Accelerator Physics at the LHC LHC Dynamic Aperture Page

Introduction

In the LHC nonlinearities of the magnetic fields limit the largest stable amplitude (dynamic aperture) of single particle motion. This is particularly true at injection energy because the multipolar errors due to persistent currents are at their maximum, the beam size is large and the particles have to survive for a long time needed to fill the LHC with 2835 bunches per beam.

Despite a decade long effort there is still no conclusive theory which could faithfully predict the long-term dynamic aperture. One is therefore obliged to do brute-force particle tracking of the full model of the LHC including all errors. Even though CERN has recently acquired a 10 processor system (NAP) with fastest floating point performance presently available only 1% of the LHC injection period can be covered by systematic numerical studies. Using early indicators, survival plot analysis and the experience from existing machines one can nevertheless give an estimate for the dynamic aperture at the end of the injection period.

The Errors

Soon you can get the error tables here: error_table

Survival Plots

The dynamic aperture is determined using the so-called survival plots (Nonlinear Dynamics, Tracking and Chaos Detection ). In these plots the survived number of turns are plotted against the amplitude measured in beam sigmas. Fig. 1 shows an example for one seed of the LHC version 4 (Ref.2 of Reports on LHC Tracking).

Fig. 1: Survival Plot LHC Version 4	Fig. 2: Dyn. Aper. Histogram for LHC Version 4

Fig.2 shows a histogram of the dynamic aperture determined for 60 representations (seeds) of the magnetic errors of LHC version 4 (Ref.3 of Reports on LHC Tracking).

Simple Scaling Law

Given the enormous effort to perform a complete tracking studies there is the interest to shortcut this effort by some kind of scaling. V. Ziemann (Ref. 2 Prediction of the Dynamic Aperture) has proposed such a scaling law. It consists of three steps:

Find the partial dynamic aperture for each multipole switched on separately.

Multiplying the strength of a multipole of order n by a factor 1/ in order to scale the corresponding partial dynamic aperture with ^{1/(n-2)}.

Finding the sum of the inverse fourth powers of the partial dynamic aperture values. The total dynamic aperture can then be calculated via ^(-0.25). This simple conjecture has been found to work surprisingly well for LHC version 4.

Ongoing Studies

A systematic study of the effect each multipole component has on the dynamic aperture. This shall be used to optimise the target error table and test the scaling law.

Investigation of the strong dependence of the dynamic aperture on the emittance ratio for LHC version 5, in particular with the use of the analysis of high order resonances.

Study of correction circuit failure modes, variants of the spool piece correction system and third order resonance compensation schemes.

Using frequency analysis to study effective resonances in tracking and experimental data.

Speed up of the beam-beam kicks using tables and approximations in view of fast tracking in collision. Special thanks to George Erskin.

Symplectified maps Dan Abell for fast tracking in particular in collision when the octants can be replaced by low order maps.

REFERENCES

Reports on LHC Tracking

Prediction of the Dynamic Aperture

Correlations with Dynamic Aperture

Computer and Tracking Tools

Sorting

Exact Tune Measurement

Spectrum Line Analysis

Dynamic Aperture Experiment

Nonlinear Dynamics, Tracking and Chaos Detection

Perturbation Theory and Resonances

Normal Form: Theory, Codes and Applications

Workshops on Nonlinear Dynamics Issues

This page has been dialed up

November 3, 1997 Frank.Schmidt@cern.ch

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