arXiv subjects

James A. Ellison

Publications and source records attributed to James A. Ellison.

9 recordsLinked to original sources

arXiv2026

Elaboration on the kinetic approach of Derbenev and Kondratenko to spin-polarized beams in electron storage rings

We present a detailed account of the kinetic approach for describing the effect of synchrotron radiation on electron and positron spin polarization in storage rings. This approach was introduced in 1974 by Derbenev and Kondratenko and was extended by us since 2001. The kinetic approach is much less frequently utilized but it is more general than the original non-kinetic approach of Derbenev and Kondratenko from 1972 since the kinetic approach is not centered on the invariant spin field. As with the non-kinetic approach the kinetic approach covers the radiative depolarization effect, the Sokolov-Ternov effect and its Baier-Katkov correction as well as the kinetic polarization effect but it enables the calculation of corrections to the original Derbenev-Kondratenko formulas and thereby provides estimates of the reliability of the latter. The kinetic approach is applicable to storage rings with energies from a few GeV up to the energies of the FCC-ee and CEPC and beyond. The kinetic approach is based on the spin-orbit Wigner functions which lead to the so-called Bloch equation for the polarization density which is a generalization of Fokker-Planck equations to spin motion. In turn, as discovered in 2019, the Bloch equation is based on stochastic ordinary differential equations which can be used to develop Monte-Carlo spin tracking codes covering the key effects beyond the radiative depolarization effect. These stochastic ordinary differential equations lead to a new viewpoint of the physical effects, in particular the kinetic polarization effect.

physics.acc-ph↗

arXiv2026

A detailed and unified treatment of spin-orbit systems using tools distilled from the theory of bundles

We return to our study \cite{BEH} of invariant spin fields and spin tunes for polarized beams in storage rings but in contrast to the continuous-time treatment in \cite{BEH}, we now employ a discrete-time formalism, beginning with the $\rm{Poincar\acute{e}}$ maps of the continuous time formalism. We then substantially extend our toolset and generalize the notions of invariant spin field and invariant frame field. We revisit some old theorems and prove several theorems believed to be new. In particular we study two transformation rules, one of them known and the other new, where the former turns out to be an $SO(3)$-gauge transformation rule. We then apply the theory to the dynamics of spin-$1/2$ and spin-$1$ particle bunches and their density matrix functions, describing semiclassically the particle-spin content of bunches. Our approach thus unifies the spin-vector dynamics from the T-BMT equation with the spin-tensor dynamics and other dynamics. This unifying aspect of our approach relates the examples elegantly and uncovers relations between the various underlying dynamical systems in a transparent way. As in \cite{BEH}, the particle motion is integrable but we now allow for nonlinear particle motion on each torus. Since this work is inspired by notions from the theory of bundles, we also provide insight into the underlying bundle-theoretic aspects of the well-established concepts of invariant spin field, spin tune and invariant frame field. Since we neglect, as is usual, the Stern-Gerlach force, the underlying principal bundle is of product formso that we can present the theory in a fashion which does not use bundle theory. Nevertheless we occasionally mention the bundle-theoretic meaningof our concepts and we also mention the similarities with the geometrical approach to Yang-Mills Theory.

physics.acc-ph↗

arXiv2021

Re-evaluation of Spin-Orbit Dynamics of Polarized e+ e- Beams in High Energy Circular Accelerators and Storage Rings: an approach based on a Bloch equation

We give an overview of our current/future analytical and numerical work on the spin polarization in high-energy electron storage rings. Our goal is to study the possibility of polarization for the CEPC and FCC-ee. Our work is based on the so-called Bloch equation for the polarization density introduced by Derbenev and Kondratenko in 1975. We also give an outline of the standard approach, the latter being based on the Derbenev-Kondratenko formulas.

physics.acc-ph↗

arXiv2020

The Bloch equation for spin dynamics in electron storage rings: computational and theoretical aspects

In this paper we describe our work on spin polarization in high-energy electron storage rings which we base on the Bloch equation for the polarization density and which aims towards the e-/e+ option of the proposed Future Circular Collider (FCC-ee) and the proposed Circular Electron Positron Collider (CEPC). The Bloch equation takes into account non spin-flip and spin-flip effects due to synchrotron radiation including the spin-diffusion effects and the Sokolov-Ternov effect with its Baier-Katkov generalization as well as the kinetic-polarization effect. This mathematical model is an alternative to the standard mathematical model based on the Derbenev-Kondratenko formulas. For our numerical and analytical studies of the Bloch equation we develop an approximation to the latter to obtain an effective Bloch equation. This is accomplished by finding a third mathematical model based on a system of stochastic differential equations underlying the Bloch equation and by approximating that system via the method of averaging from perturbative ODE theory. We also give an overview of our algorithm for numerically integrating the effective Bloch equation. This discretizes the phase space using spectral methods and discretizes time via the additive Runge-Kutta method which is a high-order semi-implicit method. We also discuss the relevance of the third mathematical model for spin tracking.

physics.acc-ph↗

arXiv2015

An Informal Summary of a New Formalism for Classifying Spin-Orbit Systems Using Tools Distilled from the Theory of Bundles

We give an informal summary of ongoing work which uses tools distilled from the theory of fibre bundles to classify and connect invariant fields associated with spin motion in storage rings. We mention four major theorems. One ties invariant fields with the notion of normal form, the second allows comparison of different invariant fields and the two others tie the existence of invariant fields to the existence of certain invariant sets. We explain how the theorems apply to the spin dynamics of spin-$1/2$ and spin-$1$ particles. Our approach elegantly unifies the spin-vector dynamics from the T-BMT equation with the spin-tensor dynamics and other dynamics and suggests an avenue for addressing the question of the existence of the invariant spin field.

physics.acc-ph↗

arXiv2014

A New and Unifying Approach to Spin Dynamics and Beam Polarization in Storage Rings

With this paper we extend our studies [1] on polarized beams by distilling tools from the theory of principal bundles. Four major theorems are presented, one which ties invariant fields with the notion of normal form, one which allows one to compare different invariant fields, and two that relate the existence of invariant fields to the existence of certain invariant sets and relations between them. We then apply the theory to the dynamics of spin-1/2 and spin-1 particles and their density matrices describing statistically the particle-spin content of bunches. Our approach thus unifies the spin-vector dynamics from the T-BMT equation with the spin-tensor dynamics and other dynamics. This unifying aspect of our approach relates the examples elegantly and uncovers relations between the various underlying dynamical systems in a transparent way.

physics.acc-ph↗

arXiv2013

Planar undulator motion excited by a fixed traveling wave: Quasiperiodic Averaging, normal forms and the FEL Pendulum

We present a mathematical analysis of planar motion of energetic electrons moving through a planar dipole undulator, excited by a fixed planar polarized plane wave Maxwell field in the X-Ray Free Electron Laser (FEL) regime. Our starting point is the 6D Lorentz system, which allows planar motions, and we examine this dynamical system as the wavelength of the traveling wave varies. By scalings and transformations the 6D system is reduced, without approximation, to a 2D system in a form for a rigorous asymptotic analysis using the Method of Averaging (MoA), a long time perturbation theory. The two dependent variables are a scaled energy deviation and a generalization of the so-called ponderomotive phase. As the wavelength varies the system passes through resonant and nonresonant (NR) zones and we develop NR and near-to-resonant (NtoR) normal form approximations. For a special initial condition and on resonance, the NtoR normal form reduces to the well-known FEL pendulum system. We then state and prove NR and NtoR first-order averaging theorems which give near optimal error bounds for the MoA approximations. The proofs are novel in that they do not use a near identity transformation and they use a system of differential inequalities. The NR case is an example of quasiperiodic averaging where the small divisor problem enters in the simplest possible way. To our knowledge the planar problem has not been analyzed with the generality we aspire to here nor has the standard FEL pendulum system been derived with associated error bounds as we do here.

physics.acc-ph↗

arXiv2003

First-Order Averaging Principles for Maps with Applications to Beam Dynamics in Particle Accelerators

For slowly evolving, discrete-time-dependent systems of difference equations (iterated maps), we believe the simplest means of demonstrating the validity of the averaging method at first order is by way of a lemma that we call Besjes' inequality. In this paper, we develop the Besjes inequality for identity maps with perturbations that are (i) at low-order resonance (periodic with short period) and (ii) far from low-order resonance in the discrete time. We use these inequalities to prove corresponding first-order averaging principles, together with a principle of adiabatic invariance on extended timescales; and we generalize and apply these mathematical results to model problems in accelerator beam dynamics, and to the Henon map.

physics.acc-ph↗

arXiv1996

Diffusion due to the Beam-Beam Interaction and Fluctuating Fields in Hadron Colliders

Random fluctuations in the tune, beam offsets and beam size in the presence of the beam-beam interaction are shown to lead to significant particle diffusion and emittance growth in hadron colliders. We find that far from resonances high frequency noise causes the most diffusion while near resonances low frequency noise is responsible for the large emittance growth observed. Comparison of different fluctuations shows that offset fluctuations between the beams causes the largest diffusion for particles in the beam core.

acc-phys↗