Elaboración de un afiche “ayudamemoria”

The PAMELA lattice [29] is a non-scaling FFAG design concept which consists of two concentric rings (Fig. 2.16): An inner ring accelerates protons from 31 MeV to 250 MeV, and carbon ions from 8 MeV/u to 68 MeV/u; the outer ring is then used to accelerate the carbon ions extracted from the first ring to 400 MeV/u. Here we will focus on the inner ring only.

The inner ring consists of 12 FDF triplet cells, which, as with the radial sector scaling FFAG, provide alternating gradient focusing whilst bending the beam both forwards and backwards. However, there are several key differences between PAMELA and the FFAG at KEK, these are:

1. In PAMELA, the field profiles are described by a Taylor series expansion of the scaling law (Eq. 2.12) that has been truncated at the decapole term. PAMELA is a non-scaling FFAG, however the non-linear field terms that are present lead to a low chromaticity.

2. The horizontal phase advance per cell of the betatron motion within PAMELA is more thanπradians. This allows for a large scaling index to be used (k= 36.721), which reduces the orbit excursion during acceleration. The first two regions of stability for Hill’s equation are demonstrated for a FDF cell in Fig. 2.18 [45]. For the purpose of the plot, we represent the focusing term in Hill’s equation byg(θ),

Figure 2.16: The PAMELA design [29]. The inner ring would accelerate through the full range of treatment energies for protons and up to 68 MeV/u for carbon ions. The

outer ring would accelerate carbon ions from 68 to 400 MeV/u.

Duodecapole Decapole Octupole 50 100 150 200 250 8.55 8.60 8.65 8.70 8.75 EkHMeVL Qh (a) Horizontal Duodecapole Decapole Octupole 50 100 150 200 250 3.5 4.0 4.5 5.0 5.5 EkHMeVL Qv (b) Vertical

Figure 2.17: The PAMELA lattice has a low chromaticity due to the inclusion of higher order field terms in Eq. 2.12. The above shows the horizontal and vertical tune profiles when the series expansion is truncated at the octupole, decapole and

duodecapole term.

whereθ indicates the variation of the focusing strength with the azimuthal angle around the ring centre:

y′′+g(θ)y= 0, z′′₋g(θ)z= 0.

We give each of the magnets within the cell an aperture of one third of the total aperture of the cell (θcell), andg(θ) is given by:

g(θ) =      g0+g1 for 0< θ < 1₃θcell,

g0−g1 for 1₃θcell < θ < 2₃θcell,

-0.6-0.4-0.2 0.0 0.2 0.4 0.6

-2

-1

0

1

g

g

Figure 2.18: Stability plot for a FDF triplet. Conventionally the phase advance of a beam per period is between 0 and πradians; the focusing strengths for which this criteria is met for both the horizontal and vertical axes are marked in lighter green in the plot, and this region is referred to as the first stability region of Hill’s equation. PAMELA is designed to operate in the second stability region of Hill’s equation, which is marked in darker green; the phase advance per period in this region is between π

and 2π radians. The regions marked blue and red give stability respectively for the horizontal and vertical axis only.

whereg0 is the average focusing strength of the entire cell for the horizontal axis

(₋g0 for the vertical axis) andg1 gives the difference for each magnet between the

average and local focusing strength. The issue with using a larger scaling index and constructing a lattice that operates with a period horizontal phase advance greater thanπradians is that the increased non-linearity of the magnetic field with radius will significantly limit the dynamic aperture of the ring. A tracking study for the PAMELA lattice [46] found a normalised dynamic aperture of approximately 30πmm mrad for the error free lattice, whilst a previous study [45] found that the inclusion of an rms alignment error of 50µm on the magnets within a lattice operating in a similar regime as PAMELA led to a reduction in the dynamic aperture of approximately two thirds. In either case, the studies indicate that the dynamic aperture of PAMELA would be large enough for the proton therapy application.

3. The magnets are rectangular (rather than sector shaped), and are placed along a straight line within in a cell (rather than along an arc). It is intended that this will help to reduce construction costs and simplify the alignment process during machine commissioning.

A requirement assumed for the PAMELA design is that it should be possible to vary the extraction energy by 40 MeV in less than 1 second; one of the obstacles to realising this within a FFAG is the shift in the horizontal closed orbit with momentum. Ex- tracting the beam horizontally from the lattice using a kicker magnet and septum was investigated, however it was found that the requirements of the beam dynamics could not be met. In particular, extraction of the low energy treatment protons (70 MeV)

needs the kicker to produce an orbit separation of 10 cm, the non-linearity of the field leads to significant amplitude detuning of the bunch at the required amplitudes and ultimately to particle loss. If it were possible to obtain the required orbit separation, then a second problem of the angular dispersion at the extraction septum poses an ad- ditional challenge [47]. For these reasons, vertical extraction was considered to be the best solution for the PAMELA lattice.

Both kicker and resonance-based extraction methods have been investigated for vertical extraction from PAMELA [48]. In comparison with the kicker-based horizontal extrac- tion scheme, the kicker strength required for vertical extraction is much lower and it is simpler to match the post-extraction transport line to the dynamics of the bunch. The resonant extraction method investigated is based on particles crossing a half-integer ver- tical tune, with the half-integer tune set to coincide with the required extraction energy by varying the F/D ratio and the resonance driven by introducing quadrupole field er- rors. Upwards and downwards crossing of the resonance have been investigated, and it was found that downward crossing gives the output that is closest to being mono- energetic. This is explained after first considering a positive tune shift with increasing particle amplitude; for upward crossing of a half-integer tune, the exact energy at which a particle crosses the half-integer tune is determined by its amplitude, with large ampli- tude particles crossing the resonance and being extracted first. For downward crossing, it is the particles at the centre of the bunch that encounter the resonance first, once the amplitude of a particle that is initially at the centre of the bunch has grown to match that of a higher amplitude particle, then both particles will have approximately the same tune and their amplitudes will grow at a similar rate to one another. Yokoi [48] highlights that these extraction methods are difficult to control, and that although extraction happens over a number of turns, it is not necessarily slow extraction (when compared to synchrotron slow extraction that can take place over a few seconds).