Intense Charged Particle Beams Research Articles

A conditional latent autoregressive recurrent model for generation and forecasting of beam dynamics in particle accelerators

Particle accelerators are complex systems that focus, guide, and accelerate intense charged particle beams to high energy. Beam diagnostics present a challenging problem due to limited non-destructive measurements, computationally demanding simulations, and inherent uncertainties in the system. We propose a two-step unsupervised deep learning framework named as Conditional Latent Autoregressive Recurrent Model (CLARM) for learning the spatiotemporal dynamics of charged particles in accelerators. CLARM consists of a Conditional Variational Autoencoder transforming six-dimensional phase space into a lower-dimensional latent distribution and a Long Short-Term Memory network capturing temporal dynamics in an autoregressive manner. The CLARM can generate projections at various accelerator modules by sampling and decoding the latent space representation. The model also forecasts future states (downstream locations) of charged particles from past states (upstream locations). The results demonstrate that the generative and forecasting ability of the proposed approach is promising when tested against a variety of evaluation metrics.

Physics-constrained machine learning for electrodynamics without gauge ambiguity based on Fourier transformed Maxwell’s equations

We utilize a Fourier transformation-based representation of Maxwell’s equations to develop physics-constrained neural networks for electrodynamics without gauge ambiguity, which we label the Fourier–Helmholtz–Maxwell neural operator method. In this approach, both of Gauss’s laws and Faraday’s law are built in as hard constraints, as well as the longitudinal component of Ampère–Maxwell in Fourier space, assuming the continuity equation. An encoder–decoder network acts as a solution operator for the transverse components of the Fourier transformed vector potential, A^⊥(k,t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hat{{\ extbf {A}}}_\\perp ({\ extbf {k}}, t)$$\\end{document}, whose two degrees of freedom are used to predict the electromagnetic fields. This method was tested on two electron beam simulations. Among the models investigated, it was found that a U-Net architecture exhibited the best performance as it trained quicker, was more accurate and generalized better than the other architectures examined. We demonstrate that our approach is useful for solving Maxwell’s equations for the electromagnetic fields generated by intense relativistic charged particle beams and that it generalizes well to unseen test data, while being orders of magnitude quicker than conventional simulations. We show that the model can be re-trained to make highly accurate predictions in as few as 20 epochs on a previously unseen data set.

Real-time visualization of the laser-plasma wakefield dynamics.

The exploration of new acceleration mechanisms for compactly delivering high-energy particle beams has gained great attention in recent years. One alternative that has attracted particular interest is the plasma-based wakefield accelerator, which is capable of sustaining accelerating fields that are more than three orders of magnitude larger than those of conventional radio-frequency accelerators. In this device, acceleration is generated by plasma waves that propagate at nearly light speed, driven by intense lasers or charged particle beams. Here, we report on the direct visualization of the entire plasma wake dynamics by probing it with a femtosecond relativistic electron bunch. This includes the excitation of the laser wakefield, the increase of its amplitude, the electron injection, and the transition to the beam-driven plasma wakefield. These experimental observations provide first-hand valuable insights into the complex physics of laser beam-plasma interaction and demonstrate a powerful tool that can largely advance the development of plasma accelerators for real-time operation.

Physics-constrained 3D convolutional neural networks for electrodynamics

We present a physics-constrained neural network (PCNN) approach to solving Maxwell’s equations for the electromagnetic fields of intense relativistic charged particle beams. We create a 3D convolutional PCNN to map time-varying current and charge densities J(r, t) and ρ(r, t) to vector and scalar potentials A(r, t) and φ(r, t) from which we generate electromagnetic fields according to Maxwell’s equations: B = ∇ × A and E = −∇φ − ∂A/∂t. Our PCNNs satisfy hard constraints, such as ∇ · B = 0, by construction. Soft constraints push A and φ toward satisfying the Lorenz gauge.

Mathematical Modeling of Intense Beams of Charged Particles in Extended Electron-Optical Systems

Mathematical Modeling of Intense Beams of Charged Particles in Extended Electron-Optical Systems

Formation of Intense Quasicylindrical Charged-Particle Beams

2D and 3D problems of the formation of intense relativistic charged-particle beams are discussed for different geometrical configurations. Both the exact solutions in the closed form and the results of the paraxial approximation are investigated. The model of elliptic beams is analyzed, and the conclusion is drawn that classical conceptions do not correspond to practical tasks of the formation and transport of these objects.

Potential of an Elliptic Cylinder with the Space Charge Uniformly Distributed over the Cross Section

The field of an intense charged-particle beam in the form of a cylinder with the elliptic cross section is discussed. The models used hitherto are not adequate.

One-dimensional self-consistent kinetic models as Vlasov equation solutions

One-dimensional self-consistent kinetic models may describe some states of intense charged particle beams. In the collisionless approximation, which is appropriate for the short current pulse duration, the kinetic distribution function may be built as a function of the motion integrals. Two situations are considered corresponding to the wide charged particle flux with the sharp front and to the sheet continuous beam freely propagating in space. In both cases, one-dimensional Vlasov equation solutions are shown to exist, which are based on algebraic functions of the motion invariants.

Solenoid transport of beams with current-dependent initial conditions

Intense charged particle beams will generally be formed with an initial correlation between their longitudinal properties, including longitudinal variations in current, and their transverse properties, including their radius and divergence. This is due to the competition between the transverse focusing fields in the beam source and the time-varying space charge forces in the beam. In DC electron guns where the current modulation is slow compared to the electron transit time, the nature of these correlations was previously shown to depend on the gun geometry, exhibiting a linear dependence of the beam radius and divergence on the beam current at the gun exit. Here, we extend the previous work to study the transport of beams with such correlation in uniform and periodic solenoid channels. For each transport channel configuration studied, the transverse envelope equation is used to calculate the envelope of 101 beam slices differing in their slice currents, as well as initial radius and divergence (due to their dependence on slice current). For each channel configuration, these calculations are performed 546 times, with each of these iterations considering a different degree of correlation between the radius and divergence, and the slice current. It is found that some degree of correlation between the initial radius and slice current actually aids in beam transport, and the required strength of correlation can be estimated with simple models. Increasing the degree of correlation between the initial divergence and slice current is generally counterproductive, and the degree of sensitivity to such correlations depends on the design of the transport channel.

Electrostatic focusing of intense charged particle beams

Methods of forming solutions of the Maxwell equations for an electric field concerning the problem of focusing intensive charged particle beams are considered.

Long path-length experimental studies of longitudinal phenomena in intense beams

Intense charged particle beams are nonneutral plasmas as they can support a host of plasma waves and instabilities. The longitudinal physics, for a long beam, can often be reasonably described by a 1-D cold-fluid model with a geometry factor to account for the transverse effects. The plasma physics of such beams has been extensively studied theoretically and computationally for decades, but until recently, the only experimental measurements were carried out on relatively short linacs. This work reviews experimental studies over the past five years on the University of Maryland Electron Ring, investigating longitudinal phenomena over time scales of thousands of plasma periods, illustrating good agreement with simulations.

Nonlinear resonance and envelope instability of intense beam in axial symmetric periodic channel

When an intense charged particle beam propagates through a given periodic focusing channel, it will experience the phenomena of nonlinear resonance, collective instability or chaotic motion with different conditions. In this paper, the collective envelope instability mechanisms are studied for symmetric beam propagation in an axially symmetric periodic channel. The beam is characterized as collectively stable if there exists a stable fixed point (SFP) located at the matched beam condition (rm,0) in (r,pr) phase space. It is found that the well-known collective envelope instability is dynamically related to the period-two orbits bifurcation of the matched SFP, meanwhile the unique stable SFP turns into an unstable saddle-node, surrounded by 1/2 resonance islands. However, higher orders of resonance (l/n, n>2) coming from period-n bifurcation will not lead to collective beam instability because a new SFP emerges immediately upon the bifurcation process. The orders of SFP bifurcation is numerically depicted by the envelope tune ν=ϕ/360, where ϕ is the eigenphase of the Poincaré tangent map T(s) in one focusing period at SFP, as functions of depressed phase advance. With strong space charge, due to these resonances from SFP bifurcation could be overlapped, mismatched beam would even show chaotic motion. For specific parameters, regular orbits, resonance islands, chaotic regions formed by resonance overlapping are clearly depicted with frequency analysis and Lyapunov spectral exponents, a method that may prove useful when extended to higher phase-space dimensions.

Propagation of numerical noise in particle-in-cell tracking

Particle-in-cell (PIC) is the most used algorithm to perform self-consistent tracking of intense charged particle beams. It is based on depositing macro-particles on a grid, and subsequently solving on it the Poisson equation. It is well known that PIC algorithms occupy intrinsic limitations as they introduce numerical noise. Although not significant for short-term tracking, this becomes important in simulations for circular machines over millions of turns as it may induce artificial diffusion of the beam. In this work, we present a modeling of numerical noise induced by PIC algorithms, and discuss its influence on particle dynamics. The combined effect of particle tracking and noise created by PIC algorithms leads to correlated or decorrelated numerical noise. For decorrelated numerical noise we derive a scaling law for the simulation parameters, allowing an estimate of artificial emittance growth. Lastly, the effect of correlated numerical noise is discussed, and a mitigation strategy is proposed.

Free-energy formula for emittance-growth estimation in intense mismatched beams

We construct a theoretical model that allows a quick estimate of emittance growth in an intense charged-particle beam initially mismatched to an external linear focusing potential. The present theory is a natural generalization of Reiser's free-energy model for coasting round beams in a uniform focusing channel. The free energy generated by a spatial mismatch, i.e. a discrepancy between the ideal beam size and an actual beam size, is calculated for an ellipsoidal bunch with an arbitrary aspect ratio. Following Reiser's prescription, we assume that the excess free energy is converted into root-mean-squared emittance growth. Multi-particle simulations are performed for comparison with theoretical predictions, which indicates that an initially mismatched bunch eventually settles into a sort of thermally anisotropic state when the mismatch is large. It is shown that the free-energy formula can explain simulation results over a wide range of parameters if the degree of the temperature anisotropy in the final state is properly incorporated into the theory.

Maximum Possible Emittance Growth in Static Analysis of Intense Charged Particle Beams with Thermal Equilibrium Distribution

Maximum Possible Emittance Growth in Static Analysis of Intense Charged Particle Beams with Thermal Equilibrium Distribution

Effect of image charge on the off-axis transport of intense beams in a small aperture quadrupole lattice

This paper investigates the first order image charge effects of circular conducting pipe on an intense off-axis continuous charged particle beam propagating through an alternating-gradient focusing channel. The equations of motion for the centroid and envelopes of slightly off-axis beam are derived and used to determine the equilibrium and stability conditions for the beam transport. The envelope oscillation frequencies of the mismatched beam are obtained by the smooth approximation method and by numerical evaluation of the linearly perturbed envelope equations. Depending on the beam intensity and focusing parameters, it is found that the frequency of the out-of-phase mode of envelope oscillation is reduced due to the presence of the conducting wall. Detailed numerical results for off-axis beam transport in a quadrupole lattice are presented and the stable regimes in the parameter space are determined numerically. It is observed that the unstable region for the envelope evolution with image charge effects depends on beam intensity and external focusing strength and increases with the off centering of the beam.

Mitigating chromatic effects on the transverse focusing of intense charged particle beams for heavy ion fusion

A final focusing scheme designed to minimize chromatic effects is discussed. The Neutralized Drift Compression Experiment-II (NDCX-II) will apply a velocity tilt for longitudinal bunch compression, and a final focusing solenoid (FFS) for transverse bunch compression. In the beam frame, neutralized drift compression causes a sufficiently large spread in axial momentum, pz, resulting in chromatic effects on the final focal spot during transverse bunch compression. Placing a weaker solenoid upstream of a stronger final focusing solenoid (FFS) mitigates chromatic effects, and significantly improves transverse focusing for relevant NDCX-II parameters.

A scaled experiment to study dynamics during longitudinal compression of intense charged particle beams

In order to investigate the beam dynamics during longitudinal compression in high power ion accelerator, we constructed a compact bunching simulator device based on electron beams with an induction voltage adder configuration. The compression ratio and the evolution of the beam current were compared with numerical simulations using a particle code. From the results, we confirmed that the space-charge force affects the beam dynamics and the compression ratio around the focal point.

Experimental verification of resonance instability bands in quadrupole doublet focusing channels

The tabletop plasma trap experiment named “S-POD” is employed to explore the stability of intense charged-particle beams focused by a series of quadrupole doublet cells. S-POD is a compact linear Paul-trap, where we generate a single-species non-neutral ion plasma that can approximately reproduce the collective motion of an intense beam focused by periodic linear forces. Unlike conventional beam-dynamics experiments relying on large-scale transport channels and accelerators, it is straightforward in S-POD to control the functional form of quadrupole beam focusing over a wide range of variation to explore a variety of quadrupole focusing lattices. We systematically measure the loss rate of trapped particles as a function of bare betatron tune to locate resonance bands in which the plasma becomes unstable. It is confirmed that a few bands of coherent resonances appear depending on the beam intensity. When there is an imbalance between the horizontal and vertical focusing, those instability bands split. Experimental results indicate that the instability band is relatively insensitive to the phase of quadrupole focusing element placement within the doublet configuration over a significant range of parameters. Experimental observations are compared with transverse slice particle-in-cell simulations carried out using the Warp code.

Adiabatic-Nonadiabatic Transition in Warm Long-Range Interacting Systems: The Transport of Intense Inhomogeneous Beams

We investigate the role of the temperature in the onset of singularities and the consequent breakdown in a macroscopic fluid model for long-range interacting systems. In particular, we consider an adiabatic fluid description for the transport of intense inhomogeneous charged particle beams. We find that there exists a critical temperature below which the fluid model always develops a singularity and breaks down as the system evolves. As the critical temperature is approached, however, the time for the occurrence of the singularity diverges. Therefore, the critical temperature separates two distinct dynamical phases: a nonadiabatic transport at lower temperatures and a completely adiabatic evolution at higher temperatures. These findings are verified with the aid of self-consistent N-particle simulations.