SIMPSONS for beam modelling in high intensity rings Shinji Machida STFC/Rutherford Appleton Laboratory 19 September 2019 Many thanks to H. Hotchi for his contribution to the code and letting me show his results. Based on the presentation at the workshop on the frontiers of Intense Beams Physics Modelling 2016.
Contents • Introduction: a couple of remarks • Single particle tracking • Space charge kick • Study examples • Summary 2
A couple of remarks as an introduction 3
Observations KEK-PS ~2000 Hotchi (J-PARC) at HB2016 4
Goals of SIMPSONS Provide as a realistic model of high intensity rings as possible to understand the observation, especially regarding beam loss of a few % or below. Predict what the crucial problem are in each ring and the way to improve the performance even reducing beam loss much below a few %. 5
Current limitation of SIMPSONS Computational algorithm is not fully optimised, only by OpenMP. It is unrealistic to simulate a time period of ~1 s self-consistently. Mainly used for 10~100 ms (real time) simulation, but frozen and hybrid models are available for longer time scale. 6
Basic structure of SIMPSONS single space particle charge tracking kick Push 6D particle coordinates At the end of each time step, within a certain time step include space charge effects including kicks from the lattice as an integrated kick. elements. For 100k ~ 1M macro particles, both processes take almost equal time 7
Single particle tracking 8
Accurate modelling of ring accelerators single particle tracking • Injection • Phase space painting with programable bump orbits. • Scattering and energy loss at charge exchange foil. • Acceleration • Lattice imperfections • Physical aperture to define beam loss 9
Emittance evolution during injection Anti-correlated painting for the first 307 turns. This wipes out detailed structure of injection beams. Random or pseudo-random (like Halton sequence) generator do not make difference. 10
Simulate multi turn injection without increasing # of macro particles 1st turn: Inject N particles. N 2nd turn: Inject another N particles. Instantly N N becomes 2N Throw away randomly selected N particles. N/2 N/2 The same number of Increase charge per particle by 2. macro particles N are 3rd turn: Inject N/2 particles. used during injection. N/2 N/2 N/2 Throw away randomly selected N/2 particles. N/3 N/3 N/3 Increase charge per particle by 3/2. n-th turn: Inject N/(n-1) particles. Throw away randomly selected N/(n-1) particles. Increase charge per particle by n/(n-1). 11
Accurate modelling of ring accelerators single particle tracking • Injection • Acceleration • RF gymnastics to capture and accelerate. • 1st and 2nd (or even higher) RF harmonics voltage. • Lattice imperfections • Physical aperture to define beam loss 12
Time dependent RF voltage and phase Single particle trajectory Manipulation of rf bucket V 2 /V 1 =0 � RF(poten:al(well((Arb.) � V 2 /V 1 =80% � 0.010 φ 2 ='100(deg � φ 2 ='50(deg � 0.005 φ 2 =0 � dp/p 0.000 -0.005 -0.010 φ ((Degrees) � -40 0 40 Change relative phase of the phase [deg.] fundamental and 2nd harmonic rf. φ 2 =#100 � 0° � 13
Accurate modelling of ring accelerators single particle tracking • Injection • Acceleration • Lattice imperfections • Imperfections of individual magnet, both static and dynamic. • Alignment errors. • Influence of neighbouring beam line components. • Physical aperture to define beam loss 14
Numerical(simula.on(setup � Simpsons !(PIC!par)cle!tracking!code!developed!by!Dr.!Shinji!Machida) ( Imperfec.ons(included:( ! Time!independent!imperfec)ons! � !!!B!Mul)pole!field!components!for!all!the!main!magnets:! !!! � !!!!!!BM!(K 1~6 ),!QM!(K 5,!9 ),!and!SM!(K 8 )!obtained!from!field!measurements! ���� B!Measured!field!and!alignment!errors! ! Time!dependent!imperfec)ons! ���� B!Sta)c!leakage!fields!from!the!extrac)on!beam!line:! !!! ���� !!!K 0,1 !and!SK 0,1 !es)mated!from!measured!COD!and!op)cal!func)ons! ���� B!Edge!focus!of!the!injec)on!bump!magnets:! !!!!!! � !!!K 1 !es)mated!from!measured!op)cal!func)ons! ���� B!BMBQM!field!tracking!errors! !!!!!! � !!!es)mated!from!measured!tune!varia)on!over!accelera)on! ���� B!1BkHz!BM!ripple! TimeBdependent!imperfec)ons! !!!!!! � !!!es)mated!from!measured!orbit!varia)on! can!be!included!easily,! ���� B!100BkHz!ripple!induced!by!injec)on!bump!magnets! because!“Simpsons”!takes!“)me”! !!!!!! � !!!es)mated!from!turnBbyBturn!BPM!data! as!an!independent!variable. � ! !Foil!scaWering:! !!! ������� Coulomb!&!nuclear!scaWering!angle!distribu)on!calculated!with!GEANT! Hotchi (J-PARC) at SC2013 15
Accurate modelling of ring accelerators single particle tracking • Injection • Acceleration • Lattice imperfections • Physical aperture to define beam loss • Each component has its own shape of aperture. 16
Single particle tracking in SIMPSONS others • Time T as the independent variable. • Easy to implement time dependent parameters such as RF voltage and phase, orbit bump for painting, ripple in magnets, etc. • Lattice is described by thin lenses created by TEAPOT. • Symplectic tracking of TEAPOT to the first order of dT . • Space charge effects are included each time step, not at fixed position. No spurious harmonic is excited. 17
Space charge kick 18
Solving Poisson equation (1) cylindrical coordinate In cylindrical coordinate system @' 2 � + @ 2 @ 2 @ 2 � ✓ ◆ @ r @ @ t 2 = − ⇢ 1 + 1 @ r � @ z 2 � − ✏ 0 µ 0 r 2 @ r ✏ 0 r Fourier decompose in azimuthal direction ∞ X � ( r, z, ' ) = � m ( r, z ) exp ( im ' ) , m = −∞ ∞ − ⇢ X ( ≡ n ( r, z, ' )) = n m ( r, z ) exp ( im ' ) . ✏ 0 m = −∞ Potential is solved separately in each mode. − m 2 ✓ ◆ 1 ∂ r ∂ ∂ r φ m ( r, z ) r 2 φ m ( r, z ) = n m ( r, z ) ∂ r r 19
Solving Poisson equation (2) mode expansion Assumption here is that the beam is nearly cylindrical. y m=0 y m=1 x x y y m=2 m=3 x x Typically up to m=8 . 20
Solving Poisson equation (3) smoothing filter Z 2 π φ m,c ( r, z ) = 1 φ ( r, z, ϕ ) cos ( m ϕ ) d ϕ , 2 π 0 Z 2 π φ m,s ( r, z ) = 1 φ ( r, z, ϕ ) sin ( m ϕ ) d ϕ , 2 π 0 Z 2 π n m,c ( r, z ) = 1 n ( r, z, ϕ ) cos ( m ϕ ) d ϕ , 2 π 0 Z 2 π n m,s ( r, z ) = 1 n ( r, z, ϕ ) sin ( m ϕ ) d ϕ , 2 π 0 Apply, if necessary, 2D Savitzky-Golay smoothing filters for charge n(r,z) and/or field d\phi(r,z)/dr, d\phi(r,z)/dz, d\phi(r,z)/dt . 21
Savitzky-Golay smoothing filter Note: Converging value depends on parameters of the filtering! 22
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Study examples with SIMPSONS 24
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