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EMMA was the first (and is currently the only) non-scaling FFAG to have been built. The aims of the EMMA project were to show that a non-scaling FFAG could operate successfully, and identify whether such machines could be suitable for applications in- cluding proton therapy and muon acceleration. The lattice design for EMMA is based on the truncation of the scaling FFAG law after the two linear terms, so that the vertical field varies radially as:

Bz =Bz,0 1 +k∆R R0 , (4.1) 61

the original EMMA design parameters. The integrated field gradient is given by the quadrupole gradient multiplied by the quadrupole effective length.

Configuration D/F doublet

Number of cells 42

D quad effective length 75.698 mm D quad offset from reference axis 36.048 mm D quad integrated field gradient 0.367 T

Short drift length 50.000 mm F quad effective length 58.782 mm F quad offset from reference axis 9.514 mm

F quad integrated field gradient -0.402 T Long drift length 210.000 mm

where Bz,0 gives the dipole field strength, k is the scaling index and R0 is a reference

radius. The resulting absence of higher order field components (e.g. sextupole) to correct chromaticity leads to the description of the machine as a linear non-scaling FFAG. The basis of the EMMA lattice is 42 DOFO cells (table 4.1), with the dipole fields that guide particles round the ring provided by introducing offsets to the transverse horizontal position of the quadrupoles that make up the cells.

The linear field design of the EMMA accelerator allows for a potentially large accep- tance (meaning that a high emittance bunch may be successfully injected onto a stable orbit), however there is also a large betatron tune range encountered during acceleration from 10 MeV/c to 20 MeV/c, and the beam will cross a number of integer tune values. If a beam crosses integer tune values quickly (i.e. through rapid acceleration of the beam), then the effects of field errors on the beam only add coherently for a relatively small number of turns, and the beam may be accelerated to the extraction momentum suc- cessfully [59]. Short acceleration cycles in a machine such as EMMA are also of interest when considering the acceleration of beams consisting of short lifetime particles such as muons.

Rapid acceleration from the injection momentum to the extraction momentum in EMMA is achieved by applying a novel acceleration method [60]. For reference, we first describe the acceleration of low energy protons within a synchrotron accelerator. The particles that make up the beam are injected into the synchrotron with some range of momenta that are distributed around a design injection momentum. A particle of the design momentum will arrive at the first rf cavity when the rf field is at a design phase. However, dispersion of the lattice (which affects the path length of a particle per turn in the accelerator) and a difference in particle speed means that a particle with some offset from the design momentum will arrive at the rf cavity either earlier or later than a design particle, and subsequently at a different phase of the rf field. Meanwhile, the frequency of the rf changes to ensure that a particle with the design momentum (referred to as a synchronous particle) arrives at the design phase (referred to as the synchronous

(a) 0.5 MV per turn. -Π 0 Π 10 12 14 16 18 20 ΦHradL p H MeV  c L (b) 1.5 MV per turn.

Figure 4.1: During acceleration, it is usually desirable to constrain particles within a stable area of longitudinal phase space, which is referred to as an rf bucket. The rf bucket is centred around a stable fixed point that corresponds to a particle having an orbital frequency that is a harmonic of the rf frequency. For EMMA, we see a low and high momentum rf bucket due to a parabolic orbital period with momentum, with the separatrices marked blue and green in Fig. 4.1a (a sample phase space trajectory is shown by a blue spot and black arrow). Given a sufficiently high accelerating voltage, the separatrices from the two rf buckets will cross and the serpentine channel opens

(Fig. (b)); in this case a particle will travel around both stable fixed points.

phase) of the rf every time the cavity is encountered. The synchronous phase is selected so that off-momentum particles are focused towards the synchronous momentum during acceleration, as a result particles can be confined to a region in longitudinal phase space that is referred to as an rf bucket (demonstrated in Fig. 4.1a). The size of the rf bucket is determined by the rf voltage and by the rate at which the rf phase changes per turn for off-momentum particles (the phase slip factor).

In EMMA, electrons have ultra-relativistic momenta, meaning that the change in particle velocity during acceleration is small. The key contribution to the change of orbital period with particle momentum is therefore the change in path length with momentum. In a radial sector FFAG, particles move to greater radii orbit with increasing momentum, but if the FFAG is also non-scaling, then the shape of particle orbit will change with momentum too. The change in orbit shape with momentum has allowed for the EMMA lattice to be optimised so that a plot of orbital period vs. momentum is parabolic with a small change in the period over the EMMA momentum range. Having a design that is near isochronous during acceleration reduces the phase slippage per turn of a particle with respect to the rf, which, for a given rf voltage, leads to an increased rf bucket size. A longitudinal phase space portrait is shown for the EMMA lattice in Fig. 4.1a for an accelerating voltage of 0.5 MV (summed over the 19 cavities within the lattice) and an rf frequency of 1.301 GHz. There are two sets of separatrices marked on

The red lines demonstrate contours of constant values of the longitudinal Hamiltonian, and we see that inside an rf bucket the contours form closed loops around a stable fixed point. A stable fixed point occurs when the rf frequency is some harmonic of the beam orbit frequency and the beam arrives at the cavity at an rf phase where the beam will see no voltage. In Fig. 4.1a we see stable fixed points at two different momenta, this is due to the parabolic orbital period, which leads to the rf frequency being at the 72nd _{harmonic of the orbital period for momenta on either side of the minimum of the}

parabola. An increase in the accelerating voltage leads to an increase in the size of the rf bucket around each stable fixed point; eventually the separatrix from the low momentum rf bucket crosses the separatrix from the high momentum rf bucket (Fig. 4.1b). In the region of overlap of the two rf buckets we see a new type of behaviour, with a particle in this region following a path in longitudinal phase space that takes the particle around both the low and high momentum stable fixed points. Acceleration along this phase space path is known as serpentine acceleration, and allows for acceleration of a beam that does not have an exactly isochronous orbital period through a large momentum range given a fixed rf frequency. In order for serpentine acceleration to be implemented successfully, it is necessary for the EMMA lattice to have a parabolic orbital period close to that obtained during optimisation of the design.

In addition to the quadrupole magnets, other components for controlling the beam include 17 vertical correctors and 19 rf cavities. Injection is achieved using a septum and two kicker magnets; the beam can be extracted using a similar set of components. Beam diagnostics tools within the EMMA ring include 82 beam position monitors (BPMs).