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Dependence of the cooling force on the electron transverse velocity MI Department Meeting 1 September 2010 Khilkevich Andrei, Belarusian State University, AD/MID Lionel Prost(Supervisor), Alexander Shemyakin Electron Cooler Electron beam generated and accelerated in a Van-de-Graaff type electrostatic accelerator (Pelletron). Cooling section consist of 10 solenoids, 2 m-long where electrons mix with anti-protons; B≈100 G There are 20 dipole correctors in each solenoid 1 Figure 1. Schematic picture of electron cooler at Fermilab Cooling force Non- magnetized model Assumptions: -the influence of the magnetic field is neglected - the electrons velocities are assumed to be described by a Gaussian distributions - the electron density and velocity spreads assumed being constant the electron density distribution is close to parabolic, the transverse velocity spread tend to increase towards the edge of a beam 2 Cooling force Non- magnetized model Vx 2 Vy2 Vz 2 Vp Ve 1 - + + F 4ne me re c Lc f Ve 2σ t 2 2σ t 2 2σl2 d 3 Ve (1) f (Ve )= 2 4 e (2) 3 3/ 2 2 Vp Ve (2π) σlσ t n e – an electron density in beam rest frame me – electron mass re – an electron classical radius V p – the velocity of pbar beam Lc – the Coulomb logarithm Ve – the velocity of electron beam f Ve – electron velocity distribution – a ratio of ring length, occupied by Cooling Section to the Recycler ring circumference Vt – transverse electron velocity l – rms longitudinal velocity spread 3 Vl – longitudinal electron velocity t – rms transverse velocity spread Purpose of studies How well could a non-magnetized model be used? F(Vt) – ? Applied oscillations y 1) Kick the beam at first solenoid, futher motion is 2A determined by longitudinal magnetic field. und (2 ) 2) Kick the beam with each dipole corrector in each v solenoid to create a helix-like trajectory with desired A amplitude and period. z t c 0 (3) 0 – rms angle spread v|| 2A 4 x osc (4) Coherent angle along the beam Figure 2. Illustration of helix motion Side benefits: possible use at BNL BNL is considering of using the electron cooling for RHIC’s low energy run possible problem : beam loss caused by recombination of heavy ions with electrons Idea (ref. Fedotov et al, PAC 2007 ) : suppress recombination by coherent oscillatory motion of electron beam, and recover the cooling efficiency by an increase of the electron beam current works only if a contribution of “far collisions” is significant(see next slides) The idea had never been tested before experimentally 5 Non-magnetized model in oscillatory case without oscillations with oscillations Typical values r_max 1.8 mm r_max r_osc pbars r_osc 0.1-0.5 mm pbars q 2 nm r_max Figures 3,4. trajectories of electron beam in both cases 6 – the impact parameter, at which the scattering angle is equal to /2 r_max – maximum impact parameter (Debae radius) r_osc – oscillation radius Non-magnetized model in oscillatory case 1 1) 0<r<r_osc , close collisions 3σ t 3σ t 3σ t L c1f (Ve )(ΔVz + vz) r_osc F(ΔVz ) = 4πn e m e re c η2 4 ∫ ∫ ∫[(vx) + (vy + U) 3σ t 3σ t -3σ t 2 2 + (ΔVz + vz) 2 ] dvxdvydvz pbars Applied oscillations are taken into account q r _ osc r_max ρ ρ⊥2 + r _ osc2 L c1 = ∫ 2 dρ = 0 ρ + ρ⊥2 ρ⊥ 2) r_osc<r<r_max , far collisions 3σ t 3σ t 3σ t L c 2 f (Ve )(ΔVz + vz) F = 4πn e m e re c η2 4 ∫ 3∫-3∫[(vx)2 + (vy)2 + (ΔVz + vz) 2 ]dvxdvydvz 2 3σ t σt σt 7 ρ + r _ max Applied oscillations are neglected r _ max ρ 2 2 Lc 2 = ∫ρ2 + ρ⊥2 dρ = ln r _ osc ⊥ ρ + r _ osc2 ref. Fedotov et al, PAC 2007 ⊥ 2 Contributions of close and far collisions to the cooling force Close collisions Far collisions Total 30 25 20 F, MeV/c*hr 15 10 5 0 8 -0.05 0.05 0.15 0.25 0.35 0.45 0.55 a, mrad Figure 5. Contributions of close and far collisions to the cooling force Realization of oscillatory trajectories x y Y BPM X BPM x goal y goal 0.060 0.040 solenoid x,y [mm] 0.020 0.000 -0.020 -0.040 BPM(X,Y) BPM(X,Y) -0.060 0 100 200 300 400 500 600 Z [cm] Figure 6. Goal and calculated trajectories of electron beam Obtained series: 1 oscillation per solenoid, und _ m ax 0.5mrad sine 1 oscillation per 2 solenoids, und _ m ax 0.4mrad 9 1 oscillation per 3 solenoids, und _ m ax 0.5mrad Undulatory motion Undulatory trajectory is created by kicking the beam with each corrector in certain direction and with certain force. In result beam three-dimensional trajectory will be helix with specified period in Z direction and amplitude in X and Y directions. BPM(X,Y) solenoid BPM(X,Y) One oscillation per two solenoids 0.4 Y BPMs Y 0.3 Trajectory, mm 0.2 0.1 X BPMs 0 -0.1 X -0.2 -0.3 -0.4 0 500 1000 1500 Z direction, cm 10 Figure 7. Electron beam differential trajectory at X and Y Drag-rate measurements In every experiment of studies the drag-rate was measured to determine the cooling force. pbars~ 1*1010 dP 0.4 4.3165 F Momentum deviation, MeV/c 0.3 dt 4.316 0.2 Errors: Voltage, MeV 4.3155 0.1 – statistical ~ 10% 0 4.315 – uncertainty of choosing -0.1 4.3145 the points for fit (small) -0.2 – uncertainty of pbar 4.314 -0.3 emittance -0.4 4.3135 11 9.20 9.21 9.22 9.23 9.24 Measurement were made time, hr with the help of a program R111 , written by Dan Figure 8. Pbar momentum change during voltage jump Broemmelsiek Results period is 2m, 17 July period is 2m, 13 July period is 4m, 17 July period is 6m, 17 July period is 1m, 27 April period is 13m, 13 July 45 – large inconsistency in 40 day-to-day measurements 35 at zero osc. angle 30 drift of the cooling section and BPM calibration F, MeV/c*hr 25 – influence of beam offset 20 2πA 15 α osc = (4) λ 10 contribution is rather weak < 6% 5 0 12 -0.05 0.05 0.15 0.25 0.35 0.45 0.55 a, mrad Figure 9. The dependence of cooling force on oscillatory angle for all series of data. Results period is 2m, 13 July period is 13m, 13 July fit for period = 13m fit for period = 2m 45 40 The fitting parameter 35 is a transverse velocity spread in an electron 30 beam 25 F, MeV/c*hr 20 15 10 5 0 13 -0.05 0.05 0.15 0.25 0.35 0.45 0.55 a, mrad Figure 10. The dependence of cooling force on oscillatory and comparison to the theory prediction. Results 0.3 mrad 0.4 mrad period = 1m period = 2m period = 2m period = 4m period = 4m period = 6m period = 6m theor dependance theor dependance period = 13m period = 13m 16.5 9 14.5 8 12.5 F, MeV/c*hr 7 F, MeV/c*hr 10.5 6 8.5 5 6.5 4 4.5 3 0 5 10 15 0 5 10 15 Period, m Period, m Figure 11. The dependence of electron-cooling force on the period of oscillations for 14 different angles. Conclusion The dependence of the measured drag rates on the oscillatory angles qualitatively agrees with the non-magnetized model For a given angle, the drag rate increases for shorter wavelengths There is a large experimental inconsistency in the measured day-to day drag rate without additional oscillation Related to drift of the cooling section and BPM calibration The max/min ratio of the drag rates is 1.7 Makes difficult to compare the results from different days Should weakly affect measurements at large oscillatory angles The measured dependence of the drag rates on the oscillation period is described by presented formulae within ~50% For angles of 0.3 – 0.5 mrad, where the effect of inconsistency in the initial value of the drag force should be low 15

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posted: | 4/13/2011 |

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