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Contents

• List of Tables
• List of Figures
• 1 A quick introduction to beam physics
  • 1. Quick memo on electromagnetism and special relativity
    • 1.1 Basic notions of Electromagnetism and charged particles
      • 1.1.1 Electromagnetic fields: Maxwell equations
      • 1.1.2 Lorentz Force
      • 1.1.3 Equation of motion
    • 1.2 Basic notions of relativity
      • 1.2.1 Relativity parameters and useful formulas
      • 1.2.2 Lorentz transformation
  • 2. Beam dynamics - linear formalism
    • 2.1 Introduction
      • 2.1.1 Coordinate system
      • 2.1.2 Equation of motion
    • 2.2 A particular case: Hill's equation - Betatron functions
    • 2.3 General resolution of the perturbated linear equation of motion
  • 3. Phase space and matrix formalism
    • 3.1 Transportation matrix
      • 3.1.1 From equations of motion to matrix formalism
      • 3.1.2 Interpretation in term of optics
    • 3.2 Beam matrix and phase space
      • 3.2.1 Phase space
      • 3.2.2 Beam emittance
      • 3.2.3 Beam matrix
      • 3.2.4 Two dimensional beam matrix - Relation with Twiss-parameters
      • 3.2.5 Relation with transport matrix
      • 3.2.6 Four dimensional beam matrix - introduction to Emittance exchange
  
  
• 2 Implementation of the space charge algorithm
  • 4. On the implementation of a 3D algorithm in Astra
    • 4.1 Wider prospects for the algorithm
      • 4.1.1 The need to adapt the meshing
      • 4.1.2 No momentum direction favorised
    • 4.2 General description of the method
    • 4.3 Detailled description of the method
      • 4.3.1 Frame definitions
      • 4.3.2 Boosting the distribution to the rest frame
      • 4.3.3 Rotation of the bunch to its principal axis
      • 4.3.4 Diagonalization algorithm
      • 4.3.5 Space charge calculation
      • 4.3.6 The 3d space charge algorithm
      • 4.3.7 Solving the equations of motion
  • 5. Space Charge calculation
    • 5.1 Poisson equation
    • 5.2 Resolution of poisson screened equation with the use of Fourier transform
    • 5.3 Resolution of poisson screened equation with the use of Green function
  
  
• 3 On the benchmarking of the space charge algorithm
  • 6. Benchmarking of the stand alone space charge algorithm without rotation
    • 6.1 Fields for a cylindrical distribution
      • 6.1.1 Theoretical expression of the potential
      • 6.1.2 Theoretical expression of the Fields for the cylinder
      • 6.1.3 Graphical comparisons
    • 6.2 Fields for an ellipsoidal distribution
      • 6.2.1 Theoretical expression of the potential for a spheroid
      • 6.2.2 Theoretical expression of the Fields for a spheroid and an ellipsoid
      • 6.2.3 Graphical comparisons of the fields
  • 7. Rms field values - Simulation vs. theory
    • 7.1 On the RMS definition
    • 7.2 Spheroidal distribution
    • 7.3 Ellipsoidal distribution
    • 7.4 Cylindrical distribution
    • 7.5 Results summary
      • 7.5.1 Fields calculation
      • 7.5.2 RMS calculation
  • 8. Benchmarking of the stand alone space charge algorithm with rotation
    • 8.1 Testing method
      • 8.1.1 Principle
      • 8.1.2 Method used
      • 8.1.3 Technical details
    • 8.2 Overview of different comparisons performed
      • 8.2.1 Exemple of a coin-shape cylinder
      • 8.2.2 Exemple of an ellipsoid at high energy
  • 9. Including our algorithm in Astra - Modifications done and problem encountered
    • 9.1 Astra conventions
    • 9.2 Organizations of the source files
    • 9.3 Learning from my mistakes
    • 9.4 Note on the sigma matrix outputed by Astra
      • 9.4.1 Analysing the previous algorithm without the source code
      • 9.4.2 Modifying the source code to output a standard sigma matrix
  
  
• 4 Study of different components found in particle accelerator
  • 10. Drift
    • 10.1 Theory without space charge
      • 10.1.1 Simple case
      • 10.1.2 Introducing transverse momentum correlation in the distribution
      • 10.1.3 Beam waist interpretation
    • 10.2 Influence of the space charge in a drift
  • 11. Dipoles
    • 11.1 Theory
      • 11.1.1 Equation of motion
      • 11.1.2 Successions of dipole - reducing the energy spread
      • 11.1.3 Fringe fields
      • 11.1.4 Effective length
    • 11.2 Dipole fields from measurement
      • 11.2.1 Coordinates conventions
      • 11.2.2 Organization of measurement data
      • 11.2.3 Field map from measurement
    • 11.3 Comparisons between dipole measurements and Astra modelization
      • 11.3.1 Modelization of dipoles in Astra
      • 11.3.2 Comparison of the different fields
      • 11.3.3 Fit with Henge's field fall-off model
  • 12. Compressors
    • 12.1 Introduction to compressors theory
    • 12.2 Adapting simulation parameters to fit reference path
    • 12.3 Maximum compression
      • 12.3.1 Introducing correlation directly in the bunch
    • 12.4 Phase space plots
      • 12.4.1 Transversal phase space, with and without space charge
      • 12.4.2 Longitudinal phase space, with and without space charge
      • 12.4.3 Projected densities for the phase space with space charge forces at the end of the compressor
      • 12.4.4 Introducing correlation with a cavity
  • 13. Emittance exchanger
    • 13.1 Adapting simulation parameters to fit reference path
    • 13.2 Emittance measurements method
  • 14. Study of a cavity
    • 14.1 Fields inside a cavity
      • 14.1.1 Longitudinal component
      • 14.1.2 Correlation introduced by a cavity between $z$ and the energy
      • 14.1.3 Determination of the cavity matrix
    • 14.2 An example of transportation matrix: five cell cavity
      • 14.2.1 Equation of motion in a pillbox
      • 14.2.2 The One cell pill box model
      • 14.2.3 Cavity Transit matrix
  
  
• 5 Preliminary steps for the simulation of full accelerators line
  • 15. A0 photo injector
    • 15.1 Specifications
    • 15.2 Geometry
      • 15.2.1 Coordinates of the principal elements
      • 15.2.2 Dogleg lattice coordinates
    • 15.3 Simulation
  • 16. ILCTA at New Muon lab (NML)
    • 16.1 About ILCTA
    • 16.2 Specifications
    • 16.3 Coordinates of the principal elements
    • 16.4 Compressor lattice coordinates
    • 16.5 Simulation
  • A. Documentation of R scripts
  • B. Jacobi diagonalization of a symmetric matrix
    • B.1 Mathematical description of the method
    • B.2 Fortran source code
  • C. The use of Green function for solving differential equations
  • D. A0 line full lattice geometry and coordinates
  • E. More exemples of the rotation algorithm benchmarking
    • E.1 Exemple of an ellipsoid at rest
    • E.2 Exemple of a sphere at low energy
  • F. Useful mathematical formulas
  • G. Longitudinal space charge - Theory vs. Simulation
    • G.1 Cylindrical distribution
    • G.2 Ellipsoidal distribution
  • H. Astra input files for the two accelerator lines
    • H.1 A0 photo injector
    • H.2 ILCTA at NML
  • I. Versions history of the 3d space charge algorithm
  
  
• Bibliography

Emmanuel Branlard
http://emmanuel.branlard.free.fr/