Selected Topics of Theory and Experiment on the Space- Charge-Dominated Beam Physics Y. Zou www.ireap.umd.edu www.ireap.umd.edu
Outline • Part I: general concepts of space-charge-dominated beams. • Part II: University of Maryland Electron Ring(UMER) and its components. • Diagnostics: BPM, energy analyzer … • Part III: selected experimental and theoretical results • Experimental study of beam energy spread evolution in intense beam • Theoretical study of beam emittance of a gridded electron gun • Experimental study of Resistive wall instability
Beam Transport in a Uniform Focusing Channel Beam envelope equation: I 2 = − γ 2 K ( 1 f ) ε 2 β γ e K 3 3 I + − − = 0 " 2 R k R 0 generalized perveance 0 3 R R external focusing External focusing force 2 a Matched Beam: k 0 Emittance Space charge force ε 2 K = + 2 k a o 3 a a ε + � 2 K Define intensity parameter ( χ ) 2a 3 a a Beam K 2 = space charge force χ = 2 a k 0 external force Space charge + emittance ω ν Betatron tune k = = − χ = χ p 1 2 Plasma frequency ν ω depression: k 0 0 0
Space-Charge Dominated vs Emittance Dominated Emittance Space-charge Plasma Dominated Dominated λ D >> a λ D << a Oscillations 1.4 Curve ω ω = χ 1.2 P 2 0 1.0 Betatron Existing 0.8 Oscillations rings Curve 0.6 k = − χ 1 0.4 k 0 Intensity HIF UMER Range 0.2 Parameter: Drivers 0.0 χ = K 0.0 0.2 0.4 0.6 0.8 1.0 Intensity Parameter ( χ ) k 2 a 2 O
University of Maryland Electron Ring UMER designed to serve as a research platform for intense beam physics • Beam Energy: 10 keV • Beam current: 100 mA • Generalized perveance 1.5 x 10 -3 • Emittance, 4x rms, norm 10 micron • Pulse Length 50 - 100 ns • Bunch charge 5 nC • Circumference 11.52 m • Lap time 197 ns • Tune Depression (k/k 0 ) >0.15
Diagnostics Available • Fast Current Monitors (2+) (rise time < 200 ps) • Beam Position Monitors (17 BPMs) • Phosphor Screens (18+ P-Screens) •End Diagnostic Chamber: – Energy Analyzer –Pepper-pot Emittance (Phase Space) Monitor –Slit-Wire Emittance (Phase Space) Monitor –Faraday Cup
UMER Diagnostics –BPM [1,2] Top Electrode 3 2 x x xy = + + 20 Ln ( V / V ) A B C Right R L 3 Left b b b Electrode Electrode θ Sensitivity term Φ 4 Relative Strength of Different Terms Nonlinear term 2 0 Bottom Electrode Coupling term -2 -4 R1 I C s 0 20 40 60 80 Electrode Angle (Degree) F is chosen to be 76.99 o to remove the coupling [1] Y. Zou etal, PAC 1999 between X and Y direction [3] . [2] B. Quinn etal, PAC 2003 [3] Y. Zou, Ph.D dissertation, UMD, 2000
UMER Diagnostics -BPM
Design of Energy Analyzer [1,2] 1 st Generation: Collimating Cylinder Parallel-Plate -10.13kV Retarding EA > 20 eV Resolution Retarding Mesh -9999.5 V 2 nd Generation Collector ~ 3 eV Resolution 10 keV Beam Grounded Housing 3 rd Generation: Res. < 1 eV [1] Y. Zou, etal, Phys. Rev. ST Accel. Beams 5(7), 2002, p. 011502. [2] Y. Cui, Y. Zou etal, to submit to RSI.
Longitudinal space-charge effect inside the Analyzer Problems: • Shift the measured mean energy towards low-energy side. • Leave a large tail at the high-energy side. • Make the FWHM of measured spectrum narrower than the true spectrum • Measured rms energy spread are different (a) Parameters: 5 keV, 135 mA 1 Curve II Relative Particle Density Curve I beam, 0.8 0.6 Curve 1: 0.2 mA beam current 0.4 inside the device 0.2 Curve 2: 2.2 mA beam current 0 inside the device 5040 5050 5060 5070 5080 5090 Beam Energy (eV)
Potential solutions for thermal beam [1] Beam energy: 5 keV, Initial beam energy spread: 10 eV ( λ = J in /J lim ) λ = 0.5 λ = 1.4 0 0 -8 V -100 V -1000 -1000 -2000 V -2000 V Potential (V) Potential (V) -2000 -2000 -3000 -4400 V -3000 -4100 -4000 -4000 -5000 V -5000 V -5000 -5000 0 0.005 0.01 0.015 0.02 0.025 0.03 0 0.005 0.01 0.015 0.02 0.025 0.03 Distance (m) Distance (m) -4960 Potential minimum Potential (V) -4980 of -5024.9 V -5000 -5020 -5040 0.0160.018 0.02 0.0220.0240.0260.028 0.03 [1] Y. Zou etal, submitted Phys. Rev. ST Acc. Beam Distance (m)
Comparison of Simulation Results and Experiments Nominal Energy : 5 keV, Current: 135 mA Experiment 1D Theory and simulation • Curve I: 0.2 mA inside the device plus 2D correction ( λ =0.062, estimated), E rms = 2.2 eV, • Curve I: λ = 0.062, E rms = 2.2 eV, FWHM=3.4 eV FWHM = 5.1eV • Curve II: 2.2 mA inside the device Curve II: λ = 1.2, E rms = 5.1eV, ( λ =0.8, estimated), E rms = 3.2 eV, FWHM = 0.49 eV FWHM=1.1 eV (c) (a) 1 1 Curve II Relative Particle Density Curve I Curve II Relative Particle Density Curve I 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 5040 5050 5060 5070 5080 5090 5040 5050 5060 5070 5080 5090 Beam Energy (eV) Beam Energy (eV)
Part III: Selected Physics Topics • Experimental study of beam energy spread evolution in intense beam • Transverse beam emittance of a gridded electron gun • Experimental study of Resistive wall instability
Experimental Study of Beam Energy Spread in Space-Charge-Dominated Electron Beam* * Y. Zou etal, to submit to Phys. Rev. STAB
Energy Spread Growth in the Intense Electron Beam • Longitudinal-transverse relaxation (intra beam scattering) [1] • Long relaxation time • Longitudinal-longitudinal relaxation [2] • Short relaxation time, ~ plasma period • Theoretical prediction for the longitudinal energy spread including both effects is given by: ∆ = + πε 2 1 / 3 1 / 2 E [( 2 qV k T ) ( C / ) qn qV ] //, 0 // 0 0 rms B • Scaling law for the energy spread due to the L-T relaxation: ( ) ( ) ∆ 1 / 2 1 / 2 ~ * / ~ * * E rms I D a J a D [1] See the reviews in Chapters 5 and 6 of M. Reiser, “Theory and Design of Charged Particle Beams”, John Willey & Sons, 1994. [2] See, for instance, A. V. Aleksandrov et al. Phys. Rev. A, 46, 6628 (1992)
Phase I Experimental Setup Feedthrough Diagnostic Chamber Electron Gun Movable First Phosphor Solenoid Energy Screen Analyzer 20 Energy Analyzer Signal (mV) 0 -20 -40 -60 -80 -100 -120 -140 -50 0 50 100 150 Time (ns) UMER Typical EA Signal Phosphor screen image
Typical Energy Spread Measurement Results 3 3 Circular: Experimental results Energy Spread (eV) Triangle: Theory Energy Spread (eV) 2.5 2.5 2 2 1.5 1.5 1 1 2.5 2.5 3 3 3.5 3.5 4 4 4.5 4.5 5 5 5.5 5.5 Beam Energy (keV) Beam Energy (keV)
Energy Spread vs Beam Energy at Different Particle Densities Calculated energy spread Beam envelope (5 keV) 20 2.8 2.6 Beam Envelope (mm) Energy Spread (eV) 15 2.4 2.2 10 2 1.8 5 1.6 1.4 EA Position EA Position 0 1.2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Distance (m) Distance (m) 3 Comparison of experimental Energy Spread (eV) 2.5 results and theory 2 1.5 UMER 1 2.5 3 3.5 4 4.5 5 5.5 Beam Energy (keV)
Energy Spread at Different Beam Currents Beam Energy : 5 keV, Sampled position: 60 nS 1.2 1 Beam current: 13 mA Relative Particle Density Beam current: 135 mA 0.8 Energy spread: 1.7 eV Energy spread: 2.1 eV 0.6 0.4 0.2 0 -0.2 5050 5060 5070 5080 5090 Retarding Voltage (V) Energy spread along the pulse (time resolved) 25 25 Beam Energy Spread (eV) 20 20 Energy Spread (eV) Beam current: 135 mA Beam current: 13 mA 15 15 10 10 5 5 0 0 0 20 40 60 80 100 0 20 40 60 80 100 Time (ns) Time(ns)
Transverse Beam Emittance Growth in a Gridded Electron Gun [1] Grid Anode (Va) Cathode (Vg) Electron beam d cg d ca Figure: Schematic of a Grid Gun [1] Y. Zou etal, to appear in NIM
Potential Distribution at Different Grid Voltages Cathode grid distance: d cg = 0.15e-3 m, d ca = 0.027 m, Va = 10000V Potential distribution Electrical field 40 Grid Scenario I 35 100000 30 H V � m L Scenario II H V L 25 d 0 l l e a i i F 20 t n l e a t c o i P 15 -100000 r Scenario III t c e l 10 E -200000 5 0.1 0.2 0.3 0.4 0.5 0 . 1 0.2 0 . 3 0.4 0.5 from Cathode H mm L Distance from Cathode H m m L Distance ( ) V 4 1/2 ∆ = + + g Field discontinuity due to 1/2 E c V c 1 2 z g 3 d the non-natural grid potential g
Emittance of Multi Beam Systems y ' x ' X m x x R Configuration space Beam trace space Effective normalized emittance can be calculated as ∆ 2 eV E GRa ε = g z 1/2 ( ) n g , 2 4 mc V g 1/2 λ 3 N ∑ = − λ 2 2 2 G 16 1 4 i i Where is geometry factor π 3 =− i N
Calculated Emittance Growth Vs. Grid Voltage Cathode grid distance: d cg = 0.15e-3 m, d ca = 0.0255 m, Beam radius: R = 4e-3 m Half opening of mesh: a = 0.075e-3 m, Anode Voltage: Va = 10000 V 30 L d a r m 25 m m H e c n a 20 t t i m E e 15 v i t c e f f 10 E d e z i l 5 a m r o N 5 10 15 20 25 30 Voltage H V L Grid Normalized effective emittance vs. grid voltage
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