1. Emittance Oscillations in Long-Pulse Induction Linacs Bruce Carlsten Los Alamos National Laboratory May 25, 2011

2. Slide 2 U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for NNSA Outline Comments about DARHT Long-term emittance decrease in long-pulse induction linacs Emittance oscillations and their thermalization Nonlinear focusing forces Discussion

3. Slide 3 U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for NNSA Long-Pulse Induction Linacs LANL and LBNL (with LLNL) collaborated on DARHT-2 accelerator. DARHT-1 was thoroughly designed and tested at ITS (only experimental demonstration of the centrifugal space-charge force) DARHT-2 was on a faster schedule THOR test facility with DARHT-2 cell

4. Slide 4 U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for NNSA DARHT Long design effort on LCLS - short turn on Short design effort on DARHT-2 – long turn on There was a lot of new engineering for DARHT-2, but also unpredicted beam physics

5. Slide 5 U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for NNSA Simulations LBNL – AMBER (Fawley and Vay) MRC – SPROP (Hughes) LANL – SLICE (Carlsten) All codes used equivalent physics – Gauss law for radial electric field, Ampere’s law for diamagnetic field, some other features Typically saw remarkable emittance decrease of about a factor of 10 (1000 mircons down to 150 microns) – very unique and interesting beam physics 90% emittances, use radial to avoid effect of solenoids 3.5-MeV, 4-kA diode

6. Slide 6 U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for NNSA DARHT Enabled Study of Exciting Beam Physics Centrifugal Space-Charge Force Measurement CSCF is extra force term: CSCF cancels potential depression: Ion-hose suppression in induction cells Halo interception in beam cleanup zone (BCUZ) (Vlasov push)

7. Slide 7 U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for NNSA Beam Physics We Want To Understand Today 1. Why does the emittance oscillate? 2. Why is there a gradual emittance decrease? 3. What causes these final stable oscillations?

8. Slide 8 U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for NNSA 1. Why Does the Emittance Oscillate? Initial nonlinearities come from: Anode hole Magnetic field nonlinearities Density non-uniformity Spherical aberration from anode hole acerbated by beam Without beamWith beam Minor hollowing of beam density Nonlinearities in initial phase space, minor wavebreaking These two set the initial conditions on beam oscillations

9. Slide 9 U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for NNSA 1. Why Does the Emittance Oscillate? Beam wants to oscillate about a stationary state Stationary state in space-charge dominated regime has uniform density – a non-uniform beam (zero emittance) will oscillate between a hollow and a peaked distribution. Below is phase space and density at first emittance maximum:

10. Slide 10 U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for NNSA 1. Why Does the Emittance Oscillate? The excess free energy (above the minimum needed for the stationary state) is left over to make the emittance. We can calculate the excess free energy by calculating the beam energy and subtracting the beam energy of the stationary state, using Reiser’s prescription for nonlinear free energy: This approach provides a surprising accurate estimate of the maximum emittance in the plots (within 5% when include correlations)

11. Slide 11 U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for NNSA 1. Why Does the Emittance Oscillate? These oscillations are same mechanism as emittance compensation, but with radial variations instead of axial variations: “Thin lens” compensation

12. Slide 12 U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for NNSA 1. Why Does the Emittance Oscillate? Emittance compensation in a drift show wavebreaking also: Focusing lens at 28 cm These particles dominate the final emittance Hanerfeld, Herrmannsfeldt, and Miller, 1989 PAC

13. Slide 13 U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for NNSA 1. Why Does the Emittance Oscillate? Emittance compensation in a uniform focusing field: (Exact same equations for the radially nonuniform case) Transverse eqn of motion for slice edge Equilibrium particle radius for a slice Do a perturbation expansion Slice expansion is oscillatory with a frequency that only depends on the external focusing Depends on axial position of slice K – external focusing, doesn’t depend on axial position of slice

14. Slide 14 U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for NNSA 1. Why Does the Emittance Oscillate? S&R used a Cauchy transform to extend this to an accelerating beam: where: Worth pointing out this solution is substantially different. There are oscillating terms, but also now a constant acceleration term in the slice divergence. S&R addresses this with an “invariant envelope” trajectory. C&P identified that rf focusing at the cathode provides control of   and , which help line up the phase space ellipses.

15. Slide 15 U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for NNSA 1. Why Does the Emittance Oscillate? If we keep the next order, we can predict when the phase space oscillations get out of phase: Expand to second order Particle oscillations out of phase after about 30 periods Final equipartitioning at 70% of maximum emittance J is current density up to r

16. Slide 16 U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for NNSA 1. Why Does the Emittance Oscillate? Boersch effect is very slow (thermalization due to Coulomb scattering), emittance growth of ~ 100 microns over 1 km at 5 MeV, 4 kA: Wavebreaking can also lead to thermalization (Reiser talks about wavebreaking in ¼ betatron period). Not true for low-emittance electron beams, wavebreaking occurs when a beam is focused if the charge density drops to < ½ of the average charge density within that point (Oscar Anderson). We can equivalently equate an emittance that is needed for particles to overcome the potential barrier of the beam and wavebreak: For a 4-kA, 4-MeV beam, this is about 10000 microns. For a 1-GeV, 100-mA, 1-mm radius H beam, this is about 0.1 micron.

17. Slide 17 U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for NNSA 2. Why is There a Gradual Emittance Decrease? The initial phase space curvature is not consistent with the initial beam density profile (doesn’t become flat). We can fix that with nonlinear forces in intense beams: Radial eqn of motion For a uniform density beam, radial variation in diamagnetic field cancels (to first order) change in potential depression This is the dominant nonlinear term, tailor beam focusing to work out phase space nonlinear “kink” We are defining divergences relative to the axial velocity on axis (r=0)

18. Slide 18 U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for NNSA 2. Why is There a Gradual Emittance Decrease? So we adjust the magnetic field profile to use the nonlinear focusing to straighten out the phase space as the beam is accelerated to 16 MeV

19. Slide 19 U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for NNSA 3. What Causes the Final Stable Emittance Oscillations? There is a beam halo from periodic wavebreaking (from every radial bounce in). Halo particles are emittance dominated with an oscillation wavenumber half the space-charge oscillation of the core: 32.78 m 53.7 micron 33.40 m 85.1 micron 34.03 m 102.8 micron 34.67 m 98.54 micron 35.30 m 53.98 micron Minimum emittance occurs when the two distributions are lined up (2:1 resonance). There is a minor trade between the focusing and minimizing the beam halo.

20. Slide 20 U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for NNSA 3. What Causes the Final Stable Emittance Oscillations? In principle, we can design a magnetic field match that leads to a nice radial profile spiraling into the axis (not oscillating) and get rid of the emittance oscillations. Let’s assume the beam is emittance dominated: Change of variables: Nice beam radial function (one of many): Leads to reasonable magnetic field profile:

21. Slide 21 U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for NNSA 3. What Causes the Final Stable Emittance Oscillations? Using the matched solution, we can squeeze the beam to get the necessary conditions into the field profile (angle in phase space), entire beam is emittance dominated: Final emittance drops to about 40 microns (10 microns rms!) Comparison: thermal emittance is 50 microns from a 0.1 eV temperature, 8-inch cathode

22. Slide 22 U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for NNSA Final Comments Non-thermalized beam preserves structure remarkably long Provides significant capability to tailor phase space using variety of nonlinear effects Insufficient diagnostics to measure these emittance Emittance dominated by downstream transport and bunch slicing