Lecture 5 beam optics and imperfections
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Lecture 5: Beam optics and imperfections. Errors in magnetic lattices dipole errors quadrupole errors Nonlinear Perturbations Resonances Dynamic aperture. Linear betatron equations of motion (recap).

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Lecture 5 beam optics and imperfections

Lecture 5: Beam optics and imperfections

Errors in magnetic lattices

dipole errors

quadrupole errors

Nonlinear Perturbations


Dynamic aperture

Linear betatron equations of motion (recap)

In the magnetic fields of dipoles magnets and quadrupole magnets (without imperfections) the coordinates of the charged particle w.r.t. the reference orbit are given by the Hill’s equations

These are linear equations (in y = x, z). They can be integrated and give

Errors in lattices

  • Transfer lines or circular accelerators are made of a series of drifts and quadrupoles for the transverse focussing and accelerating section for acceleration.

  • Real lattices will always unavoidably be different from the model lattices. This is due to

  • misalignments and rolls of the magnetic elements

  • gradient error

  • additional unwanted magnetic field multipoles (stray fields)

  • time varying magnetic errors (due to change in magnetic fields, ground motion, etc)

Errors in lattices

Dipole errors i

Dipole errors (I) of drifts and quadrupoles for the transverse

Dipole field errors arise due to calibration errors current vs magnetic fields, misalignment and rolls.


Dipole errors ii

Dipole field errors and misalignment of the magnetic elements produce orbit distortions. They generate an additional term in the Hill’s equation

Dipole errors (II)

The general solution can be found as the sum of a particular integral and the general solution of the imhogeneous equation

and reads

The effect is stronger where the  functions are larger

The orbit gets unstable if the betatron tune y is close to an integer

Orbit corrections i

With many distributed dipole errors the orbit will be distorted. The beam position monitors (BPMs) will read a non zero orbit.

Orbit corrections (I)

To correct the orbit we use a system of horizontal and vertical corrector dipoles.

We need to know what is the effect of each correctors on the orbit itself, first. Then we can use this information to correct orbit distortion.

Orbit correction ii

V distorted. The beam position monitors (BPMs) will read a non zero orbit.




The orbit response matrix collects all this information in one matrix

The orbit response matrix R is the change in the orbit at the BPMs as a function of changes in the steering magnets strength.

Orbit correction (II)

orbit RM at diamond

168*2 rows

168*2 columns

orbit reading at all BPMs

dipole corrector angle kick

Orbit correction iii

Once the matrix R is known we can correct an orbit distortion by inverting the matrix

Orbit correction (III)

The matrix R generally is not a square matrix so it might be non-invertible but we can use the Singular Value Decomposition (SVD) of the response matrix R to invert it and correct the closed orbit distortion

Quadrupole errors i

Quadrupole errors (I) distortion by inverting the matrix

Quadrupole magnet are due to calibration errors in current vs field gradient, misalignment and rolls. A misaligned quadrupole adds a dipole kick.

Quadrupole errors iii

Consider a lattice with a quadrupole distribution K(s) and a gradient error K(s)

This is still a homogeneous Hill’s equation and can be solved with the matrix method

Quadrupole errors (III)

Let us consider a localised quadrupole error and let us put

M0 the transfer matrix without error

M the transfer matrix with the error

m the transfer matrix of the localised section where the error is

m0 the transfer matrix of the same section without error

We can compute the matrix M with the relation





Quadrupole errors iv

We use the gradient error Twissparameterisation for M and M0

Quadrupole errors (IV)

and the thin lens approximation for m and m0

we get to first order in ds

Using again the Twissparameterisation for the transfer matrix we get the new phase advance

Quadrupole errors v

If the perturbation is distributed around the ring we have gradient error

Quadrupole errors (V)

this can be eventually expressed with an integral over the ring circumference

Therefore a gradient error leads to a betatron tune shift

The effect is stronger where the beta functions are larger (final focus - IPs). A gradient error changes the beta functions as well

The optics get unstable if the betatron tune Qy is close to an half-integer


The shift in the betatron tunes has to be controlled to avoid creating resonance conditions between the horizontal and the vertical tune


m Qx + n Qz = p

|m| + |n|

is called the order of the resonance

We have already met two example of resonances: integer resonances which make the closed orbit unstable and half integer resonances which make the optics unstable. More examples when we will consider nonlinear elements.

Nonlinear magnetic errors

Multipolar expansion of the magnetic field inside a magnet avoid creating resonance conditions between the horizontal and the vertical tune

Nonlinear magnetic errors

The on axis magnetic field can be expanded into multipolar components (dipole, quadrupole, sextupole, octupoles and higher orders)

Hill s equation with nonlinear terms

Including higher order terms in the expansion of the magnetic field

normal multipoles

Hill’s equation with nonlinear terms

skew multipoles

the Hill’s equations acquire additional nonlinear terms

No analytical solution available in general:

the equations have to be solved by tracking or analysed perturbatively

Example: nonlinear errors in the LHC main dipoles magnetic field

Finite size coils reproduce only partially the cos- desing necessary to achieve a pure dipole fields

LHC main dipole cross section

Multipolar errors up to very high order have a significant impact on the nonlinear beam dynamics.

Sextupole magnets

Nonlinear magnetic fields are introduced in the lattice (chromatic sextupoles)

Normal sextupole

Sextupole magnets

Normal sextupole

Skew sextupole

Example: nonlinear elements in small emittance machines (chromatic sextupoles)

Small emittance  Strong quadrupoles  Large (natural) chromaticity

 Strong sextupoles (sextupoles guarantee the focussing of off-energy particles)

strong sextupoles have a significant impact on the electron dynamics

 additional sextupoles are required to correct nonlinear aberrations

Phenomenology of nonlinear motion i

x’ (chromatic sextupoles)

Turn 1

Turn 2


Turn 3

Turn 2

Turn 1

Turn 3

The orbit in phase space for a system of linear Hill’s equation are ellipses (or circles)

The frequency of revolution of the particles is the same on all ellipses

Phenomenology of nonlinear motion (I)


The orbit in the phase space for a system of nonlinear Hill’s equations are no longer simple ellipses (or circles);

The frequency of oscillations depends on the amplitude


Nonlinear resonances

Turn 2 (chromatic sextupoles)

Turn 1

Turn 5

Turn 3

Turn 4

m = 5; n = 0; p =1

When the betatron tunes satisfy a resonance relation

the motion of the charged particle repeats itself periodically

Nonlinear resonances

If there are errors and perturbations which are sampled periodically their effect can build up and destroy the stability of motion

The resonant condition defines a set of lines in the tune diagram

The working point has to be chosen away from the resonance lines, especially the lowest order one

(example CERN-SPS working point)

5-th order resonance phase space plot (machine with no errors)

Phenomenology of nonlinear motion ii

Phase space plots of close to a 5 (chromatic sextupoles)th order resonance

Stable and unstable fixed points appears which are connected by separatrices

Islands enclose the stable fixed points

On a resonance the particle jumps from one island to the next and the tune is locked at the resonance value

region of chaotic motion appear

The region of stable motion,

called dynamic aperture, is limited by the appearance of

unstable fixed points and

trajectories with fast escape to infinity

Phenomenology of nonlinear motion (II)

Qx = 1/5

Phenomenology of nonlinear motion (III) (chromatic sextupoles)

  • The orbits in phase space of a non linear system can be broadly divided in

  • Regular orbit  stable or unstable

  • Chaotic orbit  no guarantee for stability but diffusion rate may be very small

The particle motion on a regular and stable orbit is quasi–periodic

The betatron tunes are the main frquencies corresponding to the peak of the spectrum in the two planes of motion

The frequencies are given by linear combination of the betatron tunes.

Only a finite number of lines appears effectively in the decomposition.

Tracking i

Most accelerator codes have tracking capabilities: MAD, MADX, Tracy-II, elegant, AT, BETA, transport, …

Typically one defines a set of initial coordinates for a particle to be tracked for a given number of turns.

The tracking program “pushes” the particle through the magnetic elements. Each magnetic element transforms the initial coordinates according to a given integration rule which depends on the program used, e.g. transport (in MAD)

Tracking (I)

Linear map

Nonlinear map up to third order as a truncated Taylor series

Tracking ii

A Hamiltonian system is symplectic, i.e. the map which defines the evolution is symplectic (volumes of phase space are preserved by the symplectic map)

M is symplectic transformation

Tracking (II)

non symplectic

Symplectic integrator

If the integrator is not symplectic one may found artificial damping or excitation effect

The well-known Runge-Kutta integrators are not symplectic. Likewise the truncated Taylor map is not symplectic. They are good for transfer line but they should not be used for circular machine in long term tracking analysis

Elements described by thin lens kicks and drifts are always symplectic: long elements are usually sliced in many sections.

Dynamic aperture and frequency map

Strongly excited resonances can destroy the Dynamic Aperture.

The Frequency Map Analysis is a technique introduced in Accelerator Physics form Celestial Mechanics (Laskar).It allows the identification of dangerous non linear resonances during design and operation.

Dynamic Aperture and Frequency Map

To each point in the (x, y) aperture there corresponds a point in the (Qx, Qy) plane

The colour code gives a measure of the stability of the particle (blu = stable; red = unstable)

The indicator for the stability is given by the variation of the betatron tune during the evolution: i.e. tracking N turns we compute the tune from the first N/2 and the second N/2

Engineering aperture.


E. Wilson, CAS Lectures 95-06 and 85-19 Aperture.

E. Wilson, Introduction to Particle Accelerators

G. Guignard, CERN 76-06 and CERN 78-11

J. Bengtsson, Nonlinear Transverse Dynamics in Storage Rings, CERN 88-05

J. Laskar et al., The measure of chaos by numerical analysis of the fundamental frequncies, Physica D65, 253, (1992).