โ Scribed by E.D. Courant; H.S. Snyder
No coin nor oath required. For personal study only.
The equations of motion of the particles in a synchrotron in which the field gradient index n=&(rรB) Bร 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 n 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 2& x or 2& z or & x +& z is integral, where & x and & z are the frequencies of horizontal and vertical betatron oscillations, measured in units of the frequency 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 r& 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.
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