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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Errors in magnetic lattices
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
Dipole field errors arise due to calibration errors current vs magnetic fields, misalignment and rolls.
Dipole field errors and misalignment of the magnetic elements produce orbit distortions. They generate an additional term in the Hill’s equation
The general solution can be found as the sum of a particular integral and the general solution of the imhogeneous equation
The effect is stronger where the functions are larger
The orbit gets unstable if the betatron tune y is close to an integer
With many distributed dipole errors the orbit will be distorted. The beam position monitors (BPMs) will read a non zero orbit.
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.
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 RM at diamond
orbit reading at all BPMs
dipole corrector angle kick
Once the matrix R is known we can correct an orbit distortion by inverting the matrix
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 magnet are due to calibration errors in current vs field gradient, misalignment and rolls. A misaligned quadrupole adds a dipole kick.
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
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
We use the gradient error Twissparameterisation for M and M0
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
If the perturbation is distributed around the ring we have gradient error
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.
Multipolar expansion of the magnetic field inside a magnet avoid creating resonance conditions between the horizontal and the vertical tune
The on axis magnetic field can be expanded into multipolar components (dipole, quadrupole, sextupole, octupoles and higher orders)
Including higher order terms in the expansion of the magnetic field
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.
Nonlinear magnetic fields are introduced in the lattice (chromatic sextupoles)
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
x’ (chromatic sextupoles)
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
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
Turn 2 (chromatic sextupoles)
m = 5; n = 0; p =1
When the betatron tunes satisfy a resonance relation
the motion of the charged particle repeats itself periodically
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)
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
Qx = 1/5
Phenomenology of nonlinear motion (III) (chromatic sextupoles)
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.
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)
Nonlinear map up to third order as a truncated Taylor series
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
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.
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.
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
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).