1. Self-consistent non-stationary theory of multipactor in DLA structures O. V. Sinitsyn, G. S. Nusinovich, T. M. Antonsen, Jr. and R. Kishek 13 th Advanced Accelerator Concepts Workshop, July 27 th -August 2 nd 2008, Santa Cruz, CA

2. Outline Introduction Previous work: 1-D model of multipactor Non-stationary 2D-model of multipactor Simulation results Summary and future work

3. Introduction Multipactor (MP) may occur in many situations: one- and two- surface MP, resonant and poly-phase MP, on the surface of metals and dielectrics etc. Below we consider only dielectric loaded accelerator (DLA) structures. The starting point for our work is experimental and theoretical studies of such structures jointly done by Argonne National Lab and Naval Research Lab (J. G. Power et al., PRL, 92, 164801, 2004). In the theoretical model developed during those studies, the space charge field E dc due to the total number of particles is taken into account as a parameter. We offer a simple non- stationary model where the DC field is taken into account self- consistently.

4. Previous work: 1-D model of multipactor Some basic mechanisms of multipactoring can be understood by considering 1-D radial motion of electrons. Such motion near the surface of dielectric can be described by a simple equation of electron motion: is the rf phase at the instant of emission, is the axial wavelength. The DC field acting on the electron with radial coordinate r is created by charges located at r’ < r. Correspondingly, it can be determined as When the height of electron trajectories is much smaller than the radius of the dielectric we may neglect the cylindricity and calculate DC field as

5. Equation of electron motion in normalized variables for this case can be written as: Here,,,,, where n max is the maximum electron density. 1D-model (cont.) Equation should be supplemented by initial conditions at the dielectric surface for the particle coordinate y : and particle velocity:. We assume that initially all electrons are uniformly distributed over the emission phases. Initial velocities of true secondaries, which are emitted from the dielectric surface in the process of multipactoring are randomly distributed in the interval which corresponds to kinetic energies from 0 to 20 eV.

6. 1-D model (cont.) Simulations were done for the parameters shown in the figures, V max = 250 eV.

7. Shortcomings of the 1-D model Particles with ‘high altitude’ –Depending on initial conditions, some electron sheets during their motion are not affected or affected by very weak dc field and may be subject to altitude growth without limits. What can we do about it? –We tried to artificially limit their motion by setting the highest possible height for a sheet. However, subsequent charge accumulation at this fixed level affected the saturation conditions of the process. Neglecting the effects of cylindricity –We assumed that most of the particles would stay close to the dielectric surface where the radial dependence of the rf field amplitude can be neglected. Later, our calculations demonstrated that particles may travel distances comparable with the radius of the vacuum region. Therefore, real trajectories may differ significantly from the ones calculated within the 1-D model.

8. 2-D model of multipactor in a DLA structure Equations of particle motion: RF field components for TE 01 -mode: Here r and φ are radial and azimuthal coordinates of the particle, v r, v φ and v z are its radial, azimuthal and axial velocities, respectively. The dc electric field, E dc, acting on the particle is created by cylindrical layers of charge and can be calculated by using the Gauss law.

9. 2-D model of multipactor (cont.) DC electric field: Here r d is the radius of vacuum/dielectric interface, σ n0 is the initial surface charge density of n -th cylindrical layer of charge. Summation is done over all layers with r n < r. If the operation is close to the experimental parameters ( f = 11.424 GHz, r d = 5 mm, r w = 7.185 mm, ε rd = 9.4), then, and. Also, when we may neglect the z-motion of the particle and reduce the set of equations of particle motion to Here we have used normalized variables and parameters -normalized wave amplitude and -plasma parameter; w n is the normalized weight of n-th layer.

10. 2-D model of multipactor (cont.) Each time a macro-particle leaves the surface it is assigned a random initial energy and emission angle according to the following PDFs: Here E om corresponds to the peak of the energy distribution. We used Vaughan’s empirical model to compute the secondary emission yield for particles with impact energies V i and impact angle : Here and are the parameters for the impact angle equal to 0, i.e. normal to the surface, k s is the “smoothness factor” for the surface.

11. Model implementation Particles uniformly distributed over initial emission phases. Each particle has random initial velocity and emission angle Solve equations of motion for one t- step Check particles for surface impact Get particle impact energy and calculate secondary emission yield. w n+1 = w n *δ Check particle weight Sort particles by height Calculate dc field acting on each particle w n <= w min Eliminate ‘dead’ particles Get data for total dc field, particle location and energy histograms Split particle w n >= w max ‘Kill’ particle t = t n ? Assign random velocities and emission angles to the particles END, save data files Time loop Initial number of particles in the simulations N init = 1000. Initial weight of each particle w init = 1/N init. Impact = true w min < w n < w max yes no Impact = false t >= t final ? yes no

12. Simulation results Sample particle trajectories in the presence of a) weaker and b) stronger dc field. Rf field amplitude is the same for both cases. a) b)

13. Simulation results (cont.) α=0.2 corresponds to A=50 MV/m in these simulations, δ max0 = 4. (For alumina, V max0 = 350-1300 eV and δ max0 = 1.5-9 ).

14. Simulation results (cont.) Increase in the rf amplitude did not bring to significant changes in the results. α=0.4 corresponds to A = 100 MV/m.

15. Simulation results (cont.) Effect of the change of V max0.

16. Results (cont.) Some ‘exotic’ results.

17. Summary and future work We have developed a 2D model of multipactor in dielectric-loaded accelerator structures. The model allows to analyze the effect for a reasonably small set of parameters. Simulations were done for constant rf amplitude. The effect of the rf pulse shape should be analyzed. We need to verify our results with experimental data.

18. Acknowledgement This work has been supported by the US Department of Energy (DoE).