Novel Constant-frequency Acceleration Technique for Non-scaling Muon Ffags

There is a resurgence of interest in Fixed Field Alternating Gradient (FFAG) accelerators, both in the traditional “scaling” and in novel “non-scaling” machines. We focus on the variable-tune linear-field FFAG which offers very high momentum compaction. Muons require fast acceleration to avoid decay losses; this necessitates using superconducting magnets and cavities whose field and frequency are fixed. Fixed-field acceleration implies the orbit unavoidably changes with energy; and the change in path length results in phase slip of particles relative to a fixedfrequency waveform. Nonetheless, depending on the number and location of the fixed points of motion, acceleration is possible for a limited number of turns, during which the beam crosses back and forth the crest. This is facilitated by a serpentine channel extending from injection to extraction energy when a threshold value of voltage is exceeded. Emphasis is given to quadratic dependence of path length on momentum as occurs in the nonscaling FFAGs proposed for rapid acceleration of muons. Finally, mention is made of the relevance of fixed points to longitudinal motion in an imperfectly isochronous cyclotron. SYNCHRONOUS ACCELERATION Acceleration of charged particles by electrostatic fields (Cockcroft-Walton and Van de Graaf) is limited to about 10 MeV/charge by field breakdown (arcing, dark current, etc) and the single passage through the potential difference. For larger energy gain, it is essential to employ a time varying field either to allow repeated passage through the same accelerating structure and EM-field, as in a cyclic accelerator, or through consecutive structures and ground planes as in the linear accelerator. In both cases, it is the time-varying magnetic field which overcomes the limitation inherent in ∮ E · ds = 0 for static fields. Because electrically resonant cavities are simple to build and excite, EM-fields with a sinusoidal time variation are universally employed. Thus, maintaining synchronism between the charged particles and the oscillating fields becomes of paramount importance if deceleration is to be avoided. Moreover, because the breakdown field level and energy density increases with frequency there has been a historical tendency to contemplate ever higher frequency, and this contributes to the difficulty of sychronism. The phase of the accelerating waveform is determined by the frequency and the particle arrival time, which in turn ∗TRIUMF receives federal funding via a contribution agreement through the National Research Council of Canada depends on speed and path length. In non-relativistic linacs such as the drift-tube machines (Wideroe and Alvarez), path length is used deliberately to compensate the increasing particle speed. In relativistic linacs the particle beam rides the crest of the field at constant (light) speed in travelling wave structures. In the absence of magnetic bends the dispersion in path length is very small. In both cases, dispersion in arrival times is periodically compensated by disperion in velocity (and visa vera) via synchrotron oscillations which occur when the beam is off-crest. This “phase stability” (Veskler and McMillan) is much less important in relativistic linacs. In cyclic machines too, synchronism depends on the interplay of frequency, speed and path length; but there is a wider scope of machines to consider. In the isochronous cyclotron, the increasing length of the spiral orbit is adjusted to keep synchronism with the rf field in the dees and the beam rides the crest of the wave; beam delivery is c.w. In the synchrotron, the radio-frequency is swept to match the revolution period of the centroid and the head and tail are periodically interchanged by synchrotron oscillations. In the classical MURA1 FFAG[1, 2], the central orbit spirals (in a geometrically self-similar fashion) but is not isochronous and so the rf is swept. In distinction to the synchrotron where the repetition rate is limited to of order hertz by the ramping of magnetic fields, the fixed-field accelerator may sweep the rf at repetion rates up to order kilo-hertz and thereby enjoys, in principle, a two orders of magnitude advantage in time-averaged beam current. Throughout the previous compendia of accelerators two alternate means of arranging synchronism are used. (I) There is (almost) no net arrival-time variation and acceleration proceeds on-crest, as in the ultra-relativistic linacs (where there is no path-length variation) and in the isochronous AVF-cyclotron where (for the latter) spiral path length is adjusted to exactly compensate varying orbit speed. (II) Speed or path-length variation of the beam centroid is compensated by rf sweeping, as in the synchrotron and scaling FFAG respectively. In this second case, the bunch is confined (despite a small spread of momenta and arrival-time deviations from the centroid) by synchrotron oscillations which derive from a deliberate arrangement of near-linear variation of arrival time versus momentum deviation and near-linear variation of energy increment with arrival time relative to the centroid. Of course “arrival time” depends on competition between speed and path length. For the remainder of this text, we shall confine our discussion to cyclic machines. 1The Midwestern Universities Research Association. Both methods of maintaining synchronism between particle beam and the oscillating EM-field have limitations. As is widely appreciated[3], developing a magnetic field shape which simultaneously provides isochronous orbits and resonance-free transverse focusing over a large momentum range is (probably) not possible because tune ν r = γ = E/m0c in cyclotrons. The problem is alleviated somewhat by having separated turns and employing field correction (via trim coils) on different orbits. In its extreme form the concept of separated turns leads to employing separate arcs of magnets for each turn as in the recirculating linear accelerator (RLA) – which is limited by the cost and complexity of its arcs and switchyards. The scaling FFAG[1] and synchrotron adopt the approach of constant transerse tunes and non-isochronous orbits. Though it has a large longitudinal acceptance, the transverse acceptance of the former is somewhat limited by the non-linear field profile and resonances. When operated d.c. the synchrotron lattice yields a pitifully small longitudinal acceptance due to chromaticity and horizontal dispersion. Notice, though strongly influenced by cost and technical difficulty, the matter of c.w. versus pulsed operation is a matter choice; in most cases lack of isochronism can be overcome by “brute force” instead of frequency sweeping. If the energy increment per turn is large enough, then it will be possible to achieve the final energy before an accumlated phase slip of π/2 leads to deceleration. Indeed this idea was used in Lawrence’s first cyclotron and is utilized inadvertently in some relativistic linacs where less than one synchrotron oscillation occurs during acceleration. Almost isochronous FFAG Against this historical backdrop of synchronous accelerators, the variable-tune non-scaling FFAG accelerator[4, 5] has recently emerged[8, 10] on the world stage in the context of fast muon acceleration for a Neutrino Factory and/or Muon Collider[13]. For this application, large 6dimensional acceptance is required by the diffuse beam from the muon production target; and so short is the muon decay time that the rapid on-crest acceleration is envisaged to take no longer than a few turns. In this extreme regime it is possible for the particle beam in a linear magnetic lattice to cross integer and half-integer transverse resonances without significant degredation. On the few-turn time scale, the radio frequency cannot be other than fixed; and the machine could in principle be operated c.w. Ideally such a machine would be isochronous, but this is not possible over the ±50% ∆p/p momentum range intended. The linear dependence of pathlength on momentum is set to zero, leaving the second order (quadratic) term to dominate the longitudinal dynamics of this almost isochronous FFAG. This leads to an asynchronous-type acceleration in which the beam centroid crosses back and forth and back the crest of the sinusoidal waveform during the transit from injection to extraction. Asynchronous acceleration is best understood in terms of the longitudinal phase space and its number and nature of the fixed points of motion. Ilucidating the nature of this motion leads to a general principle for acceleration over a range spanning multiple fixed points: the rf voltage must exceed the critical value to link the unstable fixed points in a zig-zag ladder of straight line segments. The direction of phase slip reverses at each fixed point, and so the criterion is simply that the voltage be large enough that another fixed point be encountered before a π phase slip has accumulated. Beyond the critical voltage value, a serpentine (possibly narrow) channel extending between injection and extraction energy (and back) opens up and the longitudinal acceptance rises from zero. This principle was employed[12] to give analytic criteria for the opening of acceleration channels in systems with parabolic, cubic and quartic dependence of path length on momentum; that is 2, 3 and 4 fixed points, respectively, for each π of rf phase. QUADRATIC TOF DEPENDENCE The case of quadratic path length dependence on momentum is important since it corresponds to that of the non-scalling FFAG type accelerator. Let the time of flight (ToF) range per cell be ∆T over the energy range ∆E, and the peak energy increment per cell be δE. Let the index n denote iteration number, En be the particle energy and tn, Tn be the absolute and relative arrival times, respectively. Let τ0 be the cell traversal time at the reference energy Er. Then tn = Tn + nτ0. The longitudinal motion in the variable-tune non-scaling FFAG accelerator may be modelled by the following simple difference equations: En+1 = En + δE cos(ωTn) (1) Tn+1 = Tn + 4(En+1 − Ē)∆T/∆E − δT2 . (2) Here Ē = (Ê + Ě)/2, ∆E = Ê − Ě and ∆T = δT1 + δT2. The time slip δT2 represents the fact that the radio-frequency is synchronous with the orbital period at Er, which is not ncessarily equal to the mean energy Ē. These equations have served as a basis for analytic and numerical studies for some time[6, 7]. Supplementary equations to describe effects of beam loading of the rf cavity have been introduced[11], but are not considered here. We introduce dimensionless variables: x = ωT and y = (E− Ē)/∆E; and dimensionless parameters: s ≡ nω∆T , a ≡ (δE/∆E)/(ω∆T ), b ≡ (δT2/∆T ); and approximate the motion by differential equations: x′ ≡ dx/ds = (2y)−b and y′ ≡ dy/ds = a cosx . (3) The injection and extraction energies Ě, Ê correspond to y = ∓1/2, respectively. The reference energy is the solution of Tn+1 = Tn, namely Er = Ē ± (∆E/2) √ b where the ratio b may take any value between 0 and 1. Choice of operating point The doublet (a, b) is the key parameter of the system. In general, the choice of operating point will depend not only on acceleration range (y̌ to ŷ) and acceptance, but also up on a compromise between dwell time (i.e. decay losses for muons), acceleration efficiency, and dispersion of arrival time (which leads to nonlinear emittance distortion); and we must consider all four quantities. Matching of the input beam depends also on (a, b). The half-period τ , the dwell time from low to high momentum is of significance particularly for decay losses.