Analytical evaluation of the second order momentum compaction factor and comparison with MAD results

The second order momentum compaction factor α₁ is a critical lattice parameter for transition crossing in hadron synchrotrons and for the operation of quasi-isochronous storage rings, which have been proposed for free electron lasers, synchrotron light sources and recently for high luminosity e⁺e^- colliders. First the relation between the momentum compaction factor and the dispersion function is established, with the “wiggling effect” included. Then an analytical expression of α₁ is derived for an ideal FODO lattice by solving the differential equation for the dispersion function. A numerical calculation using MAD is performed to show excellent agreement with the analytical result. Finally, a more realistic example, the Fermilab Main Injector, is shown not far away from the ideal lattice