| Abstract |
This PhD thesis elaborates the particle dynamics for a large number of revolutions
(long time integration) in synchrotrons (circular particle accelerators) like the Large
Hadron Collider (LHC) and its upcoming High Luminosity upgrade (HL-LHC). A fast
introduction to the CERN accelerator complex is presented in Chapter 1. In Chapter 2,
using the powerful tools of the Hamiltonian formalism and the Poisson bracket Lie algebra,
the mathematical framework that will be used in the following Chapters is developed.
For any Hamiltonian problem, the use of symplectic transfer maps is mandatory for
any long term integration study. In Chapters 3 and 4 two differed symplectic integration
methods are presented. The first one is better suited for heavy simulations like the study
of the bunch dynamics while the other one for simulations that require very good accuracy
like the single particle dynamics.
The computational cost for long term bunch dynamics analysis (e.g. study of the
space charge or beam-beam effects) can be quite large especially for machines like the
LHC and HL-LHC (more that 1016 operations per revolution are required). In order to
speed up this type of simulations, differed tricks like the parallel-computing, the use of
macro-particles and reduced lattices can be applied. In Chapter 3 a method to produce
simplified versions of the original non-linear lattices is presented. This versatile symplectic
integration scheme (effective lattice) is a sequence of linear sections that are separated
by combined non-linear kicks and can reproduce all the linear properties of the original
lattice while retains an adequate accuracy for the non-linear ones with the least possible
elements. The developed transfer map is quite ease to be adjusted for the needs of differed
rings and it is made available as a refined replacement of the simplified rotation matrix
that is often used in multi-particle studies that requiring a fast beam transport routine.
For the evaluation if the different configurations of an accelerator are operational,
a set of long term single particle tracking studies is needed. In Chapter 4 a new family
of high order symplectic integrators C S AB Aν & C SB ABν is used for the calculation of
the Hamiltonian flows in different accelerator lattices. A benefit of theses symplectic
integrators (when they can be applied) is the presence of only forward integration steps
independent of the order of accuracy. In addition to that, the C S AB Aν & C SB ABν integrators are found to be more accurate for similar or smaller integration cost than other
symplectic integration schemes that are well established in the accelerator community.
The Chapter 5 of this thesis, deals with an operational problem of the HL-LHC which
is the mitigation of the beam-beam long-range (BBLR) interactions. These BBLR kicks
can significantly reduce the performances of the collider thus, the use of DC wires as a
compensatory devise is studied. Analytical expressions for the resonance driving terms
and the tune spread with amplitude, driven by the BBLR interactions and the wires
electromagnetic field, are derived. Using these results as a guideline for the numerical
simulations, it is demonstrated that with a proper optimization of the DC wires, without
violating the machine protection restrictions, the long range beam-beam interactions can
be very well mitigated. Also, new nominal and ultimate scenarios of the HL-LHC with
smaller crossing angle are found. These scenarios are operational (good lifetime) only
with the use of wire compensators and can be used as complementary to the original ones
for improving the integrated luminosity and the operational flexibility of the machine.
Finally, the last chapter (Chapter 6) is devoted to the conclusions of this thesis.
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Non-linear dynamics modelling in accelerators with the use of symplectic integrators
