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Transport formalism. Linear matrix elements. Second order matrix elements. Truncated maps. Violation of the symplectic condition !. Lie algebraic treatment. Dragt-Finn factorization :. generators. [A. Dragt et al., Ann. Rev. Nucl. Part. Sci. 38 (1988) 455]. Linear matrix.

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transport formalism
Transport formalism

Linear matrix elements

Second order matrix elements

Truncated maps

Violation of the symplectic condition !

lie algebraic treatment
Lie algebraic treatment

Dragt-Finn factorization :


[A. Dragt et al., Ann. Rev. Nucl. Part. Sci. 38 (1988) 455]

Linear matrix

produces Tijkand higher order terms (sextupole effects)

produces third order and higher order terms (octupoles effects)

Numerical methods for nonlinear optimization : PARTICLE TRACKING,

Dynamic aperture scans, particle spectra…

tracking codes simulations to show the feasibility
Tracking codes:Simulations to show the feasibility



Importance of the benchmarking of codes


Multiparticle tracking

Optics lattice









[T. Asaka and J. Resta Lopez, CLIC-Note-637]

nanometer size beams in clic
Nanometer-Size Beams in CLIC

Nominal: σx=40.12 nm; σy=0.55 nm

Simulations: σx≈47.3 nm; σy≈0.65 nm

Beam profile at the IP:

Some problems: Residual horizontal dispersion at the IP

nanometer size beams in clic1
Nanometer-Size Beams in CLIC

Phase space at the IP:

Particles with lower energy than the nominal one (1500 GeV) contribute strongly

to the tails of the transversal phase space

chromatic effects in phase space
Chromatic effects in phase space

Chromatic aberrations study by means of tracking from matched initial ellipses at 1σ for the transversal plane X

Red line: center ellipse movement in phase space

up to third order !

chromatic effects in phase space1
Chromatic effects in phase space

Chromatic aberrations study by means of tracking from matched initial ellipses at 1σ (figure on the left)and 3σ (figure on the right) for the transversal plane Y

The particles at high position amplitude of several sigmas contribute to the

population of the long tails. For the case of the ellipses at 3σ in the vertical

phase space, it is possible to observe a strong deformation of the shape caused by the sextupoles located in the FFS.

limits of the luminosity
Limits of the Luminosity


Without SR

With SR

  • Tolerable bandwidth up to 1 % energy spread
  • The synchrotron radiation is a very important limitation factor for the
  • luminosity
beam beam effects
Beam-beam effects

Luminosity versus vertical offset

Analytic calculation considering a rigid gaussian beam:

Simulations with Guinea-Pig: it includes beam-beam effects

Disruption parameters:

Dy= 3.5 (CLIC)

Dy=19.4 (ILC)

input linac bds beam beam output

ILC integrated simulations

Input LINAC BDS Beam-Beam Output

Updated simulations:





G. White version (2005):

Input LINAC BDS Beam-Beam Output






ground motion and fb system
Ground motion and FB system

Nominal: L=2x1034 cm-2s-1

85 % of the nominal