1,546 research outputs found

### Multiparameter Fuss--Catalan numbers with application to algebraic equations

We present an exposition on the Fuss--Catalan numbers, which are a
generalization of the well known Catalan numbers. The literature on the subject
is scattered (especially for the case of multiple independent parameters, as
will be explained in the text), with overlapping definitions by different
authors and duplication of proofs. This paper collects the main theorems and
identities, with a consistent notation. Contact is made with the works of
numerous authors, including the early works of Lambert and Euler. We
demonstrate the application of the formalism to solve algebraic equations by
infinite series. Our main result in this context is a new necessary and
sufficient formula for the domain of absolute convergence of the series
solutions of algebraic equations, which corrects and extends previous work in
the field. Some historical material is placed in an Appendix.Comment: 50 pages, 4 figure

### Spin decoherence rate in a weak focusing all-electric ring

I previously derived the expression for the spin decoherence rate (for
orbital and spin motion in the horizontal plane) in a weak focusing
all-electric storage ring with a radial field (logarithmic potential, field
index $n=0$). Here I generalize the calculation to an arbitrary field index
$n\ge0$. I also solve the model for the relativistic Kepler problem (field
index $n=1$), where the solution for the orbit is known analytically, and I
verify that it confirms the solution from perturbation theory.Comment: 57 pages, 4 figures. v2: I solved the model also for off-energy
dispersion orbits. The dispersion orbits are circles and I calculated the
spin decoherence rate exactly. v3: I solved the model also for vertical
motion, for $n=0$ (logarithmic potential). I have also solved for the spin
precession the Kepler problem (inverse square law) exactly; the paper is in
press. Extra appendices have been added; in particular, Ivan Koop kindly gave
permission to reproduce his elegant analysis of motion in a vertical spiral
in a logarithmic potential. v4: Added synchrotron oscillations. arXiv admin
note: substantial text overlap with arXiv:1403.502

### Polarization at TLEP/FCC-ee: ideas and estimates

I examine various ideas for radiative polarization at TLEP/FCC-ee and
formulate some estimates for the polarization buildup time and the asymptotic
polarization. Using wigglers, a useful degree of polarization (for energy
calibration), with a time constant of about 1 h, may be possible up to the
threshold of $W$ pair production. At higher energies such as the threshold of
Higgs production, attaining a useful level of polarization may be difficult in
a planar ring. With Siberian Snakes, wigglers and some imagination,
polarization of reasonable magnitude, with a reasonable time constant (of not
more than about 1 h), may be achievable at very high energies.Comment: 38 pages, 3 figure

### Decay Lifetimes of the Vector and Tensor Polarization of a Stored Deuteron Beam

An rf solenoid was operated in the IUCF Cooler to induce a depolarizing spin
resonance for a stored polarized deuteron beam. The decay lifetimes of the
vector and tensor polarizations were found to be in the ratio $1.9:1$, as
opposed to the expected ratio of $3:1$ from standard angular momentum theory.
This report describes my investigations and attempts to formulate a theoretical
explanation of the above phenomenon.Comment: 8 pages, no figure

### Some notes on beam dynamics due to vertical oscillations in an all-electric storage ring

A document has recently been posted on the arXiv [1], describing analytical
formulas and results of particle tracking simulations, for precision tests of
numerical integration algorithms for an EDM (electric dipole moment) storage
ring. In the context of an all-electric storage ring, the authors cite
theoretical formulas by Orlov [2]. However, the reference to Orlov is to a talk
at a workshop in 2012, and is unpublished and difficult for independent
researchers to access and validate. This note rederives and generalizes some of
Orlov's principal results, using a Hamiltonian formalism, and also corrects
some details in both Orlov's note [2] and the arXiv post [1].Comment: 8 pages, no figures, fix typos in version

### Fits to data for a stored uncooled polarized deuteron beam

I perform tracking simulations to fit various measurements of the
polarization for a stored uncooled polarized deuteron beam, published in the
recent paper by Benati et. al (P. Benati {\em et al.}, {\it Phys. Rev. ST
Accel. Beams} {\bf 15}, 124202 (2012)). The collaboration kindly sent me
datafiles of the polarization measurements, and also pertinent details of the
experimental data acquisition procedure. The latter are essential to obtain
quantitative fits to the data. I describe my findings and inferences from the
data. In some cases I offer alternative interpretations of the data from that
given by Benati et. al. I also correct some mistakes in my recent paper (S. R.
Mane, {\em Nucl. Inst. Meth.} {\bf 726} 104--112 (2013)).Comment: 24 figure

### Equation for spin decoherence rate in an all-electric ring

There is a quantitative error in the derivation for the spin decoherence rate
by Talman in IPAC2012. The crucial point is a subtle confusion between the
concept of `longitudinal' as in `along the reference orbit' and `parallel to
the particle velocity.' They are $\it not$ the same direction, and the
distinction is significant for high-precision experiments to search for a
possible nonzero electric dipole moment (EDM) of a charged particle in a
storage ring.Comment: 6 pages. no figure

### Orbit and spin tracking in a radial electric field

I derive an expression for the spin decoherence rate for a smooth focusing
electrostatic ring with a radial electric field. The answer contains some
salutary lessons about spin decoherence calculations in electrostatic storage
rings: the numerical tracking results disagreed with naive analytical
calculations, but it was the tracking results which were correct, and more
careful analytical calculations (canonical transformations) were required.Comment: 13 pages, 1 figur

### Notes on orbit and spin tracking in an electrostatic storage ring

Two documents have recently been posted on the arXiv describing a numerical
integration algorithm: "symplectic orbit/spin tracking code for all-electric
storage rings" [1] and some computational results therefrom [2]. This note
comments critically on some of the claims in [1] and [2]. In particular, it is
not clear that the orbit tracking algorithm described in [1] is really
symplectic. Specifically, for electrostatic beamline elements, the so-called
"zero length elements," which are treated as position dependent kicks in the
formalism in [1], are in fact {\em not} symplectic.Comment: 8 pages, no figure

### Comment on "rf Wien filter in an electric dipole moment storage ring: The "partially frozen spin'' effect''

The Jacobi-Anger identity is employed to quantify the findings of Morse,
Orlov and Semertzidis [1]. The secular term in the longitudinal spin component
is proportional to a Bessel function. A suggestion for values of machine
parameters to optimize the secular term is presented.Comment: 5 pages, no figure

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