Multiscale Modelling, taking into account Collisions

Multiscale Modelling, taking into account Collisions Andreas Adelmann (Paul Scherrer Institut) October 28, 2017 Multiscale Modelling, taking into account Collisions October 28, 2017 Page 1 / 22

1 Motivation 2 Multiscale Nature of the Problem 3 The P 3 M algorithm 4 Disorder Induced Heating Simulation 5 Conclusion Multiscale Modelling, taking into account Collisions October 28, 2017 Page 2 / 22

Motivation I 1 model emission of ultra cold electrons 2 understand Coulomb scattering (Borsch effect) Qiang et al. [2007] 3 do we have to worry about it in next generation machines? 4 model non Gaussian tails (high intensity hadron machines) In Qiang et al. [2007] we wrote: Nano-tips with high acceleration gradient around the emission surface have been proposed to generate high brightness beams. However, due to the small size of the tip r = 10 nm, the charge density near the tip is very high even for a small number of electrons. The stochastic Coulomb scattering near the tip can degrade the beam quality and cause extra emittance growth and energy spread. Multiscale Modelling, taking into account Collisions October 28, 2017 Page 3 / 22

Motivation II Using a brute force N 2 summation we obtained the following observations: slice emittance over the bunch length 2 higher energy spread 100 higher Multiscale Modelling, taking into account Collisions October 28, 2017 Page 4 / 22

Multiscale Nature of the Problem Considering also high intensity high hadron machines (Cyclotrons & FFAGs) length scales: O(10 3 )... O(10 6 ) (m) time scales: 10 9... 10 13 (s) energy scales: ev... space charge or emittance dominated Space-Charge effects commonly treated in the mean-field or Vlasov- Poisson/Maxwell approximation. Valid mostly when timescales of interest are much smaller than relaxation timescales. In this approximation, particles move under the influence of external fields and a self-consistent force due to all the other particles. In the electrostatic case, this is the Vlasov-Poisson equation. Multiscale Modelling, taking into account Collisions October 28, 2017 Page 5 / 22

Vlasov-Poisson Equation Addressing the multi scale challange When neglecting collisions, and taking advantage of the electrostatic approximation, the Vlasov-Poisson equation describes the (time) evolution of the phase space f(x, v; t) > 0 when considering electromagnetic interaction with charged particles 1. df dt = f t + v xf + q m (E(x, t) + v B(x, t)) vf = 0. (1) Solving with ES-PIC Hockney and Eastwood, h x (t), h y (t), h z (t), M = M x M y M z SAAMG-PCG solver with geometry Adelmann et al. [2010] change M during simulation (many different field solver instances) change t adaptively Toggweiler et al. [2014] modern computational architectures Adelmann et al. [2015] 1 OPAL https://gitlab.psi.ch/opal/src/wikis/ Multiscale Modelling, taking into account Collisions October 28, 2017 Page 6 / 22

Adding (back) Collisions The Boltzmann (Landau) equation Diffusion Damping {}}{{}}{ df dt = (1) = 1 2 (f v i ) + (f v i v j ), v i 2 v i v j the quantities v i and v i v j are the average change in velocity a particle experiences due to collisions with all other particles. large-angle collisions are hard small-angle collisions are soft. Because the Coulomb force is long-ranged, in many cases, the first fluctuation correction to the mean-field approximation comes from the collective contribution of a large number of small-angle collisions. Multiscale Modelling, taking into account Collisions October 28, 2017 Page 7 / 22

The P 3 M Algorithm The P 3 M = Particle-Particle + Particle-Mesh is a efficient way to accomplish this task. high resolution from PP part: O(K 2 ), K << N, 1/(x x + ε) good performance from PM part: Φ(x) = G(x, x )ρ(x )d 3 x adjustable influence of Coulomb collisions by fixing K in choosing r c Multiscale Modelling, taking into account Collisions October 28, 2017 Page 8 / 22

The P 3 M Algorithm Interaction Splitting I The main concept behind the P 3 M algorithm is a splitting of the interaction function G(r) into a short-range contribution G pp (r) and a long-range contribution G pm (r). This splitting can be done using a Gaussian screening charge distribution G(r) = 1 r = 1 erf(αr) r } {{ } G P P + erf(αr) r } {{ } G P M The long-range interactions are computed on a grid using a particle-mesh solver and the short range interactions are evaluated using an N-body solver with appropriate cutoff.. Multiscale Modelling, taking into account Collisions October 28, 2017 Page 9 / 22

The P 3 M Algorithm Interaction Splitting II f(r) 5. 4. 3. 2. 1. 0. 0.5 1. 1.5 2. r Greens function splitting for r c = 1 and α = 2: G(r) in green, G P M (r) in red and G P P (r) in blue Multiscale Modelling, taking into account Collisions October 28, 2017 Page 10 / 22

The P 3 M Algorithm Implementation following [Hockney and Eastwood, 1988] Particle-Mesh (PM): 1 interpolate charges to mesh (CIC, NGP,...) 2 solve for potential Φ using an FFT solver (convolution with greens function) 3 compute forces by F = Φ 4 interpolate forces to particles electric field Particle-Particle (PP): 1 compute linked lists for particles in interaction radius r e, fix K 2 compute short range forces 3 update electric field Multiscale Modelling, taking into account Collisions October 28, 2017 Page 11 / 22

The P 3 M Algorithm Implementation following [Hockney and Eastwood, 1988] Particle-Mesh (PM): 1 interpolate charges to mesh (CIC, NGP,...) 2 solve for potential Φ using an FFT solver (convolution with greens function) 3 compute forces by F = Φ 4 interpolate forces to particles electric field Particle-Particle (PP): 1 compute linked lists for particles in interaction radius r e, fix K 2 compute short range forces 3 update electric field Multiscale Modelling, taking into account Collisions October 28, 2017 Page 11 / 22

Disorder Induced Heating Simulation (DIH) I B fe acc kt E acc maximum achievable accelerating field kt the temperature of photoemitted electrons, kt < 25 mev Characteristic figures: conventional Cu laser cooled FEL ERL kt (mev) 25 1 N per bunch 10 6 10 9 n 0 (m 3 ) 10 17 10 20 Multiscale Modelling, taking into account Collisions October 28, 2017 Page 12 / 22

Disorder Induced Heating Simulation (DIH) II For a near-zero temperature bunch with such high density, we expect some contribution of individual stochastic Coulomb interactions from close encounters just after emission into vacuum to add to the total effective photoemission temperature. We need to determine this amount of stochastic heating as a function of beam density and initial temperature, as well as the nature of its evolution in time. We expect this effect to be most prevalent when the electrostatic potential energy of neighboring particles is comparable to their thermal energy Multiscale Modelling, taking into account Collisions October 28, 2017 Page 13 / 22

Disorder Induced Heating Simulation (DIH) III kt the Wigner-Seitz radius. e2 4πε 0 a with a = (3/4πn 0 ) 1/3 for a given density, we expect the heating O(e 2 /4πε 0 a) heating scales with cubic root of n 0 For a rough estimate of the importance of the effect, the plasma coupling parameter Γ can be used, defined as the ratio of kt to the average pair interaction potential. Γ = e2 1 4πε 0 a kt Γ = 0.2... 2 given an electron temperature of 5 mev, and for the density ranges described above. Multiscale Modelling, taking into account Collisions October 28, 2017 Page 14 / 22

Disorder Induced Heating Simulation (DIH) IV Consequence Thus, for applications requiring a large charge density, we expect this stochastic heating to serve as a hard limit to the lowest attainable electron temperature, and limiting the maximum attainable beam brightness. Multiscale Modelling, taking into account Collisions October 28, 2017 Page 15 / 22

Disorder Induced Heating Simulation (DIH) V The heating associated with the relaxation of a random, near-zero temperature distribution of charges is well known to the ultracold neutral plasma (UNP) community. In such systems, a cold gas is laser ionized, and after a time on the order of τ = 2πω 1 p e = 2π (n 2 ) 1/2 0 m e ε 0 as a consequence, the temperature T f rises to an equilibrium non zero value i.e. T f > T i. plasma physics terminology: this effect is referred to as DIH T f can be calculated analytically Multiscale Modelling, taking into account Collisions October 28, 2017 Page 16 / 22

Problem Setup Loosely connected to Filippetto et al. [2014], for an LBL UED project and Mitchell and Qiang [2015] spherical, cold beam of radius R = 17.74 µm and charge Q = 25 fc constant focusing applied cubical domain with edge length L = 100 µm P 3 M simulation over 5 plasma periods Boundary conditions: open in x, y periodic in z M = 256 3 r c varying from 0 µm to 3.125 µm N = O(10 7 ) P core = 128 simulation over 1000 time-steps Goal compute T f Multiscale Modelling, taking into account Collisions October 28, 2017 Page 17 / 22

Simulation Results MSc thesis B. Ulmer Multiscale Modelling, taking into account Collisions October 28, 2017 Page 18 / 22

Simulation Results Coulomb collisions are responsible for beam heating! P3M can reproduce analytic results Multiscale Modelling, taking into account Collisions October 28, 2017 Page 19 / 22

Conclusion! With P 3 M collisions can be added within acceptable computing cost! The flexibility of turning on/off collisions resembles an other aspect of a multi-scale - or if you preferrer - reduced order model! P 3 M opens the door to S2E simulations with collisions! Low emittance beams at low energies can be accurately modelled Questions: how relevant is this test for: advanced photo cathodes IBS high intensity hardron machines need to better understand the r c as function of beam parameters Acknowledgements: B. Ulmer, C. Mitchell and J. Qiang Multiscale Modelling, taking into account Collisions October 28, 2017 Page 20 / 22

Bibliography I A. Adelmann, P. Arbenz, and Y. Ineichen. A fast parallel Poisson solver on irregular domains applied to beam dynamics simulations. 229(12): 4554 4566, 2010. A. Adelmann, U. Locans, and A. Suter. The Dynamical Kernel Scheduler - Part 1. ArXiv, Sept. 2015. D. Filippetto, M. M. Munoz, H. Qian, F. Sannibale, R. Wells, W. Wan, and M. Zolotorev. High repetition rate ultrafast electron diffraction at lbnl. Proceedings of IPAC2014, Dresden, Germany, page 724, 2014. R. W. Hockney and J. W. Eastwood. Computer simulation using particles. CRC Press, 1988. doi: 10.1002/phbl.19880441113. C. Mitchell and J. Qiang. A Parallel Particle-Particle, Particle-Mesh Solver for Studying Coulomb Collisions in the Code IMPACT-T. In Proceedings of IPAC2015, 2015. Multiscale Modelling, taking into account Collisions October 28, 2017 Page 21 / 22

Bibliography II J. Qiang, J. Corlett, S. Lidia, H. A. Padmore, W. Wan, A. Zholents, and A. Adelmann. Numerical study of coulomb scattering effects on electron beam from a nano-tip. In Proceedings of the IEEE Particle Accelerator Conference, 2007. M. Toggweiler, A. Adelmann, P. Arbenz, and J. Yang. A novel adaptive time stepping variant of the boris buneman integrator for the simulation of particle accelerators with space charge. JCP, 3(273): 255 267, 2014. Multiscale Modelling, taking into account Collisions October 28, 2017 Page 22 / 22