Energy Loss of Very High Charge Beams in Plasma: Theory J. Rosenzweig, N. Barov, M. Thompson, R. Yoder UCLA Dept. of Physics and Astronomy ICFA Workshop, Sardinia, July, 2002

J. Rosenzweig, ICFA 2002 Background 1: Blowout regime PWFA Modern plasma wake-field accelerators (PWFA) run in the “blow-out” regime (1991) Very nonlinear plasma electron response; total rarefaction Conditions: Measure nonlinearity in response: # of beam e-/plasma e- in cubic plasma skin-depth Plasma wake response (blowout regime PIC sim.) Fundamental quantity for theory…

J. Rosenzweig, ICFA 2002 What is Q? Ratio of beam density to plasma density Measure of field amplitude. General measure of nonlinearity (blowout). Dimensionless analysis is powerful tool… ~

J. Rosenzweig, ICFA 2002 Some notable characteristics of nonlinear plasma response Relativistic plasma motion Electromagnetic wave trapped in e - -rarefied region Artifacts in both physics and modeling –Density spike at wave turning –Noise inside of beam region –Electric field spike (sensitive to mesh) XOOPIC simulations a la D. Bruhwiler

J. Rosenzweig, ICFA 2002 Measuring the field amplitude from the spike not desirable The good focal qualities of the blow-out regime fall apart in the spike region (defocusing!) Beam loading eliminates the spike –Very little stored energy in spike region (narrow) The spike can be many times the “useful” field amplitude Beam loading in blowout PWFA

J. Rosenzweig, ICFA 2002 Recent experiments in nonlinear regime at FNAL and SLAC Energy spectra from UCLA experiment at FNAL A0 (E890), 12 MeV energy loss in 8 cm of 1E14/cc plasma (~ liquid Li!) Energy gain observed limited by spectrometer

J. Rosenzweig, ICFA 2002 Background 2: Scaling of PWFA with respect to plasma density From 1993, Rosenzweig, et al.*, noted in simulations that for blowout regime the maximum field excited in PWFA scaled as coherent Cerenkov radiation For finite bunch length, we must choose Thus we have scaling with bunch length Why is this linear “Cerenkov” scaling ~ valid in blowout? *J.B. Rosenzweig, in Proceedings of the 1992 Linear Accelerator Conference, (AECL-10728, Chalk River, 1993). *J.B. Rosenzweig, et al., Nuclear Instruments and Methods A (1998). *N. Barov, J.B. Rosenzweig, M.E. Conde, W. Gai, and J.G. Power, Phys. Rev. Special Topics – Accel. Beams (2000).

J. Rosenzweig, ICFA 2002 Recent PWFA scaling exercises S. Lee, et al., have performed scaling study in context of the “afterburner” concept Scaling with constant bunch charge, changing bunch length, Linear scaling found? Maybe… Replot

J. Rosenzweig, ICFA 2002 Background 3: Fundamental issues Energy loss in plasma — extend linear theory? Single particles –Jackson/Landau (calculate energy loss from transverse current, radiation… we will mimic) High charge, point-like bunches –Chen, et al., original PWFA Logarithmic divergence with beam transverse size Relevance to astrophysics –Intense pulse behavior –Single particle energy loss

J. Rosenzweig, ICFA 2002 Dimensionless theory of bunched beam energy loss in plasma Proceed through H-fields (connection to transverse currents) Convective force, continuity Normalize time and space to and Normalize charge and current to plasma density, speed of light Normalize fields to wave-breaking Norm. variables use tilde ~

J. Rosenzweig, ICFA 2002 Second order differential eqn. with useful relations Note we are now looking at magnetic field role in driving “electrostatic” wave… Induction

J. Rosenzweig, ICFA 2002 Linear theory Delta-function beam in assumed (Green ftn in linear world). Finite radius - disk beam. Governing equation; Integrate over , H-field is momentum kick Nonrelativistic term

J. Rosenzweig, ICFA 2002 Linear solutions Integrate to obtain longitudinal field just behind beam Same result as given by Chen, Katsouleas, Ruth, etc…

J. Rosenzweig, ICFA 2002 Limiting energy loss cases Wide beam so As expected from simple charge-plane model (1D limit) Narrow beam and

J. Rosenzweig, ICFA 2002 Comments on linear case Narrow beam is close to Cerenkov scaling, except… Logarithmic divergence with transverse beam size –Removal by Debye shielding? No, Debye shielding applies to maximum distance a charged particle’s field exists in equilibrium, not minimum distance of effect in fast transient. –Caused by divergence of transverse current induced at small impact parameters. Mitigated by relativistic saturation (transparency)?

J. Rosenzweig, ICFA 2002 Nonlinear case General response of charged particle to E and H fields for infinitesimal length beam (be very careful!) same as NR! But…density increase accompanies longitudinal kick (continuity equation) Induced radial velocity

J. Rosenzweig, ICFA 2002 Transverse current is identical to NR case! Differential equation unchanged Energy loss for a zero-length beam is the same as NR case Two effects cancel (Snowmass effect, where first presented) –Saturation of radial velocity (rel. mass increase) –Enhancement of density (Snowplow effect)

J. Rosenzweig, ICFA 2002 Check with deposited energy spread Field energy density right behind beam (only long. E), and mechanical energy density (watch for snowplow “group velocity” effect) Checks! (familiar 1/2)

J. Rosenzweig, ICFA 2002 Logarithmic divergence revisited View as Coulomb logarithm Maximum impact parameter is skin-depth Minimum impact parameter from uncertainty principle,and maximum  p Energy loss of point-like particle Quantum effects at both limits (Compton/plasmon) Fundamental for cosmic ray propagation…

J. Rosenzweig, ICFA 2002 Comparison with simulation Need to extend results to realistic finite length beam Self-consistent 2D PIC, narrow beam Examine average energy loss –Connect with  -beam –Measures efficiency of wave excitation Examine peak accelerating field –Connect with previous work (S. Lee, et al.) –Dangerous (spike!)

J. Rosenzweig, ICFA 2002 Snowplow observed in simulation MAGIC simulation shows clear snowplow. This effect does not occur in linear model.

J. Rosenzweig, ICFA 2002 Linear to extremely nonlinear cases examined, with MAGIC and OOPIC OOPIC surface plot of longitudinal E-field. Measure average deceleration, peak acceleration…

J. Rosenzweig, ICFA 2002 Simulation results Compare with predictions of linear theory Fields saturates at high charge –Snow-plow loses to relativistic “transparency” –Fields only a few times wave- breaking are possible Peak does not saturate as fast –We are misled by the spike Little field growth for Implications for experiments…

J. Rosenzweig, ICFA 2002 High charge experiments Experiments transitioning for Q~2 to near 100 Peak field still near to “scaling” for E164 E164 -> E164* loses a factor of 3 off linear scaling (this is worse if you look at the useful field, not the spike). ~

J. Rosenzweig, ICFA 2002 Conclusions Linear “scaling” of energy loss (and accelerating field) into nonlinear regime explained –Scaling persistence well into nonlinear regime –Use of spike in accel. field not a good measure Point-particle limit understood? Recent experiments already have Pulse compression critical (shown at FNAL) Next generation (at SLAC) may have –Expect fall-off in scaling