Particle Physics Reference Library

Others are in the form of an original contribution supplemented by a detailed appendix relating to recent developments in the field. He was responsible for the commissioning and energy upgrade of the CERN Large Electron-Positron Collider (LEP).

Accelerators, Colliders and Their Application

Why Build Accelerators?

It has been correctly argued that higher energy particles, which have the property of shorter wavelengths, can better reveal the structure of the nuclei that Rutherford detected. Initially, physicists used accelerators to investigate the structure of the nucleus, but continued to use higher energy accelerators to search for structure in the "fundamental" particles—protons, neutrons, and electrons—that they discovered.

Types and Evolution of Accelerators

Early Accelerators
The Ray Transformer
Repetitive Acceleration
Linear Accelerators
Cyclotrons
The Synchrotron
Phase Stability

The primary winding of the transformer marked W1 is supplied with alternative voltage from the mains. This should lead to large cost savings per unit length of the magnet system.

Exchanges in Moscow in 1944 [14] and McMillan in Berkeley to pave the way for the construction of the first synchrotrons. The images or other third-party material in this chapter are included in the chapter's Creative Commons license, unless otherwise indicated in a credit line for the material.

Beam Dynamics

Linear Transverse Beam Dynamics

Co-ordinate System
Displacement and Divergence
Bending Magnets and Magnetic Rigidity
Particle Trajectory in a Dipole Bending Magnet
Weak Focusing
Alternating Gradient Focusing
Quadrupole Magnets
The Equation of Motion
Matrix Description
Transport Matrices for Lattice Components
The Betatron Envelopes

We will see that β (later called the envelope or betatron function) is a property of the trough, not the beam. These functions (which are not the same as the parameters used in special relativity!) are a complete and compact description of the dynamics.

Fig. 2.1 Charged particle orbit in magnetic field

Coupling

Coupling Fields
Qualitative Treatment of Coupling

The angular impact p, at the inclined quadrupole transition is calculated from a similar diagram for the vertical plane and. The vertical amplitude of the betatron decreases fromw+wtowin a quarter period of the slow oscillation which takes 1/4|QH−Qv|turns.

Fig. 2.12 The magnetic field and force in a skew quadrupole

Liouville’s Theorem

Chains of Accelerators
Exceptions to Liouville’s Theorem

The shape and orientation of the ellipse are determined by the Twiss parameters at the given location. The invariance of the region of space (x,x) as we move to different points on the ring is an alternative statement of Liouville's theorem.

Momentum Dependent Transverse Motion

Dispersion
Chromaticity

The emittance, which is usually given for an electron beam, corresponds to an electron with the amplitude of σ in the Gaussian projection. Integrated over all focusing (and defocusing) elements in the ring, we achieve a change in the melody of the machine.

Fig. 2.20 The extra inward force given to a low momentum particle by the dipoles is balanced by the focusing the quadrupoles and defines a dispersion function

Longitudinal Motion

Stability of the Lagging Particle
Transition Energy
Synchrotron Motion
Stationary Buckets

If we consider the motion of a particle on the linear part of the voltage wave of an r.f. The other coordinate is the particle's arrival phase,φ, with respect to the zero crossing of the r.f.

Fig. 2.22 The limiting trajectory for a particle in a ‘moving’ or accelerating bucket when the stable phase is not zero

Non-linear Dynamics in Accelerators

Introduction

Motivation
Single Particle Dynamics
Layout of the Treatment

In many new accelerators or storage rings (eg LHC) the description of the machine with a linear formalism becomes insufficient and the linear theory must be extended to treat non-linear effects. Following a summary of the sources of non-linearities in circular machines, the basic methods for evaluating the consequences of non-linear behavior are discussed.

Variables

Trace Space and Phase Space
Curved Coordinate System

In the last part, we summarize the most important physical phenomena caused by the nonlinearities in an accelerator. In the end, the mathematical tools are very different from those discussed in this article.

Sources of Non-linearities

Non-linear Machine Elements

Unwanted Non-linear Machine Elements
Wanted Non-linear Machine Elements

Beam–Beam Effects and Space Charge

These effects can dominate the required aperture and limit the stable range of the beam. In most accelerators, the momentum-dependent focusing of the grating (chromaticity) must be corrected with sextupoles [3,4].

Map Based Techniques

They can cause significant disturbances on the same beam (space charge effects) or on the opposing beam (beam-beam effects) in the case of a colliding beam facility. The collection of all machine elements makes up the ring per beamline and it is the combination of the associated maps that is necessary for the description and analysis of the physical phenomena in the accelerator ring or beamline. It is assumed that the reader is familiar with this concept in the case of linear beam dynamics (Chapter 2) where all maps are matrices and the Courant-Snyder analysis of the corresponding one-turn map produces the desired information such as

Given that the non-linear maps can be quite complex objects, the analysis of the one-turn map should be separated from the calculation of the map itself.

Linear Normal Forms .1 Sequence of Maps

Analysis of the One Turn Map
Action-Angle Variables
Beam Emittance

This transformationA "analyzes" the complexity of the movement, it contains the structure of the phase space. Among other advantages, it can be used to "normalize" the positionx: the normalized positionx is the "non-normalized" one divided by√. The variation of the normalized positionxn is then smaller than in the non-normalized case.

These parameters are related to the moments of the beam, e.g. the beam size is directly related to the second order moment< x2>.

Techniques and Tools to Evaluate and Correct Non-linear Effects

Particle Tracking

Symplecticity

Approximations and Tools
Taylor and Power Maps

Taylor Maps
Thick and Thin Lenses
Symplectic Matrices and Symplectic Integration
Comparison Symplectic Versus Non-symplectic Integration

Potent types are not symplectic, they cannot be used. The non-symplecticity can be recovered in the case of elements with L = 0. A very explicit example of the iterative construction of a higher order map from a lower order can be found in [7]. We can apply this method to any other non-linear map and get the same integrators.

The proof of this claim and its systematic extension can be done in the form of Lie operators [12].

Fig. 3.1 Schematic representation of a symplectic kick of first order (left) and second order (right)

Hamiltonian Treatment of Electro-Magnetic Fields

Lagrangian of Electro-Magnetic Fields

Lagrangian and Hamiltonian

Hamiltonian with Electro-Magnetic Fields
Hamiltonian Used for Accelerator Physics

Lie Maps and Transformations
Concatenation of Lie Transformations

Analysis Techniques: Poincare Surface of Section
Analysis Techniques: Normal Forms

Normal Form Transformation: Linear Case
Normal Form Transformation: Non-linear Case

Truncated Power Series Algebra Based on Automatic Differentiation

Automatic Differentiation: Concept
Automatic Differentiation: The Algebra
Automatic Differentiation: The Application
Automatic Differentiation: Higher Orders
Differential Algebra: Applications to Accelerators
Differential Algebra: Simple Example

In Fig.3.3 (right) the complex integrators of order 1 and 2 as derived above are used instead. Given an algorithm, which might be a complex simulation program with several thousand lines of code, we can use techniques to "teach" the code how to calculate derivatives automatically. In the left column we get the expected result from the actual calculation of the expression.

In Example 1, a grid with 8 FODO cells is constructed and the quadrupole is implemented as a thin lens "kick" at the center of the element.

Fig. 3.4 Poincare surface of section of a particle near the 5th order resonances. Left without non- non-linear elements, right with one sextupole

Beam Dynamics with Non-linearities

Amplitude Detuning

Amplitude Detuning due to Beam–Beam Effects
Phase Space Structure

Non-linear Resonances

Resonance Condition in One Dimension
Driving Terms

Chromaticity and Chromaticity Correction
Dynamic Aperture

Long Term Stability and Chaotic Behaviour
Practical Implications

Non-linear resonances can be generated in the presence of non-linear fields and play an important role for the long-term stability of the particles. Often in the context of discussing nonlinear resonance phenomena, the concept of dynamic aperture is introduced. The same techniques can be employed to maximize the dynamic aperture, ideally beyond the limits of the physical aperture.

Satisfactory insight into the fine structure of the phase space can only be obtained by numerical calculation.

Fig. 3.11 Comparison: numerical and analytical model for one interaction point. Shown for 5σ x

Impedance and Collective Effects

Space Charge

Direct Space Charge
Indirect Space Charge

4.1) The first case corresponds to the spacecharge case where both particles move in the same direction, while the second case corresponds to the beam-beam case (see section. This means that the transverse spacecharge force is linear for small amplitudes and defocusing. Unlike the transverse case, the longitudinal spacecharge defocuses below the transition (if η < 0) and focuses above it (if η > 0).

The corresponding horizontal 2D (i.e. neglecting the longitudinal distribution) space charge force is plotted in Fig.4.2(left), and the tuning footprint in Fig.4.2(right).

Fig. 4.1 (Left) Computation of the evolution of the full bunch length vs. time for the case of the nonadiabatic theory (the adiabatic theory is not valid anymore close to transition) with and without space charge, applied to the PS nTOF bunch [10]

Wake Fields and Impedances

The situation is more involved in the case of non-axisymmetric structures (especially due to the presence of the quadrupolar wake field already discussed in Sect. If the source particle 1 and test particle 2 have the same chargeq, and in the ultrarelativistic case, the transverse wake potentials can be written (considering only the source terms for linear terms and test2 considering the source terms and the high order) The sum of the transverse drive and tuning impedances is then derived by applying the Panofsky-Wenzel theorem in the case of top/bottom and left/right symmetry [33].

In the case of a cavity, an equivalent RLC circuit with Rs longitudinal shunt impedance, C capacitance and L inductance can be used.

Fig. 4.3 Horizontal (driving) impedance of a cylindrical one-meter long LHC collimator (even if in reality the LHC collimators are composed of two parallel plates), with b = 2 mm and ρ = 10 μ m

Coherent Instabilities

Longitudinal
Transverse

4.6 (Left) Comparison between GALACLIC Vlasov Solver [63] (in red) and Laclare's approach [53] (in black) of normalized mode frequency shifts vs. The head of the group is stable and only the tail is unstable in the vertical plane. Particles lost at the tail of the cluster can be seen from the indentation in the cluster profile.

Evolution of the real (left) and imaginary (right) parts of shifts of the transverse modes (with respect to the undisturbed betatron tune), normalized by the synchrotron tune, vs.

Fig. 4.5 Signal at the pick-up electrode for three different modes shown for several superimposed turns (the red line corresponds to one particular turn), for the case of the parabolic amplitude distribution and a constant inductive impedance (exhibiting t

Landau Damping

Transverse
Longitudinal

The origin of the frequency distribution leading to Landau damping has not been considered until now. Calculation of the l.h.s is now straight forward (following Section 4.3): for a given impedance (and transverse damper), one only needs to calculate the complex mode frequency shift, in the absence of Landau damping. If it lies on the inside of the locus (the side containing the origin), the beam is stable.

In the presence of an impedance, the dipole mode coherent synchrotron frequency ωc11, which is equal to the low intensity synchrotron frequency ωs0 without synchrotron frequency spreading (due to the compensation between the incoherent and coherent melody shifts), moves closer and closer to the incoherent band (stable region).

Fig. 4.15 Simplified model/example of Ref. [86], describing the mode-coupling in the “short- “short-bunch” regime, i.e

Two-Stream Effects (Electron Cloud and Ions)

Electron Cloud Build-Up in Positron/Hadron Machines
The Electron Cloud Instability
Mitigation and Suppression

With this type of beam, even after extensive machine scrubbing, the undesirable effects of the electron cloud have remained visible on the beam and machine equipment. Moreover, in some cases it has been found that the stationary value of the electron cloud density does not increase monotonically with forest intensity. Three effects are suspected to be responsible for the long memory and longevity of the electron cloud.

The numerical modeling of the interaction between an electron cloud and a particle group is discussed in Sect.4.7.

Fig. 4.21 Schematic of electron cloud build-up in the LHC beam pipe during multiple bunch passages, via photo-emission (due to synchrotron radiation) and secondary emission (Courtesy of F

Beam–Beam Effects

Introduction
Beam–Beam Force

Elliptical Beams
Round Beams

Incoherent Effects: Single Particle Effects

Beam–Beam Parameter
Non-linear Effects
Beam Stability
Beam–Beam Limit

Studies of Head-on Collisions at the LHC

PACMAN Bunches

Head-on Beam–Beam Tune Shift
Effect of Number of Head-on Collisions
Crossing Angle and Long Range Interactions

Long-Range Beam–Beam Effects
Opposite Sign Tune Shift
Strength of Long-Range Interactions
Footprint for Long-Range Interactions

Studies of Long Range Interactions in the LHC

Dynamic Aperture Reduction Due to Long-Range Interactions For too small separation, the tune spread induced by long-range interactions can
Beam–Beam Induced Orbit Effects

Coherent Beam–Beam Effects

Coherent Beam–Beam Modes

Compensation of Beam–Beam Effects

Electron Lenses
Electrostatic Wire
Möbius Scheme

First, we briefly discuss the immediate effect of the nonlinearity of the beam-beam force on a single particle. This can be approximated by inserting a quadrupole that produces the same tuning shift at the position of the beam-beam interaction. This in turn changes the strength of the beam-beam interaction and the parameters must be found in a self-consistent form.

Another finding is that for the same intensity value, the beam–beam parameter ξ saturates.

Fig. 4.22 Tune shift (non-linear detuning) as a function of the amplitude (left) and 2-dimensional tune footprint (right)

Numerical Modelling

The Electromagnetic Problem
Beam Dynamics

Métral, et al.: Kicker impedance measurements for the future CERN proton synchrotron multiplex extraction, Proc. Rumolo et al., A simulation study on the energy dependence of the TMCI threshold at CERN-SPS, Proceedings of EPAC 2006, Edinburgh, Scotland. Herr et al., Observations of beam-to-beam effects at high intensities at the LHC, in Proceedings of the 2011 Particle Accelerator Conference, San Sebastion (2011).

Shiltsev, et al., Compensation of beam-to-beam effects in the Tevatron electron beam collider, in Proceedings of the 1999 Particle Accelerator Conference, New York, p.

Fig. 4.33 Example of numerical simulation with collective effects: schematic view of the interaction of a single bunch with a longitudinal wake field at several locations in an accelerator ring

Interactions of Beams with Surroundings

The Interactions of High Energy Particles with Matter

Basic Physical Processes in Radiation Transport Through Matter
Simulation Tools

FLUKA
GEANT4
MARS15
MCNP
PHITS
Simulation Uncertainties

Practical Shielding Considerations

Radiation Attenuation
Shielding of Electromagnetic Showers
Shielding of Neutrons

Their ionization and electromagnetic cascades determine the core of the energy deposition distribution. Below is a brief overview of the most commonly used multipurpose codes around booster applications. Descriptions of intranuclear cascades include implementations of the Binary and Bertini cascade models (the latter has been significantly reworked and is not at all related to the original Bertini model).

Further uncertainties due to the assumptions used in the description of the geometry and of the materials studied.

Lifetimes, Intensity and Luminosity