Space Charge Effects in Linacs CERN-School High Intensity - PowerPoint PPT Presentation

Space Charge Effects in Linacs CERN-School High Intensity Limitations, 2015 November 2-11, 2015 Ingo Hofmann GSI Darmstadt / TU Darmstadt 1

Overview � This lecture focuses on direct space charge � p or heavy ion high intensity linacs at non- or weakly relativistic energies � electrostatic interaction – ignorable image charge effects � several mechanisms also relevant to circular accelerators! limited relevance to space charge at injection of e - linacs � � Introduction to envelopes and space charge � Space charge resonances & instabilities � nearly all sources of emittance growth are of resonant nature (why?) � discuss three main criteria for linac design � Mismatch, errors, halo � Beam loss � Summary 2

Overview on high power linacs Crucial issue: � hands-on maintenance requires beam loss < 1W/m � control of beam power loss at level 10 -6 for MW beam power Average Intensity GeV C. Prior, HB2010 3

Levels of description in linacs Analytical basis: Reiser’s book Envelope dynamics with linear space charge in design linear optics Multi-particle beam dynamics in idealized linear verification of (nonlinear) optics with nonlinear space charge design Multi-particle beam dynamics in optics with beam halo random errors and loss prediction 4

Calculation of direct space charge force E z – non-uniform density x E z – uniform density z � bunches usually close to spherical (within factor of 2) � � image charges usually negligible (pipe far away) � forces E x,y,z = linearly increasing with amplitudes in uniform bunch � in non-uniform bunch non-linear E x,y,z not negligible � major source of ε growth − 3 qN ( 1 f ) x 3 qNf z = = E , E x z πε + πε 4 ( r r ) r r 4 r r r 0 x y z x 0 x y z r for uniform ellipsoid with semi - axi x, y, z 5

Sacherer's r.m.s envelope equations and the equilibrium problem in 2d (infinitely long) beams � Linear force (lattice + space charge) predicts rms emittances are constant! • with space charge exact self-consistent solution is 2D – KV • equivalent to envelope equation (transversely uniform density, infinitely long) � for non-KV distribution the r.m.s envelope equations still hold – in good approximation! (Sacherer, ~1973) non-uniform density leads � nonlinear space charge force • • surprisingly r.m.s. envelope equations still very good approximation - if emittances constant! applies ~ also to 3D case of "bunched beam" ! • 6

Rms envelope equations - valid under assumption of constant emittances - 2 ε − 3 K ( 1 f ) ' ' x + κ − − = a ( s ) a 0 x x x 3 + ( a a ) a a = x y z rms beam sizes : a r / 5 x x , y , z x , y , z 2 2 ε 2 2 2 ε = − rms emittances : x x' - xx' 3 K ( 1 f ) x y ' ' + κ − − = a ( s ) a 0 space charge parameter : y y y 3 + ( a a ) a a x y z y qN = K 2 3 2 πε β γ 2 20 5 mc ε 3 Kf 0 ' ' z + κ − − = a ( s ) a 0 z z z 3 a a a x y z When are the rms emittances constant? ε 99%, ε 99.99% equally important! numerous studies: Struckmeier and Reiser, Part. Accel. 14 (1984) ..............Li and Zhao, PRSTAB 17 (2014) 7

Linac beam dynamics is different! - varying structures, focusing and tunes – Linac: Linac: 8 GeV 2 MW H - proton driver @ FNAL � single pass � optics ~ linear � space charge potential − nonlinear − periodically varying � � resonances may exist � often transient and not separable � avoid by design – if possible Circular tune diagram Circular: � many turns � optics nonlinear effects matter � space charge potential ~ a correction � � many resonances exist − avoid or compensate 8

Example of linac structure effect on beam dynamics - varying structures and focusing – concern: emittance increase, halo, beam loss & activation Proposal of a sc 8 GeV H - proton driver for Fermilab (Project X) P. Ostroumov (ANL), 2006 9

How to characterize space charge strength? • lattice: k 0x , k 0y, k 0z describe lattice reduced by space charge to k x , k y, k z (k 2 ~ force) • • “tune depression” k x /k 0x or k z /k 0z relative importance of space charge; • “convention” in p linacs: k x /k 0x < 0.7 ~ “space charge dominated”: effective force ~ reduced to half by space charge • k x /k 0x � 0 strict space charge limit • =0 is “cold” beam with zero emittance 10

Idealized “demo lattice” - for simplicity periodic cells / RF gaps + well-separated resonant effects F/2 – O – D – O – F/2 with symmetric RF gaps 11

How to get more space charge dominated? downwards k 0xy -ramp – envelope model “demo lattice” k z k xy k 0xy weakened � k 0xy � 0 weaker focusing � � beam size grows � � more space charge k xy /k 0xy more depressed !! dominated � although absolute space charge force weaker! 12

Application to intrabeam stripping serious issue in H - high power linacs � � � � cure: expand beam! 7 6 • 2010: SCL losses can be caused by Intra Beam Stripping of H - (Valeri Lebedev, 5 B, T/m 4 FNAL) 3 • By lowering SCL quads’ field gradients the Design 2 losses were reduced to an acceptable level. Minimal Losses 03/04/2011 1 • Weaker focusing – more space charge 0 dominated 0 5 10 15 20 25 30 35 SCL Quad Index SCL Losses for Production Optics, 30 mA 100 - H 90 Protons 80 70 Losses, Rad/C 60 50 40 30 20 10 0 0 50 100 150 200 250 source: J. Galambos et al. BLM Position, m 13

Equilibrium - Resonance – Instability - sources of emittance growth – any accelerator - deviation from stable equilibrium = „mismatch“ small deviations � response bounded by initial value • • return to initial position if „damping“ exists – here particles � energy into „damping particles“ • periodic kick instability resonant excitation small deviations � runaway • increasing amplitude • no return to initial position • limited by de-tuning or loss � instability (also resonant) • Beam: potential from magnets/RF and self-consistent electric field all 3 involve resonant mechanisms – also in linac! 14

Full particle-in-cell simulation TRACEWIN code for linac design and verification TRACEWIN: design and verification • − http://irfu.cea.fr/Sacm/logiciels/. • Grid-based Poisson solver “inside” bunch • analytical continuation outside − model halo particles accurately far away from core • free boundary: − ignore image charges – direct space charge dominant # simulation particles ~ 10 7 • − worry about loss at level 10 -6 “error studies”: statistics with ~ 10 3 error seeded linacs • − � effect on beam loss • limited spatial resolution − � noise needs to be checked 15

Full particle-in-cell (PIC) in “demo-lattice” 100 0 downwards k 0xy -ramp – demonstration of main resonant effects initial mismatch k 0x,y,z k x,y,z 90 degree resonance crossing 2k xy -2k z =0 crossing 6k xy =360 0 rms emittances initialization 5% growth 16

Sources of emittance growth in linacs in principle also relevant to circular accelerators Non-resonant “Classical” resonances Initial density profile mismatch 1. Structure resonances • if starting with non-selfconsistent initial • driven by periodically modulated space charge force � resonance condition distribution • evolves very fast: ~¼ plasma period (typically 2. Anisotropy < 1 betatron period) • driven by energy (emittance or "temperature") difference between degrees of freedom • is a difference resonance - only exchange of Resonant instability by periodic emittances (rings: “Montague resonance”) structure Resonant halo formation “90 degree” stopband envelope instability” driven by rms mismatch � periodic force from • exponential growth from initial noise space charge • involves a resonance condition • pushes particles into a halo • requires time (distance) to develop • also caused by random errors in magnet optics Distinction instability – resonance sometimes confused Not all equally serious 17

Initial density profile mismatch – rms matched! “un-matched” nonlinear field energy � � � � emittance growth • discovered in 1980's under "nonlinear field energy" − 1D: Wangler et al., IEEE Trans. Nucl. Sci. NS·32, 2196 (1985) − 3D: Hofmann and Struckmeier, Part. Accel. 81, 69 (1987) • always present at injection of a space charge dominated beam • reason: space charge repulsion wants to flatten the beam the more the closer to space charge limit (k/k 0 � 0) (self-consistent solution including non-parabolic space charge potential ) • “Plasma effect” known as “Debye shielding” – a non-resonant effect (only one here!) matched density profiles (schematic – Gaussian distribution): increasing space charge effect � profile flattening extreme space charge limit k/k 0 � 0 emittance dominated moderate k/k 0 ~1 space charge (vanishing emittance – “cold” beam) 18