[Paper Review] Dynamics of Particles in Non Scaling FFAG Accelerators

James Jones, Bruno Muratori+2 more|arXiv (Cornell University)|Apr 14, 2009

Nuclear Physics and Applications4 references3 citations

TL;DR

This paper develops a complete Hamiltonian framework for modeling particle dynamics in non-scaling Fixed-Field Alternating Gradient (FFAG) accelerators, focusing on energy-dependent reference orbits, higher-order dispersion functions, and longitudinal dynamics. It demonstrates that higher-order phase slip coefficients are essential for accurate modeling of serpentine acceleration, with a detailed application to the EMMA electron FFAG prototype at Daresbury Laboratory.

ABSTRACT

Non scaling Fixed-Field Alternating Gradient (FFAG) accelerators have an unprecedented potential for muon acceleration, as well as for medical purposes based on carbon and proton hadron therapy. They also represent a possible active element for an Accelerator Driven Subcritical Reactor (ADSR). Starting from first principle the Hamiltonian formalism for the description of the dynamics of particles in non scaling FFAG machines has been developed. The stationary reference (closed) orbit has been found within the Hamiltonian framework. The dependence of the path length on the energy deviation has been described in terms of higher order dispersion functions. The latter have been used subsequently to specify the longitudinal part of the Hamiltonian. It has been shown that higher order phase slip coefficients should be taken into account to adequately describe the acceleration in non scaling FFAG accelerators. A complete theory of the fast (serpentine) acceleration in non scaling FFAGs has been developed. An example of the theory is presented for the parameters of the Electron Machine with Many Applications (EMMA), a prototype electron non scaling FFAG to be hosted at Daresbury Laboratory.

Motivation & Objective

• To establish a rigorous Hamiltonian formalism for particle dynamics in non-scaling FFAG accelerators, which lack constant tunes and require new theoretical tools.
• To derive the energy-dependent stationary reference orbit using synchrobetatron formalism, accounting for path length variations due to energy deviation.
• To investigate the role of higher-order dispersion functions in longitudinal dynamics and their impact on beam stability during fast acceleration.
• To develop a complete theory of serpentine (fast) acceleration in non-scaling FFAGs, crucial for muon and medical therapy applications.
• To validate the theory using realistic parameters from the EMMA electron FFAG prototype at Daresbury Laboratory.

Proposed method

• Formulates the relativistic Hamiltonian in a natural coordinate system attached to the curved reference orbit, incorporating electromagnetic potentials and curvature.
• Applies the synchrobetatron framework to determine the energy-dependent reference orbit and derive energy-dependent Twiss parameters and betatron tunes.
• Introduces higher-order dispersion functions to model path length variation with energy deviation, essential for longitudinal dynamics.
• Derives the longitudinal Hamiltonian including contributions from higher-order phase slip coefficients, showing their necessity for accurate modeling.
• Constructs the one-period transfer matrix and shift vector for the linearized dynamics, using trigonometric and hyperbolic functions of betatron and synchrotron frequencies.
• Analyzes phase stability using a reduced Hamiltonian form, solving the linearized equations of motion with Bessel functions to describe beam oscillations.

Experimental results

Research questions

• RQ1 How can a consistent Hamiltonian formalism be developed for non-scaling FFAGs with varying betatron tunes and energy-dependent orbits?
• RQ2 What is the role of higher-order dispersion functions in determining path length variation and longitudinal beam dynamics?
• RQ3 How do higher-order phase slip coefficients affect the stability and efficiency of fast (serpentine) acceleration in non-scaling FFAGs?
• RQ4 What is the stability of the longitudinal phase space under serpentine acceleration, particularly near the separatrix (H₀ = 0)?
• RQ5 How well does the developed theory describe the beam dynamics in the EMMA electron FFAG prototype?

Key findings

• The Hamiltonian formalism successfully describes the energy-dependent reference orbit and betatron oscillations in non-scaling FFAGs, with Twiss parameters and tunes varying with energy.
• Higher-order dispersion functions are essential for modeling path length changes due to energy spread, and their inclusion is critical for accurate longitudinal dynamics.
• The theory shows that higher-order phase slip coefficients must be included to correctly describe the longitudinal beam dynamics during fast acceleration.
• A complete theory of serpentine acceleration is developed, with phase stability analyzed using a reduced Hamiltonian and solutions expressed in terms of Bessel functions of order 1/6.
• For the EMMA prototype, phase stability is confirmed with a tolerance of ±1.3° in phase and ±0.1 MeV in energy, indicating robustness for experimental implementation.
• The derived transfer matrix and shift vector provide a complete linearized model for beam dynamics, enabling precise simulation and control of particle trajectories.