Report

ELENA Tracking studies P.Belochitskii, O.Berrig With thanks to: C.Carli, L.Varming Jørgensen, G.Tranquille BMAD – an alternative tracking program BMAD is a program from Cornell by D.Sagan. It is symplectic ( which basically means energy conservation: http://en.wikipedia.org/wiki/Symplectic _integrator), and has the added capability of tracking through user defined fields. Since the electron cooler is described by a field map (and the space charge from the electron beam also) it would have been a fine candidate for ELENA. However there were some issues with the running of the program (that has been solved later) which at the time discouraged us from using BMAD. Plot of the ELENA ring; made by BMAD MADX does not calculate correct tunes for off-momentum particles Sextupoles ON; Solenoids OFF; EBEAM OFF; Tracking with MADX Qhor MADX TRACK MADX TWISS 0.280 MADX 0.278 0.276 0.274 0.010 0.005 0.005 0.010 PP Sextupoles ON; Solenoids OFF; EBEAM OFF Qhor 0.280 PTC_TRACK PTC_TWISS 0.279 PTC 0.278 Even though tunes, chromaticities, etc. are not calculated correctly in MADX, the tracking in MADX is the same as the tracking in PTC up to second order (but only for onmomentum particles; for offmomentum particles there are errors in MADX) See: http://cern-acceleratorsoptics.web.cern.ch/cernacceleratorsoptics/OtherInfo/MADX_PTC_Bend ingMagnet_difference_14Nov.doc 0.277 All simulations were therefore done with PTC 0.276 0.275 0.010 0.005 0.005 0.010 PP How to simulate the space charge generated by the electron beam? The electron beam makes de-focusing of the antiprotons in both planes; This process cannot be simulated by PTC. MADX however, can do simulations with an arbitrary MATRIX element – but only for thin lens optics. We made a model of ELENA with thin lens, however the beta-functions were not perfect. Comparizon Thick and Thin lens optics Because of the following arguments: • In order not to spend too much time optimizing the thin-lens beta-functions (which would in any case always represent an approximation). • The MATRIX element of MADX only simulate the effect on the protons that are inside the radius of the electron beam, but not the protons that are outside. • MADX is only precise to second order. beta function 14 12 betax thin 10 betx thick 8 6 4 2 0 0 5 10 15 20 25 30 s Comparizon Thick and Thin lens optics beta function 14 12 betay thin 10 bety thick 8 We decided to simulate the electron cooler with Mathematica and the rest of the ring with PTC 6 4 2 0 0 5 10 15 20 25 30 s It was decided to run without the main sextupoles magenta=10mm, green=20mm, red=30mm, blue=40mm, orange=50mm Without sextupoles With sextupoles The simulations on this page were done with the electron cooler simulated with MADX solenoids (the electron beam was not included). pxn pxn Horizontal 0.04 0.04 0.02 0.02 0.04 0.02 0.02 0.04 xn 0.04 0.02 0.02 0.02 0.04 0.04 pyn Vertical xn The strength of the sextupoles was nominal, i.e. set to make the chromaticity equal to zero, so that off-momentum particles will not touch neighboring resonances 0.04 0.02 0.02 0.04 pyn 0.04 0.04 0.02 0.02 0.02 0.04 yn 0.04 0.02 0.02 0.02 0.02 0.04 0.04 0.04 yn Since the sextupoles brought too much non-linearity (leading to loss of particles with 50mm amplitude) we decided to temporarily switch off the main sextupoles in order to concentrate on resonances from the rest of the machine (especially the electron cooler) Philosophical interlude (1) There are no linear magnets, not even quadrupoles and bending magnets. In reality they contain non-linear elements like sextupoles and octupoles and up to any higher order mode. This is not only because the fringe fields at the end of the magnets are non-linear, but even the magnets themselves. See: http://zwe.home.cern.ch/zwe/talks/cern_nonlinear.pdf A bending magnet contains e.g. sextupolar components (the larger the bending angle, the larger the sextupolar component). This was already discovered by Karl Brown in 1982: http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-r-075.pdf : X2-y2 is a sextupolar component: HORIZONTAL motion: D[P[x, y], x] VERTICAL motion: D[P[x, y], x] x·y is a sextupolar component, see J.Jowett: \\cern.ch\dfs\Projects\ .. \MultipoleFields.nb For a bending magnet,h is equal to 1 divided by the bending radius: h=1/r Compensation of sextupolar component in the main magnets NO compensation sextupole Sextupoles OFF; Solenoids ON 100 Gauss ; EBEAM OFF No Mathematica Qx 2.22 Qy 1.205 Momentum offset 0.000 HOR phase space diagram With compensation Sextupoles OFF; Solenoids ON 100 Gauss ; EBEAM OFF No Mathematica Qx 2.22 Qy 1.205 Momentum offset 0.000 HOR phase space diagram pxn 0.03 pxn 0.03 1. mm 1. mm 6. mm 6. mm 11. mm 11. mm 16. mm 16. mm 21. mm 21. mm 26. mm 0.02 26. mm 0.02 31. mm 0.01 0.03 0.02 0.01 31. mm 0.01 0.01 0.02 xn 0.03 0.03 0.02 0.01 0.01 0.01 0.01 0.02 0.02 0.03 0.03 0.02 xn 0.03 Philosophical interlude (2) Any machine with non-linear elements will couple the horizontal and vertical planes and lead to lines in the tune diagram that should be avoided: Avoid tunes that fulfill the equation: where: l,m and r are integers See page 69 in http://cds.cern.ch/record/212880/files/CERN-92-01.pdf See also: http://cern.ch/bruening/CAS/Driven_Resonances.ppt and: http://cds.cern.ch/record/1694484/files/CERN-2014-002.pdf Qx=2.22 Qy=1.205 Qx=2.26 Qy=1.29 Sextupoles OFF; Solenoids ON 100 Gauss ; EBEAM OFF Real field in solenoids Qx 2.22 Qy 1.205 Momentum offset 0.000 VER phase space diagram Sextupoles OFF; Solenoids ON 100 Gauss ; EBEAM OFF Qx 2.26 Qy 1.29 Momentum offset VER phase space diagram pyn 0.03 Qx=2.28 Qy=1.30 Sextupoles OFF; Solenoids ON 100 Gauss ; EBEAM OFF Qx 2.28 Qy 1.30 Momentum offset VER phase space diagram Real field in solenoids 0.000 Real field in solenoids 0.000 pyn 0.03 pyn 0.03 1. mm 1. mm 1. mm 6. mm 6. mm 11. mm 11. mm 0.03 0.02 0.02 0.02 0.02 0.01 0.01 0.01 0.01 0.01 0.02 yn 0.03 0.03 0.02 0.01 0.01 0.02 yn 0.03 0.03 0.02 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.02 yn 0.03 Biggest problem. Vertical emittance is growing – why? No shielding – No correctors With shielding – With correctors Observe first that there is coupling between the horizontal and vertical planes (Qx-Qy = 1). The signatures of this resonance are seen in the FFT plots (next slide) and in the variations of Wx and Wy versus turn; when one of them is on the crest, the other is in the trough - but amplitude of variations is different for x and y-planes. On top of that, in the vertical plane Wy is growing with number of turns! Sextupoles OFF; Solenoids ON 100 Gauss ; EBEAM OFF Real field in solenoids Qx 2.26 Qy 1.29 Momentum offset 0.000 Wxn xn^2 pxn^2 Versus turn no. Sextupoles OFF; Solenoids ON 100 Gauss No Shielding Scaled ; EBEAM OFF Real field in solenoids Qx 2.26 Qy 1.29 Momentum offset 0.000 Wxn xn^2 pxn^2 Versus turn no. Wxn mm mrad Wxn mm mrad 120 80 1. mm 100 1. mm 6. mm 6. mm 11. mm 80 60 16. mm 60 11. mm 40 40 20 20 200 400 600 n 1000 800 200 400 600 800 1000 n Sextupoles OFF; Solenoids ON 100 Gauss ; EBEAM OFF Real field in solenoids Qx 2.26 Qy 1.29 Momentum offset 0.000 Wyn yn^2 pyn^2 Versus turn no. Sextupoles OFF; Solenoids ON 100 Gauss No Shielding Scaled ; EBEAM OFF Real field in solenoids Qx 2.26 Qy 1.29 Momentum offset 0.000 Wyn yn^2 pyn^2 Versus turn no. Wyn mm mrad 70 Wyn mm mrad 100 60 1. mm 1. mm 6. mm 80 6. mm 50 11. mm 11. mm 16. mm 40 60 30 40 20 20 10 200 400 600 800 n 1000 200 400 600 800 1000 n Biggest problem. Vertical emittance is growing – why? No shielding – No correctors With shielding – With correctors No essential difference in the tune spectra, except for the presence of the linear coupling resonance Qx-Qy = 1. But coupling cannot increase the emittance in one plane and not in the other! Biggest problem. Vertical emittance is growing – why? Other observations: 1. The electron cooler field without shielding and without correctors is a theoretical calculation followed by a Mathematica interpolation at a given particle position. The electron cooler field with shielding and with correctors is calculated by OPERA with a given mesh; for tracking we again use Mathematica with interpolation at a given particle position. 2. The curl (=rotB) of the electron cooler field without shielding and without correctors is much smaller than for the electron cooler field with shielding and with correctors Summary: solenoid solenoid solenoid solenoid solenoid (without shielding and without correctors) (without shielding and without correctors) (without shielding and without correctors) (with shielding and with correctors) (with shielding and with correctors) with with with with with 2.5 5 10 3 5 mm mm mm mm mm spacing; spacing; spacing; spacing; spacing; the the the the the rot rot rot rot rot B B B B B amplitude amplitude amplitude amplitude amplitude is is is is is 0.0000022 0.0000230 0.0001600 0.0090000 0.0650000 3. The vertical emittance growth depends on the working point. The vertical emittance grows slower for WP (2.26 ; 1.29) than for WP (2.28 ; 1.30). Conclusion (1) A. We found that the main bending magnets have strong sextupolar components. We compensated for these by placing sextupoles (with opposite polarity) around the main bending magnets. B. We simulated several working points C. We simulated off-momentum particles Problems: 1. The vertical emittance is growing. Hypothesis: Numerical imprecision in the electron cooler field (simulated with OPERA; which is a standard program in CERN for simulating magnets). 2. Comparison of MADX solenoid field with a Mathematica solenoid (with hard edges). Big differences were observed – is it because of fringe fields that are automatically added in MADX. But in that case I would assume that the fringe fields depends on the aperture of the solenoid and since the aperture is not part of the MADX model, it seems there is an error ? Conclusion (2) Things to do: 1. Understand the problem with the vertical emittance growth 2. Possibly solve the problem with the vertical emittance growth by making an extremely fine mesh in OPERA 3. Do the simulations with space charge, in order to see the effect of the electron beam. 4. Documentation – how to use Mathematica tracking inside PTC. It is a fairly simple and modular setup. Nice things to do: 1. Fully compensate the main bending magnets. The compensation with the sextupoles significantly reduced the coupling in the machine; however quite some coupling remains, especially at large amplitudes. Is the remainder of the coupling linear ? 2. Do more investigations on why a hard edge solenoid does not give the same result as a MADX solenoid 3. Run ELENA as a transfer line. See what compensation gives the best closed orbit solution 4. Decomposition of the magnetic field in the electron-cooler field into thin lenses