala University - 14 - M. Holz

Test Application: Simple Ring FODO-cell with 22 . 5 ◦ QF QD bending angle → 16 cells B B Introduction Basic periodic structure Accelerator Physics Space Charge Model Beam Core Dynamics Spectator Turn-by-turn data: Particle Dynamics (equilibrium) beam sizes Conclusion emittance growth space charge tune shifts 14/09/17 FREIA, Uppsala University - 15 - M. Holz