The Cockcroft Institute
Post-graduate
Lecture Courses: Academic Year 2005-6
Core Course
Transverse Dynamics
(timetable)
Tutor : Dr B Holzer (DESY)
Syllabus
1. Revision (see An
Introduction to
Accelerator
Science ):
Equation of motion (homogeneous): basic
concepts; the linear lattice; magnetic multipoles;
Matrix formalism: how trajectories
overlap; trajectory and the definition of tune;
The Twiss parameters alpha, beta, gamma
and the phase space parameter epsilon: the mathematical definition of
these parameters; meaning of epsilon in Phase space; calculation of
sigma and sigma' from the phase space area; Liouville's theorem
2. Linear Optics and Lattice Design:
Matrix expressed as function of alpha,
beta, gamma;
Stability of a lattice or cell;
Calculation of lattice parameters: what is a FoDo; how does beta depend
on phi; how do we design a storage ring;
Example: a mini beta insertion
3. The "not so ideal " world:
Liouville during acceleration:
adiabatic shrinking;
Beams with momentum spread; the
inhomogeneous equation solution; the dispersion function;
General particle trajectory: x =
x_beta + x_d;
Single element dispersion;
Boundary conditions: dispersion in a
FoDo; periodic dispersion;
Momentum compaction factor
4. Errors:
Sources of magnet errors;
Tolerances; stringent requirements;
examples from different machines: dipole lengths, current stability of
magnets, multipole contributions;
Dipole error; closed orbit distortion;
comparison with periodic dispersion; sensitivity of the beam determined
by square root of beta
5. Quadrupole error:
Recapitulation of 1) and 2);tune and
working diagram; resonance denominator as a general problem (from 4);
Tune shift from quadrupole errors;
Beat from quadrupole errors;
Chromaticity; definition and meaning; examples: typical values;
Chromaticity correction using sextupoles;
Dynamic
aperture: how to treat sextupoles in tracking codes;
Simulation; sextupole in a FoDo ring;
effect of 6poles in phase space; resonant extraction close to 3rd
integer resonance
6. Resonances:
Floquet transformation;circular
diagram; resulting equation of a simple harmonic oscillator;
The effect of a dipole error in phase
space;
A random dipole error distribution in a
storage ring (numbers from a
real machine: SPS or HERA);
Inhomogeneous equation of motion; the
driving term of the orbit
distortion; the solution from the harmonic oscillator;
The second order stop band:tune shift;
stop band width
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