Theory of the alternating-gradient synchrotron (Reprinted from Annals of Physics, vol 3, pg 1-48, 1958)

The equations of motion of the particles in a synchrotron in which the field gradient index n = - (r/b)partial derivative B/partial derivative r varies along the equilibrium orbit are examined on the basis of the linear approximation. It is shown that if n alternates rapidly between large positive and large negative values, the stability of both radial and vertical oscillations can be greatly increased compared to conventional accelerators in which ir is azimuthally constant and must lie between 0 and 1. Thus aperture requirements are reduced. For practical designs. the improvement is limited by the effects of constructional errors. these lead to resonance excitation of oscillations and consequent instability if 2v(x), or 2v(x), or v(x) + v(z) is integral, where v(x) and v(z) the frequencies of horizontal and vertical betatron oscillations. measured in units of the fnqucncy of revolution. The mechanism of phase stability is essentially the same as in a conventional synchrotron. but the radial amplitude of synchrotron oscillations is reduced substantially. Furthermore, at a "transition energy" E-1 approximate to v(x)Mc(2) the stable and unstable equilibrium phases exchange roles. necessitating a jump in the phase of the radiofrequency accelerating voltage. Calculations indicate that the manner in which this jump is performed is not very critical. (C) 1958 Academic Press.