Lattice Design in High-energy Particle Accelerators

This lecture gives an introduction into the design of high-energy storage ring lattices. Applying the formalism that has been established in transverse beam optics, the basic principles of the development of a magnet lattice are explained and the characteristics of the resulting magnet structure are discussed. The periodic assembly of a storage ring cell with its boundary conditions concerning stability and scaling of the beam optics parameters is addressed as well as special lattice insertions such asdrifts, mini beta sections, dispersion suppressors, etc. In addition to the exact calculations that are indispensable for a rigorous treatment of the matter, scaling rules are shown and simple rules of thumb are included that enable the lattice designer to do the first estimates and get the basic numbers 'on the back of an envelope'.Comment: 40 pages, contribution to the CAS - CERN Accelerator School: Advanced Accelerator Physics Course, Trondheim, Norway, 18-29 Aug 2013. arXiv admin note: substantial text overlap with arXiv:1303.651

Introduction to Transverse Beam Dynamics

In this chapter we give an introduction to the transverse dynamics of the particles in a synchrotron or storage ring. The emphasis is more on qualitative understanding rather than on mathematical correctness, and a number of simulations are used to demonstrate the physical behaviour of the particles. Starting from the basic principles of how to design the geometry of the ring, we review the transverse motion of the particles, motivate the equation of motion, and show the solutions for typical storage ring elements. Following the usual treatment in the literature, we present a second way to describe the particle beam, using the concept of the emittance of the particle ensemble and the beta function, which reflects the overall focusing properties of the ring. The adiabatic shrinking due to Liouville's theorem is discussed as well as dispersive effects in the most simple case.Comment: 19 pages, contribution to the CAS-CERN Accelerator School: Ion Sources, Senec, Slovakia, 29 May - 8 June 2012, edited by R. Bailey, CERN-2013-00

Higher-order Modes and Heating

This chapter gives a basic introduction to the problem of wake fields created in the beam-surrounding environment and the resulting heating effects of machine components. The concepts are introduced and scaling rules derived that are exemplified by several observations from operation of the LHC and other machines.Comment: 14 pages, contribution to the CAS-CERN Accelerator School: Superconductivity for Accelerators, Erice, Italy, 24 April - 4 May 2013, edited by R. Baile

Introduction to Longitudinal Beam Dynamics

This chapter gives an overview of the longitudinal dynamics of the particles in an accelerator and, closely related to that, the issue of synchronization between the particles and the accelerating field. Beginning with the trivial case of electrostatic accelerators, the synchronization condition is explained for a number of driven accelerators like Alvarez linacs, cyclotrons and finally synchrotrons and storage rings, where it plays a crucial role. In the case of the latter, the principle of phase focusing is motivated qualitatively as well as on a mathematically more correct level and the problem of operation below and above the transition energy is discussed. Throughout, the main emphasis is more on physical understanding rather than on a mathematically rigorous treatment.Comment: 15 pages, contribution to the CAS-CERN Accelerator School: Ion Sources, Senec, Slovakia, 29 May - 8 June 2012, edited by R. Bailey, CERN-2013-00

First Evaluation of Dynamic Aperture at Injection for FCC-hh

In the Hadron machine option, proposed in the context of the Future Circular Colliders (FCC) study, the dipole field quality is expected to play an important role, as in the LHC. A preliminary evaluation of the field quality of dipoles, based on the Nb$_{3}$Sn technology, has been provided by the magnet group. The effect of these field imperfections on the dynamic aperture, using the present lattice design, is presented and first tolerances on the b$_3$ and b$_5$ multipole components are evaluated

Simulation of the Pion Decay Channel of a Neutrino Factory

In the pion decay channel of a neutrino factory the particles are transported in a solenoidal magnetic field, a 1.8 T constant field in the CERN reference scenario. Increasing the field strength decreases the transverse emittance of the decay muons. To define a lower limit on the achievable u emittance, the reference scenario is compared with a study case where pions decay in a field of 20 T. This simulation shows that in any realistic scenario the u emittance will only reduced by less than 21 in each transverse plane as compared to the reference scenario, and at the expense of an increase in longitudinal emittance

State Complexity of Reversals of Deterministic Finite Automata with Output

We investigate the worst-case state complexity of reversals of deterministic finite automata with output (DFAOs). In these automata, each state is assigned some output value, rather than simply being labelled final or non-final. This directly generalizes the well-studied problem of determining the worst-case state complexity of reversals of ordinary deterministic finite automata. If a DFAO has $n$ states and $k$ possible output values, there is a known upper bound of $k^n$ for the state complexity of reversal. We show this bound can be reached with a ternary input alphabet. We conjecture it cannot be reached with a binary input alphabet except when $k = 2$, and give a lower bound for the case $3 \le k < n$. We prove that the state complexity of reversal depends solely on the transition monoid of the DFAO and the mapping that assigns output values to states.Comment: 18 pages, 3 tables. Added missing affiliation/funding informatio