Collective effects (accelerator physics)

Last updated March 20, 2019

Charged particle beams in a particle accelerator or a storage ring undergo a variety of different processes. Typically the beam dynamics is broken down into single particle dynamics and collective effects. Sources of collective effects include single or multiple inter-particle scattering and interaction with the vacuum chamber and other surroundings, formalized in terms of impedance.

A charged particle beam is a spatially localized group of electrically charged particles that have approximately the same position, kinetic energy, and direction. The kinetic energies of the particles are much larger than the energies of particles at ambient temperature. The high energy and directionality of charged particle beams make them useful for applications.

Particle accelerator device to propel charged particles to high speeds

A particle accelerator is a machine that uses electromagnetic fields to propel charged particles to very high speeds and energies, and to contain them in well-defined beams.

Storage ring Type of particle accelerator

A storage ring is a type of circular particle accelerator in which a continuous or pulsed particle beam may be kept circulating typically for many hours. Storage of a particular particle depends upon the mass, momentum and usually the charge of the particle to be stored. Storage rings most commonly store electrons, positrons, or protons.

The collective effects of charged particle beams in particle accelerators share some similarity to the dynamics of plasmas. In particular, a charged particle beam may be considered as a non-neutral plasma, and one may find mathematical methods in common with the study of stability or instabilities. One may also find commonality with the field of fluid mechanics since the density of charged particles is often sufficient to be considered as flowing continuum.

Plasma is one of the four fundamental states of matter, and was first described by chemist Irving Langmuir in the 1920s. Plasma can be artificially generated by heating or subjecting a neutral gas to a strong electromagnetic field to the point where an ionized gaseous substance becomes increasingly electrically conductive, and long-range electromagnetic fields dominate the behaviour of the matter.

A non-neutral plasma is a plasma for which the total charge is sufficiently different from zero, so that the electric field created by the un-neutralized charge plays an important or even dominant role in the plasma dynamics. The simplest non-neutral plasmas are plasmas consisting of a single charge species. Examples of single species non-neutral plasmas that have been created in laboratory experiments are plasmas consisting entirely of electrons, pure ion plasmas, positron plasmas, and antiproton plasmas.

The stability of a plasma is an important consideration in the study of plasma physics. When a system containing a plasma is at equilibrium, it is possible for certain parts of the plasma to be disturbed by small perturbative forces acting on it. The stability of the system determines if the perturbations will grow, oscillate, or be damped out.

Another important topic is the attempt to mitigate collective effects by use of single bunch or multi-bunch feedback systems.

Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to feed back into itself. The notion of cause-and-effect has to be handled carefully when applied to feedback systems:

Simple causal reasoning about a feedback system is difficult because the first system influences the second and second system influences the first, leading to a circular argument. This makes reasoning based upon cause and effect tricky, and it is necessary to analyze the system as a whole.

Types of collective effects

Collective effects can include emittance growth, bunch length or energy spread growth, instabilities, or particle losses.^[1] There are also multi-bunch effects.

Emittance is a property of a charged particle beam in a particle accelerator. It is a measure for the average spread of particle coordinates in position-and-momentum phase space and has the dimension of length or length times angle. As a particle beam propagates along magnets and other beam-manipulating components of an accelerator, the position spread may change, but in a way that does not change the emittance. If the distribution over phase space is represented as a cloud in a plot, emittance is the area of the cloud. A more exact definition handles the fuzzy borders of the cloud and the case of a cloud that does not have an elliptical shape.

Instability Wikimedia disambiguation page

In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds. Not all systems that are not stable are unstable; systems can also be marginally stable or exhibit limit cycle behavior.

Formalisms for treating collective effects

The collective beam motion may be modeled in a variety of ways. One may use macroparticle models, or else a continuum model. The evolution equation in the latter case is typically called the Vlasov equation, and requires one to write down the Hamiltonian function including the external magnetic fields, and the self interaction. Stochastic effects may be added by generalizing to the Fokker–Planck equation.

The Vlasov equation is a differential equation describing time evolution of the distribution function of plasma consisting of charged particles with long-range interaction, e.g. Coulomb. The equation was first suggested for description of plasma by Anatoly Vlasov in 1938 and later discussed by him in detail in a monograph.

In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field, defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations in classical mechanics. The integral curves of a Hamiltonian vector field represent solutions to the equations of motion in the Hamiltonian form. The diffeomorphisms of a symplectic manifold arising from the flow of a Hamiltonian vector field are known as canonical transformations in physics and (Hamiltonian) symplectomorphisms in mathematics.

In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well. It is named after Adriaan Fokker and Max Planck, and is also known as the Kolmogorov forward equation, after Andrey Kolmogorov, who independently discovered the concept in 1931. When applied to particle position distributions, it is better known as the Smoluchowski equation, and in this context it is equivalent to the convection–diffusion equation. The case with zero diffusion is known in statistical mechanics as the Liouville equation. The Fokker–Planck equation is obtained from the master equation through Kramers–Moyal expansion.

Software for computation of collective effects

Depending on the effects considered and the modeling formalism used, different software is available for simulation. The collective effects must typically be added in addition to the single particle dynamics, which may be modeled using a tracking code. See article on Accelerator physics codes.

A charged particle accelerator is a complex machine that takes elementary charged particles and accelerates them to very high energies. Accelerator physics is a field of physics encompassing all the aspects required to design and operate the equipment and to understand the resulting dynamics of the charged particles. There are software packages associated with each such domain. There are a large number of such codes. The 1990 edition of the Los Alamos Accelerator Code Group's compendium provides summaries of more than 200 codes. Certain of those codes are still in use today although many are obsolete. Another index of existing and historical accelerator simulation codes is located at

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Particle physics is a branch of physics that studies the nature of the particles that constitute matter and radiation. Although the word particle can refer to various types of very small objects, particle physics usually investigates the irreducibly smallest detectable particles and the fundamental interactions necessary to explain their behaviour. By our current understanding, these elementary particles are excitations of the quantum fields that also govern their interactions. The currently dominant theory explaining these fundamental particles and fields, along with their dynamics, is called the Standard Model. Thus, modern particle physics generally investigates the Standard Model and its various possible extensions, e.g. to the newest "known" particle, the Higgs boson, or even to the oldest known force field, gravity.

Stellar dynamics is the branch of astrophysics which describes in a statistical way the collective motions of stars subject to their mutual gravity. The essential difference from celestial mechanics is that each star contributes more or less equally to the total gravitational field, whereas in celestial mechanics the pull of a massive body dominates any satellite orbits.

The particle-in-cell (PIC) method refers to a technique used to solve a certain class of partial differential equations. In this method, individual particles in a Lagrangian frame are tracked in continuous phase space, whereas moments of the distribution such as densities and currents are computed simultaneously on Eulerian (stationary) mesh points.

A particle beam is a stream of charged or neutral particles, in many cases moving at near the speed of light.

Plasma acceleration is a technique for accelerating charged particles, such as electrons, positrons, and ions, using the electric field associated with electron plasma wave or other high-gradient plasma structures. The plasma acceleration structures are created either using ultra-short laser pulses or energetic particle beams that are matched to the plasma parameters. These techniques offer a way to build high performance particle accelerators of much smaller size than conventional devices. The basic concepts of plasma acceleration and its possibilities were originally conceived by Toshiki Tajima and Prof. John M. Dawson of UCLA in 1979. The initial experimental designs for a "wakefield" accelerator were conceived at UCLA by Prof. Chan Joshi et al. Current experimental devices show accelerating gradients several orders of magnitude better than current particle accelerators over very short distances, and about one order of magnitude better at the one meter scale.

Two-photon physics, also called gamma–gamma physics, is a branch of particle physics that describes the interactions between two photons. Normally, beams of light pass through each other unperturbed. Inside an optical material, and if the intensity of the beams is high enough, the beams may affect each other through a variety of non-linear effects. In pure vacuum, some weak scattering of light by light exists as well. Also, above some threshold of this center-of-mass energy of the system of the two photons, matter can be created.

The AWAKE facility at CERN is a proof-of-principle experiment, which investigates wakefield plasma acceleration using a proton bunch as a driver, a world-wide first. It aims to accelerate a low-energy witness bunch of electrons from 15-20 MeV to several GeV over a short distance (10m) by creating a high acceleration gradient of several GV/m. Particle accelerators currently in use, like CERN's LHC, use standard or superconductive RF-cavities for acceleration, but they are limited to an acceleration gradient in the order of 100 MV/m.

The two-stream instability is a very common instability in plasma physics. It can be induced by an energetic particle stream injected in a plasma, or setting a current along the plasma so different species can have different drift velocities. The energy from the particles can lead to plasma wave excitation.

A relativistic particle is a particle which moves with a relativistic speed; that is, a speed comparable to the speed of light. This is achieved by photons to the extent that effects described by special relativity are able to describe those of such particles themselves. Several approaches exist as a means of describing the motion of single and multiple relativistic particles, with a prominent example being postulations through Dirac equations of single particle motion.

The electron-cloud effect is a phenomenon that occurs in particle accelerators and reduces the quality of the particle beam.

The Nonequilibrium Gas and Plasma Dynamics Laboratory (NGPDL) at the Aerospace Engineering Department of the University of Michigan is headed by Professor Iain D. Boyd and performs research of nonequilibrium gases and plasmas involving the development of physical models for various gas systems of interest, numerical algorithms on the latest supercomputers, and the application of challenging flows for several exciting projects. The lab places a great deal of emphasis on comparison of simulation with external experimental and theoretical results, having ongoing collaborative studies with colleagues at the University of Michigan such as the Plasmadynamics and Electric Propulsion Laboratory, other universities, and government laboratories such as NASA, United States Air Force Research Laboratory, and the United States Department of Defense.

Plasma Modeling refers to solving equations of motion that describe the state of a plasma. It is generally coupled with Maxwell's Equations for electromagnetic fields or Poisson's Equation for electrostatic fields. There are several main types of plasma models: single particle, kinetic, fluid, hybrid kinetic/fluid, gyrokinetic and as system of many particles.

Gyrokinetics is a theoretical framework to study plasma behavior on perpendicular spatial scales comparable to the gyroradius and frequencies much lower than the particle cyclotron frequencies. These particular scales have been experimentally shown to be appropriate for modeling plasma turbulence. The trajectory of charged particles in a magnetic field is a helix that winds around the field line. This trajectory can be decomposed into a relatively slow motion of the guiding center along the field line and a fast circular motion, called gyromotion. For most plasma behavior, this gyromotion is irrelevant. Averaging over this gyromotion reduces the equations to six dimensions rather than the seven. Because of this simplification, gyrokinetics governs the evolution of charged rings with a guiding center position, instead of gyrating charged particles.

Impedance in Accelerator Physics is a quantity that characterizes the self interaction of a charged particle beam, mediated by the beam environment, such as the vacuum chamber, RF cavities, and other elements encountered along the accelerator or storage ring.

References

↑ A. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators, Wiley Publishers, 1993. Available here.

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[1] A. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators, Wiley Publishers, 1993. Available here.