John M. Jowett (links to my home page)
Accelerator Physics Group, SL Division, CERN, CH-1211 Geneva 23In the first part of the talk, I gave an introduction to the dynamics of electrons (or positrons) in storage rings. As distinct from other particles such as protons, the dynamics of electrons is characterised by the fact that they emit synchtrotron radiation when bent in magnetic fields.
Accordingly, I started with a review of the basic results from electrodynamics that give the essential properties of this radiation. Within the domain of parameters of typical storage rings, the frequency spectrum calculated from classical electrodynamics can be re-interpreted as the energy distribution of emitted photons. Since the photons are actually emitted at random with energies chosen randomly in this distribution, it can be used to calculated statistical properties such as the mean photon emission rate, or the mean and standard deviation of the photon energy. These properties can be used in two ways:
I then went on to review the coordinate system and Hamiltonian for a storage ring, using the ring azimuth s as independent variable. This was mainly intended to fix notation. To keep the presentation simple, I treated the simplest non-trivial case of a flat ring with upright (non-skew) quadrupoles, sextupoles and discrete RF cavities.
Radiation reaction forces can be added to the Hamiltonian equations of motion. These then become stochastic differential equations (technically, they are to be interpreted in the Stratonovich sense). In the primitive coordinates of a storage ring, they take a particularly symmetric form.
Since these equations are no longer Hamiltonian, Liouville's Theorem no longer applies. It is replaced by Robinson’s Theorem which can be proved very straightforwardly and generally from these equations of motion (the theorem is actually much more general than can be seen from the original derivation). It shows how the rate of compression of 6-dimensional phase space (or the damping) is related to the synchrotron radiation power.
In the case of LEP at 90 GeV, the horizontal component of the closed orbit reaches amplitudes of several mm. Since, moreover, the orbit has the opposite sign and comparable magnitude for the other beam, there is a substantial separation of orbits and differences of optics between two beams which must co-exist in the same ring.
Further canonical transformations can be constructed to find the normal modes of linear oscillations around the closed orbit. In the formulation given in this talk, the generating functions contain unspecified functions. Imposing natural conditions of decoupling between the modes, these functions turn out to be just the familiar beta and dispersion functions so commonly used in accelerator practice. In the simplest case the normal modes are just the usual betatron and synchrotron motions.
Since canonical transformations of the Hamiltonian parts of the equations of motion are so useful, it is useful to have corresponding rules for transforming the additional dissipative terms. The derivations of such rules using generating functions were sketched.
Inclusion of the radiation terms gives rise to radiation damping and quantum fluctuations in each of the normal modes. One of the main purposes was to show how all the parameters related to these (damping times, energy spread, emittance, etc.) can be derived consistently from the equations of motion.
For any stochastic equation of motion, an equivalent Fokker Planck equation gives the evolution of the particle distribution function in time. In the simplest cases, the equilibrium distribution is exponential in the normal mode actions. The mean values of these quantities are just the emittances.
In this section of the talk I briefly described how the radiation effects can be implemented in a particle tracking program (MAD) and showed a few examples of non-linear dynamics with radiation. There are several levels of inclusion of radiation effects in MAD. Experimenting with these has shown, for example, that the naive belief that particles are more stable when radiation damping is switched on is not true in general.
Note that a significant part of the talk was based on lectures that I gave at the US Particle Accelerator School in SLAC in 1985. In fact I used parts of the text of this reference as a quick way to make the transparencies for the middle part of the talk. Since I did not want to over-burden the talk with too much detail, you can also refer to this talk for further details of the formalism and basic theory.
Some of the examples covered in the last part of the talk have not been published anywhere else.
I would like to thank Helmut Mais and Sig Martin for the invitation to speak at this workshop. With some help from the ambience of the Physikzentrum Bad Honnef, they created a stimulating and congenial workshop. It was genuinely conducive to learning and exchange of information among people working in somewhat different fields yet sharing a common interest in mathematical physics.