6 Design and Principles of Synchrotrons and Circular Colliders
209
Fig. 6.3 Basic element of a high-energy storage ring: the FODO cell
so-called FODO cell: A magnet structure consisting of focusing and defocusing
quadrupole lenses in alternating order. In between the focusing elements the dipole
magnets are located and any other machine elements like orbit corrector dipoles,
multipole correction coils or diagnostic instruments can be installed.
In Fig. 6.3 the optical solution of such a FODO cell is plotted: The graph shows
the β-function in the two transverse planes (red curve for the horizontal, green curve
for the vertical plane). In the lower part of the plot the position of the magnet
lenses, the lattice, is indicated schematically. In first order the optical properties
of such a lattice are determined only by the parameters of the focusing (F) and
de-focusing (D) quadrupole lenses. In between these two quadrupole magnets only
lattice elements are installed that have zero (“O”) or negligible influence on the
transverse particle dynamics. Hence the acronym FODO for such a structure. Due
to the symmetry of the cell the solution for the β function is periodic (in general
such a FODO cell is the smallest periodic structure in a storage ring) and it reaches
its extreme values at the position of the quadrupole lenses. As a consequence, at
these locations in the arc, the beam will reach its maximum dimension σ =
√
εβ,
and the aperture need will be highest.
Accordingly, the “Twiss” parameter α, which is the derivative of β is generally
zero in the middle of the FODO quadrupoles. Based on the thin lens approximation
a number of scaling laws and rules can be established to understand the properties
of such a FODO structure [2]: How do we arrange the strength and position of the
quadrupole lenses in the lattice to obtain a certain beta-function? How does the cell
length influence the phase advance of the particle trajectories? How do we guarantee
that, turn by turn, a stable particle oscillation is obtained?
209
Fig. 6.3 Basic element of a high-energy storage ring: the FODO cell
so-called FODO cell: A magnet structure consisting of focusing and defocusing
quadrupole lenses in alternating order. In between the focusing elements the dipole
magnets are located and any other machine elements like orbit corrector dipoles,
multipole correction coils or diagnostic instruments can be installed.
In Fig. 6.3 the optical solution of such a FODO cell is plotted: The graph shows
the β-function in the two transverse planes (red curve for the horizontal, green curve
for the vertical plane). In the lower part of the plot the position of the magnet
lenses, the lattice, is indicated schematically. In first order the optical properties
of such a lattice are determined only by the parameters of the focusing (F) and
de-focusing (D) quadrupole lenses. In between these two quadrupole magnets only
lattice elements are installed that have zero (“O”) or negligible influence on the
transverse particle dynamics. Hence the acronym FODO for such a structure. Due
to the symmetry of the cell the solution for the β function is periodic (in general
such a FODO cell is the smallest periodic structure in a storage ring) and it reaches
its extreme values at the position of the quadrupole lenses. As a consequence, at
these locations in the arc, the beam will reach its maximum dimension σ =
√
εβ,
and the aperture need will be highest.
Accordingly, the “Twiss” parameter α, which is the derivative of β is generally
zero in the middle of the FODO quadrupoles. Based on the thin lens approximation
a number of scaling laws and rules can be established to understand the properties
of such a FODO structure [2]: How do we arrange the strength and position of the
quadrupole lenses in the lattice to obtain a certain beta-function? How does the cell
length influence the phase advance of the particle trajectories? How do we guarantee
that, turn by turn, a stable particle oscillation is obtained?