Particle tracking in circular accelerators using the exact Hamiltonian in SixTrack

Abstract:

Particle motion in accelerators is in general complex. Tracking codes are developed to
simulate beam dynamics in accelerators. SixTrack is a long lived particle tracking
code maintained at CERN, the European Organization for Nuclear Research.

A particle accelerator consists of a large number of magnets and other electromagnetic
devices that guide the particle through the accelerator. Each device defines its own
equation of motion, which often cannot be solved exactly. For this purpose, a
number of approximations are introduced in order to facilitate the solution and to
speed up the computation.

In a high-energy accelerator, the particle has small transverse momentum components.
This is exploited in the small-angle approximation. In this approximation the
equations of motion are expanded to a low order in the transverse momentum
components. In low-energy particle accelerators, or in tracking with large momentum deviations, this
approximation is invalid.

The equations of motion of a particle passing through a field-free region in an accelerator,
a so called drift space, has been implemented in the SixTrack code. The equations of
motion are derived from the exact Hamiltonian, keeping the non-linear term unexpanded.
This solution of the drift is called the exact drift space. Previously, the drift space
has been solved using the small-angle approximation. This solution of the drift is called
the expanded drift space.
The new implementation is a step towards a more realistic, and more general, tracking code.
The drift space contains the bulk of the small-angle approximation in a tracking code,
it is therefore the most important element to address.

The new drift space implementation is applied in two simulation studies on the
Large Hadron Collider (LHC). In the first, particle losses in the
collimation system of the machine are studied. The collimation system is a collection
of protective devices, used to protect the rest of the accelerator from particles spiraling out of
the machine. The application of the exact drift space in this simulation shows a small,
but insignificant, variation compared to the expanded drift.
Of the total 14·10⁶ tracked particles,
about 12·10⁶ are absorbed in the collimators for each model. The total number
of particles lost in other locations of the ring are about 12·10³ for both models.
The most dangerous losses are losses in the superconducting magnets, called cold losses.
For the exact drift, the number of cold losses were 4471. This is a short increase from
the expanded drift, where the number of cold losses were 4446. These results do not
show that the exact drift space is necessary in collimation studies for the LHC.
It should still be an improvement to consider for future machine protection studies.

The second simulation study on the LHC is an investigation of thetune variation as a
function of the momentum deviation of the particle. The tune is a measure of the number
of oscillations a particle makes during one complete turn around the accelerator. The number
of oscillations must avoid certain values to not induce a resonance in the motion,
causing the motion to be unstable. The momentum deviation, δ, is a measure of
the momentum of a particle compared to an ideal reference particle.
The horizontal- and vertical tunes were calculated for a range of values for δ,
both with the exact- and expanded drift space. As expected the deviation between
the models grows with a larger momentum deviation. The maximum differences in the simulation
were obtained for δ=-4·10^-3, where the exact model results in a tune value
larger by 3·10^-5 for the horizontal tune and 1.5·10^-5 for the vertical tune.
These tune shifts are small, and for regular tracking simulations in the LHC they are
insignificant. However, in simulations where very high-order resonance effects are considered,
these tune shifts could start to become important.