Invariant Spin Field Research Articles

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-[Formula: see text] and spin-[Formula: see text] 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.

Pre-existing betatron motion and spin flipping with RF fields in storage rings

It seems to be a common perception that pre-existing, non-spin-resonant, vertical, betatron motion in storage rings has little influence on spin-flipping with external, resonant, radio-frequency, magnetic fields. While this is often the case, care is needed, and useful insights are obtained by studying the geometry of the invariant spin field (ISF). I illustrate this by numerical means for protons in the COSY ring and I suggest a test that could be carried out at COSY.

Spin motion at and near orbital resonance in storage rings with Siberian Snakes I: at orbital resonance

In this paper, 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 polarization at and near so-called snake resonances in proton storage rings.

Adiabatic invariance of spin-orbit motion in accelerators

It has been predicted and found experimentally that the polarization direction of particles on the closed orbit of a circular accelerator can be manipulated, without a noticeable reduction of polarization, by means of a slow variation of magnetic fields. This feature has been used to avoid imperfection resonances where the spin precession frequency is close to a multiple of the circulation frequency. As a first step we show that this property is related to an adiabatic invariant of spin motion. The proof is relatively simple since it involves only two frequencies, the spin-rotation frequency and the particle's rotation frequency on the closed orbit. The invariant spin field (ISF) describes a periodic polarization state of a beam's phase-space distribution. This ISF leads to a very useful parametrization of coupled spin and orbit dynamics. We prove that this ISF gives rise to an adiabatic invariant of spin-orbit motion. This proof is much more complicated since the orbital frequencies are involved. Because of this adiabatic invariance, a beam's spin field follows slow changes of the accelerator's ISF that can occur during a slow acceleration cycle. This feature is essential when high-order spin-orbit resonances are crossed, since it allows polarization that has been reduced at the resonance condition to be recovered, to a large degree, after the resonances have been crossed.

A critical analysis of the conventional theory of spin resonances in storage rings

We present a critical assessment of the conventional theory of spin resonances in storage rings. We show that an application of the rigorous formula for the locations of spin-resonant orbits, using the nonperturbative off-axis solution for the spin tune, frequently fails to correctly identify the locations of depolarization. We use recently discovered analytical solutions for the invariant spin field and spin tune in several storage ring models, to obtain a general overview of the spin dynamics and also of the polarization. We calculate the long-term polarization and the depolarizing spin resonances for each model. In particular, we present a detailed examination of the so-called “Snake resonances”. We also compare our analytical solutions against the results of numerical simulations. We consider various measures of the degree of long-term polarization and point out fallacious tacit assumptions in the conventional literature. We also present some new mathematical results linking the so-called sine-Bessel functions to the complete elliptic integral of the first kind.

Miles: a new nonperturbative formalism to calculate the invariant spin field in circular accelerators

We describe a new nonperturbative algorithm called Miles to calculate the invariant spin field in circular accelerators. It is a map-based algorithm, based exclusively on the field-theoretic transformation of a vector field under the flow in the orbital phase-space. We employ the method to derive the exact analytical solution for a model of a planar ring with one Siberian Snake and a single resonance driving term. This model is in some sense the classic problem in the field. The solution contains new types of mathematical functions, which we call “sine-factorial” and “sine-Bessel” functions. We also display the exact analytical solutions for several other storage ring models, including rings with two or more Snakes. We compare our method against other formalisms, including numerical tracking programs.

Analytical solutions for the invariant spin field for model storage rings

We present nonperturbative analytical expressions for the invariant spin field for several storage ring models. In particular, we solve the important models of a ring with one Snake and a single resonance driving term, and a ring with two Snakes and a single resonance driving term. We also treat several other models, all of which contain Siberian Snakes. Our solutions contain some novel features, e.g. in some cases the polarization does not point along the direction of the closed-orbit spin quantization axis. We also include vertical resonance driving terms, and consider the contributions of sextupoles and higher order multipoles to the resonance driving terms, and argue that these can play a significant role in some circumstances. We offer some brief remarks on the so-called Snake resonances. We relate our results to observations of higher-order depolarizing spin resonances for polarized proton beams in a real ring, and offer some suggestions as to how our ideas might be verified.

Polarized protons in HERA

Polarized proton beams at HERA can currently only be produced by extracting a beam from a polarized source and then accelerating it in the three synchrotrons at DESY. In this paper, the processes which can depolarize a proton beam in circular accelerators are explained, devices which could avoid this depolarization in the DESY accelerator chain are described, and specific problems which become important at the high energies of HERA are mentioned. At HERA's high energies, spin motion cannot be accurately described with the isolated resonance model which has been successfully used for lower energy rings. To illustrate the principles of more accurate simulations, the invariant spin field is introduced to describe the equilibrium polarization state of a beam and the changes during acceleration. It will be shown how linearized spin motion leads to a computationally quick approximation for the invariant spin field and how to amend this with more time consuming but accurate non-perturbative computations. Analysis with these techniques has allowed us to establish optimal Siberian Snake schemes for HERA.