In practice
First, a set of two "pinger" magnets, the fields of which rise and fall in less time than it takes the electrons to travel round the ring, can deliver horizontal and vertical kicks respectively to the beam, displacing it with independently adjustable horizontal and vertical amplitudes. Second, beam-position monitors synchronized with the pinger magnets can measure the transverse centre of charge of the electron beam for each turn round the storage ring.
In one experiment, David Robin and Christoph Steier from the ALS and Jacques Laskar and Laurent Nadolski from the Bureau des Longitudes collected data for a set of 25 initial transverse momenta. They set the small-amplitude betatron tunes close to the setting for user operation and adjusted magnet parameters so that the storage ring was as close as possible to the ideal 12-fold symmetry.
Under these conditions they found that the frequency map contained two strongly excited resonances that are "unallowed". These do not appear in the frequency map for the numerical model of the ideal storage ring but do show up in the frequency map calculated using a machine model with realistic errors. It appears that small coupling errors in the storage-ring lattice broke the 12-fold periodicity sufficiently to excite the unallowed resonances.
The excellent agreement between the map that was based on experimental data and that which was obtained with calibrated numerical models leads the collaborators to propose the use of frequency map analysis as a tool to improve both numerical models and the behaviour of actual storage rings, including parameters that are important to synchrotron radiation users, such as lifetime and injection efficiency.
Further reading
H S Dumas and J Laskar 1993 Global dynamics and long-time stability in Hamiltonian systems via numerical frequency analysis Phys. Rev. Lett. 70 2975.
J Laskar 1989 A numerical experiment on the chaotic behaviour of the Solar System Nature 338 237.
J Laskar 1990 The chaotic motion of the solar system: a numerical estimate of the size of the chaotic zones Icarus 88 266.
Y Papaphilippou 1999 Frequency maps of LHC models Proceedings of the 1999 Particle Accelerator Conference (IEEE, New York) 1554-1556. See also http://accelconf.web.cern.ch/AccelConf/top-page.html.
D Robin et al. 2000 Global dynamics of the Advanced Light Source revealed through experimental frequency map analysis Phys. Rev. Lett. 85 558.