V. The Semiclassical Foldy-Wouthuysen Transformation and the Derivation of the Bloch Equation for Spin-1/2 Polarised Beams Using Wigner Functions

A semiclassical Foldy--Wouthuysen transformation of the Dirac equation is used to obtain the radiationless Bloch equation for the polarisation density.Comment: 7 pages. No figures. Latex. Paper 5 of a set of 5. others are physics/9901038 physics/9901041 physics/9901042 physics/990104

Spin decoherence in electron storage rings --- more from a simple model

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 $\partial \hat n/\partial\eta$ [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 $\partial \hat n/\partial\eta$ 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 $\partial \hat n/\partial\eta$ by another derivative as suggested in the Epilogue of [1], while giving a heuristic justification for the new derivative.Comment: 17 page

Optimal Axes of Siberian Snakes for Polarized Proton Acceleration

Accelerating polarized proton beams and storing them for many turns can lead to a loss of polarization when accelerating through energies where a spin rotation frequency is in resonance with orbit oscillation frequencies. First-order resonance effects can be avoided by installing Siberian Snakes in the ring, devices which rotate the spin by 180 degrees around the snake axis while not changing the beam's orbit significantly. For large rings, several Siberian Snakes are required. Here a criterion will be derived that allows to find an optimal choice of the snake axes. Rings with super-period four are analyzed in detail, and the HERA proton ring is used as an example for approximate four-fold symmetry. The proposed arrangement of Siberian Snakes matches their effects so that all spin-orbit coupling integrals vanish at all energies and therefore there is no first-order spin-orbit coupling at all for this choice, which I call snakes matching. It will be shown that in general at least eight Siberian Snakes are needed and that there are exactly four possibilities to arrange their axes. When the betatron phase advance between snakes is chosen suitably, four Siberian Snakes can be sufficient. To show that favorable choice of snakes have been found, polarized protons are tracked for part of HERA-p's acceleration cycle which shows that polarization is preserved best for the here proposed arrangement of Siberian Snakes.Comment: 14 pages, 16 figure

Quasiperiodic spin-orbit motion and spin tunes in storage rings

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.Comment: 62 pages, 1 figure. A slight extension of the published versio