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4 (13) We have the important and simple result that for stable motion the focal length of the quadrupole lenses in the FODO has to be larger than a quarter of the length of the cell. 3 Scaling optical parameters of a lattice cell After the discussion on stability in a lattice cell and the first estimates and calculations of the optical functions α,β,γ and φ, we would like to concentrate a little more on a detailed analysis of a FODO with these parameters. We can calculate the β-function that corresponds to the periodic solution –– provided we know the strength and length of the focusing elements in the cell.
In the upper part, the beta function in the horizontal (x) and vertical (y) plane is plotted (or the square root proportional to the beam size), in the middle part the position of the lattice elements is shown, and the lower part shows the so-called dispersion function in x and y. We will discuss these parameters in some detail in this paper. Fig. 2: Lattice and beam optics in a part of a typical high-energy accelerator. The curves in the upper part refer to the square root of the beta function, the lower part shows the dispersion function.
To make things more difficult, each line has a finite width, proportional to the strength of the imperfection which drives it. In some cases we must compensate the imperfections with correction multipoles to reduce this width. But before discussing resonances and their correction in another contribution to these proceedings , a word about chromaticity. 8. CHROMATICITY This steering of Q depends on careful regulation of quadrupole and dipole power supplies. In fact, much of the setting up time of a large accelerator is devoted to tune Q to be constant as the fields and energy rise.