High Energy and Nuclear Physics

Accelerator

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Introduction

Simulating the dynamics of a beam of particles in a circular accelerator can be accomplished by modeling the particles as a "herd" of macroparticles with each macroparticle containing numerous individual particles. The overarching reason for grouping individual particles in this way is due to the limitations of current computer hardware. Even with the use of macroparticles, current hardware is stressed to its limits. Particles are propagated through the accelerator lattice through the use of transfer maps that model the accelerator modules, i.e. magnets, RF cavities etc. These transfer maps consist of matrices in six dimensional particle phase space and higher order transformations. High intensity beams introduce space charge effects that are caused by macroparticles interacting with each other and also interactions between the macroparticles and the walls of the vacuum chamber. These effects can even dominate in certain instances with subsequent formation of a beam halo. Parallel computing can allow the practical tracking of great numbers of macroparticles.

Accelerator Components

The physics of an accelerator is based on the study of the relativistic equations of motion for a particle in an electric and magnetic field:

Eq.1

with the definitions

Eq.2

and are the charge and mass of the particle respectively, its rest mass, its total energy, the electric field, the magnetic field, and the speed of light. Equation 1 shows that to effectively accelerate a particle on a curved trajectory, the electric field must have a non vanishing component parallel to the particle velocity and magnetic field perpendicular to it.

Electro-magnetic fields are created by machine elements or modules, arranged in the lattice of the accelerator. The lattice contains magnets that provide magnetic fields perpendicular to the orbit and radio frequency cavities that provide electric fields parallel to the orbit.

Accelerator Models

Accelerator models are comprised of two components. Initially, the models are based on single particle dynamics (no particle-particle interaction). These results then form the input to the multi particle dynamics component of the models, which deal with particles interacting with each other as well as the environment, i.e. chamber walls.

Single Particle Dynamics - A particle is represented by a vector in 6 dimensional phase space:

Eq.3

and are the transverse coordinates, and are the components of the transverse momentum, is the deviation of the given particle in time, and is its deviation in longitudinal momentum. During the course of the simulation, r is transformed from one accelerator module to the next, along the machine lattice by a map. Each map must obey the symplecticity conditions to insure that the relevant physical invariants are conserved. One important invariant is the Hamiltonian of the system, a function of the particle coordinates and of the electro-magnetic field through which they move:

Eq.4

where is the electric potential and the magnetic potential. is representative of the total energy of the system. To first order, a symplectic map is a matrix that represents a rotation of in phase space. Various Optics codes are utilized to perform the chain of transformations along the accelerator lattice. It is here that the various physics parameters of the machine are input. Well known optics codes include Mad, Marylie and Teapot.

Multi Particle Dynamics and Tracking with Space Charge - A randomly generated herd of macroparticles is injected into the machine lattice generated by the optics code and the vector defined in Eq. 3 is transformed through each module. For high intensities, when space charge forces are present, each macroparticle’s coordinates are transferred from one machine element to the next as individual particles. The particles are binned on a mesh to establish the charge density (P). Space charge forces are then calculated and applied to each particle which, in turn, produces an angle kick described by

Eq.5

Figure 1

A herd of 1000 macroparticles after injection

Numerical Modeling of Space Charge - The numerical solution of the space charge forces is among the most computer intensive problems in particle tracking. The particle beam in a circular accelerator is generally a long structure that occupies a large fraction of the entire length of the accelerator chamber. "Bunches" may be present in this beam depending on the harmonic mode of the accelerating electric field. The size of a bunch is on the order of mm to cm in the transverse dimension x and y as a perturbation to the position of a symmetric beam and meters in the longitudinal dimension z. The particles in the beam are of like electric charge and therefore repel one another. Since they are in motion they are current elements which, since they are moving in the same direction, attract. The net force can be shown to be repulsive but decreasing as:

Eq.6

This implies that as the velocity of the particles approaches that of light, the space charge forces vanish. Space charge treatment involves the inversion of the Poisson problem in the presence of boundaries. The potential field is calculated by solving the Poisson differential equation:

Eq.7

Poisson Solver based on LU Decomposition - Recent efforts have focused on solving Eq. 7 using LU Decomposition techniques. Let us discretize the charge density and the potential at the corners of a mesh (i, j) and write Eq. 7 as follows:

Eq.8

Let the source point be denoted by Q = (i, j) and the Field point be denoted by P = (k, l). The inversion of the Poisson equation is then performed yielding:

Eq.9

L can then be constructed using a finite difference expression of the second partial derivatives as follows:

Eq.10

where = 0, 1 and the grid spacing is normalized to 1. This creates a matrix of size . The force field can then be calculated directly from the potential field by differentiating numerically using central difference schemes.

Conclusions and Results

Parallel performance has been analyzed using the MPI implementation of ORBIT run on the CSC Linux Cluster (CLC). The performance tests have shown a linear increase in speed with increasing numbers of processors as shown in Figure 2.

Figure 2

Performance results of ORBIT runs utilizing increasing numbers of processors. The Performance Quotient is calculated by dividing the base runtime (1 calculating processor) by the test runtime. The results show a linear speed-up over 33 processors.

Another topic of interest is beam losses due to collisions with the walls of the vacuum chamber. Analysis has shown a loss rate of 1E-4 particles with walls of radius 3-sigma of the particle distribution. These losses are dependent on the machine parameters input at the start of the program.

Figure 3

Distribution of a herd of macroparticles. The region ends at a distance of 3-sigma and thus represents the wall of the vacuum chamber. The beam losses at this boundary are roughly 1E-4 macroparticles.

Work is now commencing on including effects of the walls on the forces as well as implementation of graphics and visualization into ORBIT.

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Last Modified: January 31, 2008
Please forward all questions about this site to: Claire Lamberti

One of ten national laboratories overseen and primarily funded by the Office of Science of the U.S. Department of Energy (DOE), Brookhaven National Laboratory conducts research in the physical, biomedical, and environmental sciences, as well as in energy technologies and national security. Brookhaven Lab also builds and operates major scientific facilities available to university, industry and government researchers. Brookhaven is operated and managed for DOE’s Office of Science by Brookhaven Science Associates, a limited-liability company founded by Stony Brook University, the largest academic user of Laboratory facilities, and Battelle, a nonprofit, applied science and technology organization.

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