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 spaceimplementation 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 14·10⁶tracked particles, about 12·10⁶are 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 the tune 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^-5for 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.