arXiv subjects
Publications and source records attributed to D. P. Barber.
We present a set of notes, meant for quick reference, on radiative spin polarization, computer algorithms and spin matching in electron storage rings.
I extend and update earlier work, summarised in [1], whereby the invariant polarisation-tensor field (ITF) for deuterons in storage rings was introduced to complement the invariant spin field (ISF). Taken together, the ITF and the ISF provide a definition of the equilibrium spin density-matrix field which, in turn, offers a clean framework for describing equilibrium spin-1 ensembles in storage rings. I show how to construct the ITF by stroboscopic averaging, I give examples, I discuss adiabatic invariance and I introduce a formalism for describing the effect of noise and damping.
We argue that the invariant tensor field introduced in [1] is unique under the condition that the invariant spin field is unique, and thereby complete that part of the discussion in that paper.
This is an addendum to the paper "Some models of spin coherence and decoherence in storage rings" by one of the authors [1] in which spin diffusion in simple electron storage rings is studied. In particular, we illustrate in a compact way, a key implication in the Epilogue of [1], namely that the exact formalism of [1] delivers a rate of depolarisation which can differ from that obtained by the conventional treatments of spin diffusion which rely on the use of the derivative $\partial \hat n/\partialη$ [2,3,4]. As a vehicle we consider a ring with a Siberian Snake and electron polarisation in the plane of the ring (Machine II in [1]). For this simple setup with its one-dimensional spin motion, we avoid having to deal directly with the Bloch equation [5,6] for the polarisation density. Our treatment, which is deliberately pedagogical, shows that the use of $\partial \hat n/\partialη$ provides a very good approximation to the rate of spin depolarisation in the model considered. But it then shows that the exact rate of depolarisation can be obtained by replacing $\partial \hat n/\partialη$ by another derivative as suggested in the Epilogue of [1], while giving a heuristic justification for the new derivative.
A short overview about the potential of polarized beams at future colliders is given. In particular the baseline design for polarized beams at the ILC is presented and the physics case for polarized $e^-$ and $e^+$ is discussed. In order to fulfil the precision requirements spin tracking from the source to the interaction point is needed. Updates concerning the theoretical calculations as well as their implementation in simulation codes are reported.
Here, and in a sequel, we invoke the invariant spin field to provide an in--depth study of spin motion at and near low order orbital resonances in a simple model for the effects of vertical betatron motion in a storage ring with Siberian Snakes. This leads to a clear understanding, within the model, of the behaviour of the beam polarisation at and near so--called snake resonances in proton storage rings.
Snake ``resonances'' are classified in terms of the invariant spin field and the amplitude dependent spin tune. Exactly at snake ``resonance'' there is no continuous invariant spin field at most orbital amplitudes.
We present an in-depth analysis of the concept of spin precession frequency for integrable orbital motion in storage rings. Spin motion on the periodic closed orbit of a storage ring can be analyzed in terms of the Floquet theorem for equations of motion with periodic parameters and a spin precession frequency emerges in a Floquet exponent as an additional frequency of the system. To define a spin precession frequency on nonperiodic synchro-betatron orbits we exploit the important concept of quasiperiodicity. This allows a generalization of the Floquet theorem so that a spin precession frequency can be defined in this case too. This frequency appears in a Floquet-like exponent as an additional frequency in the system in analogy with the case of motion on the closed orbit. These circumstances lead naturally to the definition of the uniform precession rate and a definition of spin tune. A spin tune is a uniform precession rate obtained when certain conditions are fulfilled. Having defined spin tune we define spin-orbit resonance on synchro--betatron orbits and examine its consequences. We give conditions for the existence of uniform precession rates and spin tunes (e.g. where small divisors are controlled by applying a Diophantine condition) and illustrate the various aspects of our description with several examples. The formalism also suggests the use of spectral analysis to ``measure'' spin tune during computer simulations of spin motion on synchro-betatron orbits.
We show how, beginning with the Fokker--Planck equation for electrons emitting synchrotron radiation in a storage ring, the corresponding equation for spin motion can be constructed. This is an equation of the Bloch type for the polarisation density.
The exact solution of the Dirac equation and the spectrum of electron quasi-energies in a superposition of the field of a circularly polarized electromagnetic wave and a homogeneous magnetic field parallel to the direction of wave propagation, are found taking into account the anomalous magnetic moment. It is found that taking account of the anomalous magnetic moment removes the spin degeneracy and that for intense fields the levels change radically. The shift of the radiation frequency due to the intensity of the wave field is found. This shift can be considerable.
I review recent developments in spin dynamics in electron storage rings and accelerators.
The first attainment of longitudinal spin polarisation in a high energy electron (positron) storage ring is described.
We illustrate the use of the invariant spin field for describing permissible equilibrium spin distributions in high energy spin polarised proton beams.}
The relationship between the level of spin polarisation caused by Unruh radiation as calculated by Bell and Leinaas and that obtained from the Derbenev-Kondratenko formalism is explained.
A semiclassical Foldy--Wouthuysen transformation of the Dirac equation is used to obtain the radiationless Bloch equation for the polarisation density.
This article provides a unified introduction to the theory of electron and proton spin polarisation in storage rings and it provides a common starting point for the written versions of the four talks that I gave at Monterey