Determination of the absolute luminosity at the LHC

166 Pages·2017·8.64 MB·English

by  S. White

Determination of the absolute luminosity at the LHC
S. White
To cite this version:
S. White. Determination of the absolute luminosity at the LHC. High Energy Physics - Experiment
[hep-ex]. Université Paris Sud - Paris XI, 2010. English. ￿NNT: ￿. ￿tel-00537325￿
HAL Id: tel-00537325
https://theses.hal.science/tel-00537325
Submitted on 18 Nov 2010
HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est
archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents
entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non,
lished or not. The documents may come from émanant des établissements d’enseignement et de
teaching and research institutions in France or recherche français ou étrangers, des laboratoires
abroad, or from public or private research centers. publics ou privés.
LAL 10-154
Septembre 2010

THÈSE

présentée le 11 octobre 2010

par

Simon WHITE

pour obtenir le grade de

Docteur ès Sciences
de l’Université Paris-Sud 11

Determination of the absolute luminosity
at the LHC

Soutenue devant la commission d’examen composée de :

M. H. Burkhardt
M. W. Fischer
Mme. V. Halyo Rapporteur
M. O. Napoly Rapporteur
M. P. Puzo Directeur de thèse
M. G. Wormser Président

Contents
Introduction 7
1 BeamDynamics 9
1.1 BasicsofAcceleratorPhysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.1 CoordinateSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.2 MagneticField . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.3 Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.4 Quadrupoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.5 Acceleration Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 BetatronMotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.1 TransferMatrixandStability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.2 Courant-SnyderParametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 TransverseEmittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.1 Courant-SnyderInvariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.2 BeamEmittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Beam-beamInteractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.1 TheBeam-beamForce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.2 Beam-beamParameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.3 Long-rangeInteractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5 Luminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.1 Head-onCollisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.2 OffsetCollisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5.3 CrossingAngle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5.4 CrossingAngleandOffsetBeams . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.5.5 Hourglass Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.5.6 LinearCoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.5.7 Integrated Luminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.5.8 MethodsforLuminosityCalibration . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 AbsoluteLuminosityFromMachineParameters 33
2.1 TheVanDerMeerMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1.1 ConceptofLuminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1.2 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.1.3 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.4 GaussianBeams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.1.5 DoubleGaussianBeams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.1.6 CrossingAngle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3
4 CONTENTS
2.1.7 Hourglass Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.1.8 Hourglass andCrossingAngle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.1.9 LinearCoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.1.10 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.2 DiscussionoftheUncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2.1 StatisticalAccuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2.2 BeamDisplacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2.3 BeamCurrentTransformers(BCT) . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2.4 Beam-beamEffects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2.5 Pile-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 FromInjectiontoCollisionatHighEnergy 51
3.1 TheLHCInjectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 TheLargeHadronCollider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3 LHCCommissioningandOperation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4 TheLHCCrossingScheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.2 SeparationBumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.3 CrossingAngle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.4.4 HysteresisEffects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.5 BringingtheBeamsIntoCollision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.5.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.5.2 HowFastCanWeGoIntoCollision? . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.5.3 OptimizingtheCollapsingTimeviaOpticsRematching . . . . . . . . . . . . . . . 61
3.5.4 Beam-BeamEffectsWhileBringingtheBeamsintoCollisions . . . . . . . . . . . . 62
3.5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.6 IROpticsOptimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.6.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.6.2 β MeasurementsforInjectionOptics . . . . . . . . . . . . . . . . . . . . . . . . . 67
∗
3.6.3 β KnobMeasurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
∗
3.6.4 OutlookforSqueezedOpticsandConclusions . . . . . . . . . . . . . . . . . . . . 70
4 LHCInstrumentation 73
4.1 BeamPositionMonitors(BPM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1.1 InsertionRegionBPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 TransverseEmittanceMeasurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2.1 WireScanners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2.2 Synchrotron LightMonitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3 IntensityMeasurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.4 LuminosityMeasurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.4.1 TheIonization Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4.2 TheCdTeDetectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.4.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.4.4 SimulationResultsfortheCdTeDetector(IR2andIR8) . . . . . . . . . . . . . . . 80
4.4.5 SimulationResultsfortheIonization Chamber(IR1andIR5) . . . . . . . . . . . . 81
4.4.6 SimulationandMeasurementsat350GeV . . . . . . . . . . . . . . . . . . . . . . . 82
4.4.7 FirstResultswithBeam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
CONTENTS 5
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5 ExperimentalResultsfromthe2009RHICProtonRun 87
5.1 TheRelativisticHeavyIonsCollider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 BeamParametersforthe2009PolarizedProtonRun . . . . . . . . . . . . . . . . . . . . . 87
5.3 OverviewoftheMeasurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.4 DataAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.4.1 BeamPosition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.4.2 IntensityMeasurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.4.3 CrossingAngleandHourglass Effect . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.4.4 BeamProfile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.5 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.6 Beam-beamDeflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6 ExperimentalResultsfromtheLHC 101
6.1 Implementation andProcedurefortheLHC . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.1.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.1.2 OrbitBumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.1.3 MachineProtection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.1.4 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2 FirstCollisionsandOptimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.2.1 450GeVCollisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.2.2 LuminosityOptimizationat450GeV . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2.3 3.5TeVCollisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.2.4 LuminosityOptimizationat3.5TeV . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2.5 FirstExperiencewithHighIntensity . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3 LuminosityCalibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.3.1 MeasurementsSummary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.3.2 BeamProfileandFitMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.3.3 HysteresisEffects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.3.4 BumpCalibrationandLinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.3.5 CrossingAngleMeasurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.3.6 Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.3.7 Emittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.3.8 IntensityMeasurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.3.9 ComparisonwithOpticsMeasurements . . . . . . . . . . . . . . . . . . . . . . . . 122
6.3.10 FilltoFillConsistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.3.11 Conclusions andOutlookforFutureMeasurements . . . . . . . . . . . . . . . . . . 123
7 TowardsHigherPrecision: TheHigh-β Experiments 125
∗
7.1 WhyHigh-β Optics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
∗
7.2 High-β ExperimentsintheLHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
∗
7.3 AnalyticalEstimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.4 TOTEM90mOptics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.5 TOTEMveryhigh-β Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
∗
7.5.1 BaselineSolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.5.2 AlternativeSolutionwithQ4On . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6 CONTENTS
7.5.3 ComparisonofthePerformanceforPhysics . . . . . . . . . . . . . . . . . . . . . . 133
7.5.4 Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.5.5 Compatibilityat5TeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.6 ATLASVeryHigh-β Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
∗
7.6.1 OpticsforPhysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.6.2 InjectionOpticswithQ4Inverted . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.6.3 Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.7 CommissioningandRunningScenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.7.1 EarlyRunning: 3.5TeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.7.2 Veryhigh-β Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
∗
Conclusion 143
A SoftwareforLuminosityOptimizationandCalibration 145
A.1 Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
A.2 LuminosityCalibrationRoutine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
A.3 LuminosityOptimizationRoutine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
A.4 SteeringRoutine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
A.5 OnlineAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
B CouplingAngleCalculation 151
Bibliography 154
Acknowledgments 161
Re´sume´ 163
Introduction
Forparticle colliders, the most important performance parameters are the beam energy and the luminosity.
High energies allow the particle physics experiments to study and observe new effects. The luminosity
describes the ability of the collider to produce the required number of useful interactions or events. It is
defined as the proportionality factor between the event rate, measured by the experiments, and the cross
section of the observed event which describes its probability to occur. The absolute knowledge of the
luminositythereforeallowstheexperimentstomeasuretheabsolutecrosssections.
TheLargeHadronCollider(LHC)wasdesignedtoproduceprotonprotoncollisionsatacenterofmass
energy of 14TeV. This energy would be the highest ever reached in a particle accelerator. The knowledge
andunderstandingofparticlephysicsatsuchhighenergyisbasedonsimulationsandtheoreticalpredictions.
As opposed to e+ e colliders, for which the Bhabba scattering cross section can be accurately calculated
−
and used forluminosity calibration, there are no processes with well known cross sections and sufficiently
high production rate to be directly used for the purpose of luminosity calibration in the early operation of
theLHC.
The luminosity can also be expressed as a function of the number of charges per beam and the beam
sizes at the interaction point. Using this relation, the absolute luminosity can be determined from machine
parameters. Thedetermination of the absolute luminosity from machine parameters is an alternative to the
crosssectionbasedcalibrationandprovidescomplementaryinformationtothefragmentationmodel. Inthe
LHC,itwasproposedtousethemethoddevelopedbyS.VanDerMeerattheISR[1]toprovidealuminosity
calibration basedonmachineparameterstothephysicsexperimentsduringthefirstyearofoperation.
Theworkpresented inthis thesis started in2007. Atthetime,theLHCwasexpected tostart operating
in 2008 and to produce collisions at the design center of mass energy of 14TeV. Some of the studies and
simulations intended as a preparation for luminosity calibration were done for this original design energy.
After a very successful start-up in 2008 issues were found that required a major repair and consolidation
which resulted in an extended shutdown period of one year. Operation resumed in 2009 with a reduced
targetcenterofmassenergyof7TeVandthefirstcollisions wereproducedinMarch2010.
This shutdown period was used to extend the scope of this thesis to more general studies such as lu-
minosity optimization, optics studies and operation in collision. It also allowed for a collaboration with
BNL.Luminosity calibration measurements were performed at the RHIC collider in 2009 as a preparation
forLHCstart-up. The RHICcollider is in some sense very similar to the LHCand most of the experience
acquired during this collaboration could directly be applied to the LHC. Differences still exist and beam
dynamics orinstrumental effects have tobe considered whileanalyzing theRHICdata which do not apply
totheLHC.Theworkpresented heretherefore includes moregeneral considerations notdirectly relatedto
thecalibrationoftheluminosityattheLHC.
Chapter 1 of this thesis is intended as an introduction to general accelerators physics concepts and
definitions that will be used in the following chapters. The principles of transverse beam dynamics are
explained as well as some basic notions related to beam-beam interactions. General expressions of the
luminosity are derived including complications such as the presence of a crossing angle or the hourglass
effect.
7
8 INTRODUCTION
Chapter 2 focuses on the Van Der Meer method. The principle of the method and implications of the
effects introduced in Chapter 1 are discussed. Most of these effects are small and well controllable under
specific beam conditions. Initial estimates on the expected uncertainty related to luminosity calibration in
theLHCarediscussed.
Chapter3and4giveanoverviewoftheCERNacceleratorcomplexfocusingontheLHCanditsinstru-
mentation. Beamdynamicsandopticsstudiesrelatedtotheoptimizationofthecollisionsandmoregenerally
oftheinteraction regionsareshownaswellastrackingsimulationsfortheLHCluminositymonitors.
Chapter 5 and 6 present the results obtained at the LHC and RHIC during luminosity calibration mea-
surements. Adetailedanalysisofthesystematicsuncertaintiesassociatedtothemeasurementandproposals
forfutureimprovementsarediscussed.
Chapter6alsodescribesmorespecificallytheprocedureandimplementationofthetoolsforluminosity
optimizationandcalibration attheLHCaswellasthefirstexperiencewithoperationincollision.
Finally, in Chapter 7, an alternative method for luminosity calibration is introduced. Dedicated optics
arerequiredforthismeasurement. Anoverviewofthestudyandperformanceoftheseopticsispresented.
MypersonnalworkcanbefoundinsomeofthederivationsoftheluminositypresentedinChapter 1,in
Chapter 2, in the second part of Chapter 3 and Chapter 4 and in the last three Chapters of this thesis. The
luminosity scan software was written as part of this thesis and was used to collide and optimize the LHC
beamsforthefirsttime. Itisnowusedonaregularbasisandrepresentsmymostsignificantcontribution to
LHCoperation.
Chapter 1
Beam Dynamics
This Chapter aims at introducing some general concepts of beam dynamics and defining common param-
eters and formalism that will be used in this thesis. General equations of the motion of the particles in an
accelerator will be derived as well as a definition of the beam-beam interactions. More specifically, the
concept ofluminosity anditscalculation undervarious conditions willbedetailed asanintroduction tothe
followingchapters.
1.1 Basics of Accelerator Physics
A charged particle with charge q, momentum!p and velocity!v in the electromagnetic fields (!E,!B) experi-
encestheLorentz’s force !F:
d!p
!F =q(!E+!v !B)= . (1.1)
× dt
In an accelerator, the charged particles gain energy by their interaction with the electric field !E. The
magnetic force!v !B is perpendicular to both!v and !B. The trajectory of a charged particle will be curved
×
when it passes through adipole magnet. Atrelativistic velocities an electric field E and amagnetic fieldB
havethesameeffectforE =cB. Amagneticfieldof1Twouldthenbetheequivalentofanelectricfieldof
3.108V.m 1. Producing such an electric field is far beyond technical limits for current magnet designs, as
−
aresult wealways usemagneticfieldstosteerthebeams. Thephysical fundamentals ofbeamsteering and
focusingarecalledbeamoptics.
1.1.1 CoordinateSystem
Wecandefine acoordinate system shown inFigure 1.1to describe the path of the particles in which swill
describe the longitudinal direction along the reference orbit. x and y will define the transverse plane and
furthermorethedeviationfromthereference. Locallythetrajectoryhasaradiusofcurvatureρ. Inacircular
accelerator the elements the beam is passing through can be straight or curved, this coordinate system is
therefore curvilinear. The trajectory of the reference particle!r is the one that has null x and y coordinates
0
foralls. Theparticletrajectoryaroundthereferenceorbitcanbeexpressedas:
!r=!r (s)+xxˆ(s)+yyˆ(s), (1.2)
0
wherexˆandyˆaretheunitvectorsinthetransverse plane.
9