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Theory of accelerators. I. Electrostatic Machines II. Cyclotrons III. Linacs IV. Synchrotrons V. Colliders VII. Synchrotron Radiation Sources VIII. Other Applications. →. →. →. →. Centripetal Force. Lorentz force.

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Theory of accelerators

by Ted Wilson

Theory of accelerators

I. Electrostatic Machines

II. Cyclotrons

III. Linacs

IV. Synchrotrons

V. Colliders

VII. Synchrotron Radiation Sources

VIII. Other Applications

Basic concepts

Centripetal Force

Lorentz force

Fixes the relation between magnetic field and particle’s energy


Magnetic rigidity

LHC: ρ = 2.8 km given by LEP tunnel!

Basic concepts

Charged particles are accelerated, guided and confined by electromagnetic fields.

- Bending: Dipole magnets - Focusing: Quadrupole magnets - Acceleration: RF cavities

In synchrotrons, they are ramped together synchronously to match beam energy.

- Chromatic aberration: Sextupole magnets

Lhc injector complex
LHC injector complex

450 GeV

1.4 GeV

26 GeV

The large hadron collider lhc
The Large Hadron Collider (LHC)


B field




F force







Two-in-one magnet design

LHC: B = 8.33 T ⇒ E = 7 TeV










Transverse focusing is achieved with quadrupolemagnets, which act on the beam like an optical lens.

Linear increase of the magnetic field along the axes (no effect on particles on axis).

Focusing in one plane, de-focusing in the other!



Alternating gradient lattice




Alternating gradient lattice

An illustrative scheme

(LHC: 2x3 dipoles per cell)

One can find an arrangement of quadrupole magnets that provides net focusing in both planes (“strong focusing”).

Dipole magnets keep the particles on the circular orbit.

Quadrupole magnets focus alternatively in both planes.

Coordinate system

Transverse equation of motion

Hill’s equation

K(s) describes the distribution of focusing strength along the lattice.

G. Hill, 1838-1914

Transverse equation of motion

Magnetic field [T] :

Field gradient [T m-1] :

Normalized grad. [m-2] :

and its solution

Lhc layout and accelerator systems
LHC layout and accelerator systems

Eight arcs and eight straight sessions:

Point 1: Atlas, LHCf

Point 2: Alice, injection

Point 3: Momentum cleaning

Point 4: RF

Point 5: CMS, TOTEM

Point 6: Beam Dumps

Point 7: Betatron cleaning

Point 8: LHCb, injection

Lhc design parameters
LHC design parameters

- These are the key parameters of a collider – the LHC

- Why are they important for physics?

- What is the basic theory which limits each one of them?

Luminosity single bunches

Imagine a blue particle colliding with a beam of cross section area - A

Probability of collision for an interaction is

For N particles in both beams

Suppose they meet f times per second at the revolution frequency

Event rate

Luminosity (single bunches)

Make big

Make small



s section area -


Acceleration is performed with electric fields fed into Radio-Frequency (RF) cavities. RF cavities are basically resonators tuned to a selected frequency.

In circular accelerators, the acceleration is done with small steps at each turn.

LHC: 8 RF cavities per beam (400 MHz), located in point 4

At the LHC, the acceleration from 450 GeV to 7 TeV lasts ~ 20 minutes (nominal!), with an average energy gain of ~ 0.5 MeV on each turn. [Today, we ramp at a factor 4 less energy gain per turn than nominal!]

Buckets and bunches

ΔE section area -

LHC bunch spacing = 25 ns = 10 buckets ⇔ 7.5 m

RF bucket


2.5 ns

450 GeV 7 TeV

RMS bunch length 11.2 cm 7.6 cm

RMS energy spread 0.031% 0.011%

Buckets and bunches

The particles oscillate back and forth in time/energy

The particles are trapped in the RF voltage:

this gives the bunch structure

RF Voltage

2.5 ns


Synchroton radiation
Synchroton section area - Radiation

  • Electrons accelerating by running up and down in a radio antenna emit radio waves

  • Radio waves are nothing more than Long Wavelength Light-

Synchroton light sources
Synchroton section area - Light Sources

When the electron speed gets close to the speed of light,

e radiation comes out only in a narrow forward cone;

a laser-like concentrated stream

This 300 MeV electron section area - synchroton at the General Electric Co. at Schenectady, built in the late 1940s. The photograph shows a beam of synchrotron radiation emerging.

Synchrotron light sources

Spring 8, a synchrotron light source located in Japan. section area -

Synchrotron Light Sources

This intricate structure of a complex protein molecule structure

has been determined by reconstructing scattered synchrotron radiation

Linac coherent light source and the european union x ray free electron laser fourth generation

Engines of Discovery section area -

Linac Coherent Light Source and the European Union X-Ray Free Electron Laser(Fourth Generation)

FELs, invented in the late 1970’s at Stanford are now becoming the basis of major facilities in the USA (SLAC) and Europe (DESY) .They promise intense coherent radiation. The present projects expect to reach radiation of 1 Angstrom (0.1 nano-meters, 10kilo-volt radiation)

Basis of muon collider
Basis of muon collider section area -

Spallation Neutron Sources (SNS) section area -

1GeV protons

mean current 1 mA

= 1.4 MW of power

In a

0.7 microsecuond burst

Cost is about 1.5 B$

An overview of the Spallation Neutron Source (SNS) site at Oak Ridge National Laboratory.

Unstable Isotopes and their Ions section area -

The Rare Isotope Accelerator (RIA) scheme. The heart of the facility is composed of a driver accelerator capable of accelerating every element of the periodic table up to at least 400 MeV/nucleon. Rare isotopes will be produced in a number of dedicated production targets and will be used at rest for experiments, or they can be accelerated to energies below or near the Coulomb barrier.

Energy amplifier
Energy amplifier section area -

Proton Drivers for Power Reactors section area -

A linac scheme for driving a reactor. These devices can turn thorium into a reactor fuel, power a reactor safely, and burn up long-lived fission products.

I have not mentioned
I have not mentioned section area -


Chip manufacture

Art and archaeology

National Security

Surface treatment

Etc. etc….


Author’s e-mail: section area - [email protected]

“Engines of Discovery”:

“Particle Accelerators”