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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 [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 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 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
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