Nonlinear effects in alternating-gradient synchrotrons

Abstract Nonlinear effects in alternating-gradient synchrotrons are responsible for resonances in the betatron-oscillation mechanism over and above those which arise in linear theory. These effects are here investigated by means of a perturbation procedure which leads directly to practical formulas. The larger part of the paper is given to “static” theory which ignores the acceleration process and concomitant synchrotron oscillation. Results are represented diagrammatically in three ways: by invariant contours in the phase plane; by “configuration diagrams” which are convenient transformations of the preceding diagrams; and by curves which relate the amplitudes of closed orbits to the characteristic phase. The resonances are associated with the following critical relations between the characteristic phases Θx, Θy, which denote the phase-change of radial and vertical oscillations, respectively, over one revolution: rxΘx + ryΘy = 2kπ, where rx, ry, k are integers. In an ideal machine, composed of N identical sections, the above resonances are restricted to values of k which are integral multiples of N; the remaining resonances must therefore be attributed to departures of the machine from its ideal form due to constructional inaccuracies. For this reason, an analysis of nonlinear effects is necessary for the evaluation of tolerances. The resonance with “coordinates” rx, ry, is due principally to fluctuations in the nth radial derivative of the magnetic field strength, ∂n H y/∂xn, where n = |rx| + |ry| − 1. The following general observations may be made: (i) The relation |rx| + |ry| ≤ 3 determines values of the characteristic phases in some neighborhood of which betatron oscillations of indefinitely small amplitude will grow to finite, possible large, amplitude. If |rx| + |ry ≥ 5, there is no such neighborhood. Resonances associated with |rx| + |ry| = 4 may have the characteristics of either of the preceding categories, depending upon the “quartic phase-independent term” Q, which is due principally to ∂n H y/∂x3, but there is reason to expect that in practice its properties will be those of the second category. Static theory therefore suggests that resonances for which |rx| + |ry| ≥ 4 may be ignored. (ii) Among the coupling resonances, for which rx, ry are both nonzero, those for which rx, ry have opposite signs lead only to an exchange of betatron-oscillating energy, which in alternating-gradient synchrotrons is harmless. (iii) Since deviations of the magnetic field which violate the (mirror) symmetry of the accelerator will be slight compared with those which are compatible with this symmetry, a resonance associated with an even value of ry and odd value of rx is more serious than that for which these values are interchanged. Calculation of the effect of fluctuations in the nonlinear contribution to the magnetic field may be based upon first-order perturbation theory. However, the consideration of a specific example shows that the combined influence of linear fluctuations with the average nonlinear contribution may be of comparable importance. Calculation of such effects involves second-order perturbation theory, formulas for which are derived in an appendix. Some attempt is made to consider the “dynamic” problem which takes account of the influence upon the betatron-oscillation mechanism of synchrotron oscillation and progressive saturation of the magnets. The two extremes of slow and rapid traversal of a resonance are discussed: the former should be appropriate to the possible crossing of low-order resonances near ejection due to magnet saturation; the latter should be appropriate to the inevitable crossing of high-order resonances during the complete acceleration cycle, and to the possible crossing of low-order resonances due to the large amplitude of synchroton oscillation near injection. Slow traversal of a resonance will give drastic expansion of at least part of the beam for at least one of the two directions, but the effect may be reduced by increasing Q. The effect of repeated rapid traversal may, under certain conditions, be approximated to a diffusion process which leads to progressive expansion of the entire beam; this effect is independent of Q and receives contributions from resonances of all orders. According to the simple theory of this paper, this effect is sufficiently serious to warrant more thorough investigation.