Issue 76

Beam Dynamics Newsletter

3.14 RACCAM: An Example of Spiral Sector Scaling FFA Technology

François Méot, Brookhaven National Laboratory C-AD, Upton, NY, USA.


The RACCAM project prototyped a spiral sector, scaling FFA dipole. It was the first of its kind and a major component in a rapid-cycling compact ring design for a multiple beam proton-therapy installation. A brief description introduces the project and the merits of spiral, scaling FFA technology. The article then focuses on the magnet design, field measurements and their use in beam dynamics simulations carried out to validate the design.

3.14.1 The RACCAM method

The RACCAM project (Recherche en ACCélérateurs et Applications Médicales) covered the period 2006-2009 and was funded by the French Agence Nationale de la Recherche  [1]. The RACCAM partnership comprised CEA Saclay and Grenoble CNRS-IN2P3/LPSC laboratories, the SIGMAPHI magnet company, oncologists from Grenoble university hospital, the AIMA cyclotron company at the MEDICYC anti-cancer clinic in Nice, and was later joined by IBA. A number of technical reports were produced in which details of the project and its technical achievements can be found  [2, 3]. The project owed much to long term collaborations with Prof. Y. Mori and his team at the Kyoto University Research Reactor Institute (KURRI, since renamed KURNS).

The concept of Fixed Field alternating gradient Accelerators (FFA) was developed in the 1950’s and 60’s during the early years of the strong focusing and phase stability studies  [4]. FFAs benefit nowadays from advancements in 3-D magnet design tools and high gradient RF technologies, and from 6-D beam dynamics simulations. They can call on extensive R&D and experimentation carried out in Japan with electron and proton machines  [5].

Drawing on experience at other labs that have looked at spiral FFA design or construction  [6, 7], RACCAM prototyped a normal-conducting spiral scaling FFA dipole as a sector of a 10-period ring with the potential for high dose delivery. The ring design was based on fixed-field, synchro-cyclotron style, rapid-cycling techniques, and had the potential for compactness using modern magnet technologies. The magnet design used OPERA for the 3-D field calculations and the Zgoubi ray-tracing code  [8] for 6-D beam dynamics validation. The magnet was fabricated and measured at SIGMAPHI. Field measurements were ultimately used to demonstrate constant tunes and large dynamical acceptance and so validate the magnet design and construction.

The different stages of the RACCAM development are addressed in what follows, with the aim of showing the merits and potential of rapid-cycling, spiral sector, scaling FFA technology.

Protontherapy application

A feature of a rapid-cycling FFA machine is that it can deliver a high dose rate, potentially far beyond 5 Gy.litre/minute, in a space-charge free regime. This beam delivery regime is well suited for slice-to-slice energy variation so allowing 3-D conformational irradiation. It allows bunch-to-bunch energy variation and thus 3-D tumor motion tracking. Other potential advantages include the use of non-pulsed power supplies, simple RF systems, simultaneous beam delivery to several treatment rooms and different particle species. In addition, a fixed field accelerator is associated with conservative construction methods and operation, high stability and reliability, low maintenance, and potential for low treatment cost.

With the aim of exploiting FFA potential, the most challenging issues are likely to be: (i) variable energy over a 100 ms time scale; (ii) dose rate of 5 Gy.litre/minute and beyond; (iii) bunch-to-voxel beam delivery method; (iv) fast bunch-to-bunch energy variation over an energy range, \( \Delta E \), of a few 10’s of MeV; (v) multiple port simultaneous extraction.

Table 1:  Parameters of the proton-therapy accelerator installation.

Energy, variable MeV 70 to 230
Injection energy, variable MeV 5.5 to 18
Average intensity nA \( >100 \)
Extraction efficiency % \( >95 \)
Extraction mode single-bunch
Irradiation mode bunch-to-voxel
Repetition Rate Hz \( > 100 \)
Number of beam extraction ports 1 to 5
Repetition rate per port Hz 20 to 100

The general specifications for the installation are summarized in Table 11. In terms of repetition rate and bunch filling, the parameters and properties on which the RACCAM study was based are the following.

  • - The reference irradiation volume considered is a \( 10\times 10 \times 10 \, \textrm {cm}^3 \) cube, comprising \( 20\times 20\times 20 \), \( 0.5\times 0.5 \times 0.5 \, \textrm {cm}^3 \) elementary voxels. The reference irradiation is thus performed by scanning a series of twenty, 5 mm thick slices, 400 voxels each.

  • - As a consequence, in a single-pass painting mode the repetition rate needs be 8000 voxels per minute, i.e. \( \sim \!133\, \)Hz.

  • - In a 10 cm depth spread-out Bragg peak distribution, the energy deposition is \( \sim \!0.07\, \)J per \( 10^{11} \) protons, independent of energy.

  • - The bunch charge needed is a maximum for the deepest slice, \( \sim \!20 \)% of the total in the present reference cubic volume. There it reaches \( \sim \!10^9 \) protons per bunch, for 5 Gy/minute in the 1 litre volume considered.

  • - The time necessary for a change of beam energy from one slice to the next needs to be small, much less than one minute. Referring to the experience at PSI’s superconducting medical cyclotron, a conservative \( 100\, \)millisecond time scale is considered. This suggested the prototype should be constructed with a laminated yoke.

1 Parameters of an “optimal medical proton ring” similar to those in Table1 were derived at the FFAG07 Workshop in Kyoto, see Ref.  [7],

A spiral FFA accelerator system

The principle of the RACCAM multi-beam delivery FFA is sketched in Fig. 1. The ensemble includes a \( \rm H^- \) variable extraction energy cyclotron, a short transfer line to the FFA ring and a multiturn injection system. The ring has two RF cavities for \( f_\mathrm {rep}>100 \) Hz repetition rate, and \( N=2\text { to }5 \) extraction ports that deliver beams simultaneously (at a rate of \( f_\mathrm {rep}/N \) at each port).


Figure 1: Layout of the RACCAM variable energy FFA ring, with its inner cy- clotron injection system, two cavities (red boxes) for high repetition rate and multiple beam extraction systems for simultaneous beam delivery to several treatment rooms.

Simultaneous beam delivery to several rooms

A detailed layout of a dedicated treatment center, using the multi-port simultaneous extraction system to serve several rooms, has been devised and is shown in Fig. 2. The symmetry of the building is adapted to the configuration of the accelerator and extraction system. Each beam line out of the FFA ring is directed to a treatment room, where beam can be delivered independently of the other rooms at a rate ensuring 2 Gy/minute delivery and beyond.

The FFA ring is located in the basement of the building. The beam lines have vertical bends so as to reach the level of the treatment rooms on the upper floor. Patient preparation rooms and other medical and administrative premises are located in the central area of the building, as well as between the treatment rooms.

The potential advantages of the multi-port polygonal arrangement for beam extraction and delivery, compared to a classical rectangular, single-extraction layout, include:

  • - potential for a greater number of treatments, Fig. 3;

  • - potential for more beam time in a single room;

  • - independent ion species delivery for radiobiology R&D, accelerator and beam developments;

  • - minimizing building and architectural footprint as well as distances between the various medical zones;

  • - minimizing the distances to be covered by patients and medical staff;

  • - reducing the man-power needed to operate the treatment center;

  • - lowering building construction and operation costs;

  • - improving architectural aspects such as lighting, radiation shielding, etc.

These considerations meet the objectives of reducing treatment cost, and improve the economic effectiveness of hadron-therapy so that it compares favourably with (X-ray) Intensity Modulated Radio-Therapy (IMRT). It is estimated that the cost of a treatment session is roughly one half of a conventional proton-therapy session, bringing it down to less than twice the cost of an RX session  [9].

The hospital and its accelerator installation gain flexibilty, and can, for example, target a specialized type of treatment, meaning a reduced energy range, common to all the rooms. Energy adjustments are thus easier and can even be based on a small scale degrader system. There can be a low energy installation - \( \sim \! 70 \) MeV in the case of eye treatment - or a higher, \( \rm \sim \! 180 \) MeV installation for head-neck treatment. Such a “specialized energy” hospital may be another advantage of multi-room, simultaneous delivery, and may bring additional reductions in construction and operating costs.


Figure 2: A \( 2\pi /N \)-symmetric proton-therapy center based on a multi-extraction spiral sector FFA ring. The ring is located beneath the treatment floor; its \( N=5 \) extraction ports deliver beams to each of the five treatment rooms simultaneously. The building is 58 m in diameter for a 230 MeV installation including gantries (a center specializing in eye tu- mor treatment for instance would have a smaller accelerator installation, simpler beam lines and less shielding).


Figure 3: Simultaneous beam delivery to multiple treatment rooms increases beam delivery efficiency (sessions per day) linearly with the number of rooms. The lines labelled RACCAM_1, _2 and _3 differ mostly by the time the patient spends in the treatment room for pre- and post-irradiation procedures, respectively 41, 34 and 22 minutes. By comparison, sequential beam delivery from a single extraction accelerator (“REF”, red curve) indicates a maximum of about 150 sessions per day, from 4 or more rooms.

3.14.2 RACCAM prototype spiral sector dipole

When spiral sector magnet studies started on RACCAM, various ways were seen to achieve the \( B\propto r^K \) scaling law. Coil shaping methods had been studied and implemented in the past  [6, 10], and would eventually also be studied in the context of this project  [11]. However RACCAM opted for an innovative gap-shaping technique for compactness and lower power consumption. This section briefly introduces the theoretical aspects and design principles used.

A spiral sector ring is characterized by a mid-plane field of the form  [12]

(1) \begin{equation} \label {EqSLawB} B(r)=B_0\,\left (\frac {r}{r_0}\right )^K\,\mathcal {F}\left
(\tan (\zeta )\ln \frac {r}{r_0}-N\theta \right ) \end{equation}

with \( \theta \) the azimuthal angle, \( r \) the local radius measured from the center of the ring (Fig. 4), \( K \) the field index, \( B_0 \) the field at an (arbitrary) reference radius \( r_0 \), \( \zeta \) the spiral angle, and \( N \) the number of cells. \( \mathcal {F} \) is essentially an AVF (Thomas focusing) style of azimuthal periodic form factor, featuring (possibly negative) field valleys between high field hills for a large flutter, as discussed below. A radial dependence of the field of the form \( B(r)/B_0 =(r/r_0)^K \) ensures zero chromaticity (horizontal tune independent of energy), a key property of scaling optics (where orbits scale with momentum), and constant vertical tune (ignoring fringe field constraints, addressed below). The RACCAM dipole acquires the \( r \)-dependence of the field from gap shaping, Fig. 5, based on a variable gap size of the form

(2) \begin{equation} \label {EqGap} g(r)\approx g_0(r_0/r)^\kappa . \end{equation}

The prototype sector dipole has \( g_0=4\, \)cm at the maximum orbit excursion \( r_0=3.46\, \)m (Table 2) and, from the OPERA optimization, \( \kappa =5.2 \approx K \).


Figure 4:  Diagram of a spiral sector FFA cell and its geometrical parameters, nomenclature as in Table 2. The origin of the polar system is at the center of the ring.



Figure 5:  Two-dimensional model of the upper half-yoke, gap and coil in OPERA. This 2-D study gives the proper gap shape to satisfy \( B(r)/B_0=(r/r_0)^K \) for a given K over the useful field region (namely, \( 2.9\leq R\leq 3.3 \) m, in the prototype dipole).


The resulting \( r \)-dependent momentum, the momentum compaction and the transition \( \gamma \) can be written, respectively,

\[ \frac {p(r)}{p_0}=\left (\frac {r}{r_0}\right )^{K+1},\qquad \alpha =\frac {\Delta r}{r}/\frac
{\Delta p}{p}=\frac 1{K+1},\qquad \gamma _\mathrm {tr}=\sqrt {1+K}\,. \]

The local radius \( r \) and curvature radius \( \rho \) (Fig. 4) are related by

(3) \begin{equation} \label {EqRRho}r\sin (A/2)=\rho \sin (\pi /N) \end{equation}

where \( A \) is the sector angle of the dipole. A value of \( N=10 \) was adopted in the RACCAM prototype approach; however a greater number would allow simultaneous beam delivery from more extraction systems.


Figure 6: Half-yoke and pole during the magnet construction and assembly at SIGMAPHI, 2008. The wide variation of the chamfer over the radial extent of the pole can be seen on its exit edge (left).



Figure 7: Field densities, from the OPERA 3-D model. The model is complete here: it includes variable chamfers and field clamps, both major ingredients in achieving constant vertical tunes over the 40 cm radial extent of the good field region \( 2.9 \leq R \leq 3.3 \) m. The center of the ring is to the left, the gap decreases to the right.

The logarithmic spiral field boundaries of the magnet (Fig. 6) satisfy \( r=r_0\exp (\theta /\tan \zeta ) \). The fringe fall-off is characterized by the flutter

\[ F=\frac {\overline {B^2}}{\overline {B}^2}-1\qquad \text {which tends to}\qquad \frac {r}{\rho
}-1 \]

in the case of a hard-edge model. The greater the flutter (closer to hard-edge), the greater the vertical focusing. A rough approximation to the horizontal and vertical tunes given by  [12, Eq. 5.2]

(4) \begin{equation} \label {EqTunes} Q_r \approx \sqrt {1+K},\qquad Q_y \approx \sqrt
{-K+F^2(1+2\tan ^2\zeta )} \end{equation}

is helpful as a first approach to understanding the respective effects of varying \( K \), \( \zeta \) or \( F \).

Table 2 summarizes parameter values in the case of the RACCAM dipole.

Table 2:  Parameters of the prototype spiral sector (Figs. 6- 9), and an FFA ring (Fig. 1).

Prototype gap shaping spiral sector
Deviation (\( 2\pi /N \)) deg 36
Reference radius (\( r_0 \)) m 3.46
Field index (\( K \)) 5.00
Spiral angle (\( \zeta \)) deg. 53.7
Sector angle (\( A \)) deg. 12.24
Packing factor (\( p\!f \)) 0.34
Principal FFA ring
Energy range, high MeV 70–180
Energy range, low MeV 6.3–70
Packing factor (\( p\!f \)) 0.34
Maximum field at \( r_0 \) T 1.7
Number of cells (\( N \)) 10
\( Q_r \) 2.76
\( Q_y \) for \( 15 \rightarrow 180 \) MeV \( 1.55\rightarrow 1.60 \)
Transition gamma (\( \gamma _\mathrm {tr} \)) 2.45
Orbit excursion m \( 2.794 \leq r \leq 3.460 \)

In the table “Energy range” indicates what is achievable during operation of the ring, from injection at \( E_{min} \) to extraction at \( E_{max} \). This is dependent on the field setting, with 1.7 T at \( r_0 \) being the “high” case. “Orbit excursion” stands for the radial span of the orbit, from \( E_{min} \) to \( E_{max} \).

The field index \( K \) has been taken large enough to limit the radial beam excursion \( r_\mathrm {xtr}-r_\mathrm {inj} \) from the injection to extraction orbits, which follows from

(5) \begin{equation} \label {EqDeltaRmag} r_\mathrm {xtr}-r_\mathrm {inj}=r_0\left (1-\left (\frac
{p_\mathrm {inj}}{p_\mathrm {xtr}}\right )^{1/(K+1)}\right )\,, \end{equation}

thus limiting the extent of the magnet. On the other hand, \( K \) has been taken small enough that the spiral angle \( \zeta \) remains below 55 degrees (\( K \) and \( \zeta \) act in opposing ways on the vertical tune, see Eq. (4)). There are three main reasons for this: (i) simplifying the magnet design and construction; (ii) allowing room between magnets in the ring; and (iii) yielding weak non-linear field components to get large enough dynamic aperture (DA). These various constraints and optimizations resulted in \( Q_r\approx 2.76 \) for all energies, while \( Q_y \) varies from 1.55 at injection to 1.60 at extraction.

A note on the vertical tune.

The wedge vertical focusing term in the linear approximation can be written \( y'/y_0 = \tan (\epsilon -\psi )/\rho \) with \( \rho \) the local curvature radius and \( \epsilon =2\pi (1-p\!f)/N\mp \zeta \), the upstream and downstream wedge angles. \( p\!f= \)magnetic length / orbit length is the packing factor. \( \psi \) is a correction term for the fringe field extent and is written

\[ \psi =I_1(r)\ \frac {\lambda (r)}{\rho (r)}\ \frac {1+\sin ^2\epsilon }{\cos \epsilon }\qquad
\textrm {with}\qquad I_1=\int \frac {B_y(s)(B_0-B_y(s))}{B_0^2}\ \frac {ds}{\lambda }\,. \]

Since the (\( r \)-dependent) fringe field extent \( \lambda (r) \) is proportional to the gap height \( g(r) \) (Eq. (2)), and given that \( \rho \) and \( r \) are proportional to one another (Eq. (3)), the vertical focusing is \( r \)-dependent and thus so is the vertical tune \( Q_y \). The role of the field clamps in the prototype (Figs. 7 and 9), which decrease the fringe extent in the large gap region, and of the variable width chamfer (Fig. 6) that increases the fringe extent in the smaller gap region, is to compensate that effect and substantially reduce the variation of \( Q_y \) to \( \Delta Q_y\lesssim 0.05 \) (Table 2) over the full orbit excursion in the ring (\( 2.794\leq r\leq 3.460 \)).


Figure 8: RACCAM spiral dipole during assembly, field clamps not yet mounted.



Figure 9: RACCAM dipole during Hall probe measurements using an XYZ ta- ble. The field clamps along the coils (hidden) are visible on the right. Their presence helps reduce the fringe field extent in the large gap region.

The electro-mechanical parameters of the fabricated prototype are summarized in Table 3. Note that the “Good field region” does not cover the full orbit excursion in the ring; this is for cost saving purposes.

Table 3: Electro-mechanical parameters of the prototype spiral sector dipole.

Yoke shape Parallelepiped
Lamination thickness (mm) 1.5
Gap shape \( \propto 1/r^{\kappa }, \ \kappa \approx 5.2 \)
Gap at 3.46 m (cm) 4
Gap at 2.794 m (cm) 11.6
Overall dimension L\( \times \) W\( \times \) H (mm) 2913 \( \times \) 579 \( \times \) 1230
Good field region (m) \( 2.9 \leq r \leq 3.3 \)
Total weight of magnet (t) 18
PS voltage (V) 159
PS current (180 MeV operation) (A) 200
Total water flow (litres/min) 12.13
Water temperature, in/out (\( ^\circ \)C) 24/44
3.14.3 Field measurements

The RACCAM magnet prototype has been designed, constructed and measured to validate the gap shaping method, and the computation and fabrication methods, including thorough comparison of the field derived both. With that in mind, a series of 3D field map Hall probe measurements were performed, using an XYZ table, Fig. 9, over a total period of 600 hours  [13, 14].


Figure 10: Typical measurements: \( B_y(\theta ) \) at constant radius, on 5 different radii across the good field region.

Mid-plane field measurements have been performed along arcs at constant radius (with center at the origin of the polar frame, Fig. 4) where the field is expected to be constant in the body of the magnet, according to the scaling law, Eq. (1). The azimuthal step of the mesh was 0.2\( ^\circ \); the radial values considered were \( R_0\pm 55 \) mm in steps of 11 mm (11 arcs) with \( R_0= 2750,\ 2900, \ 3125,\ 3300 \) and \( 3450 \) mm so ensuring coverage of the good field region of the prototype (2.9 to 3.3 m), Fig. 10.

The measurements were performed at three different intensities:

  • - \( I_\mathrm {max}=225 \) A yielding \( B(r_0)=1.933 \) T at the extraction radius \( r_0=3.46 \) m, a theoretical extraction energy of 227 MeV, well beyond the 180 MeV RACCAM scope;

  • - \( I80 = 0.8\times I_\mathrm {max}=180 \) A yielding \( B(r_0)=1.606 \) T, extraction energy 162 MeV; and

  • - \( I60 = 0.6\times I_\mathrm {max}=135 \) A yielding \( B(r_0)=1.227 \) T, extraction energy 98 MeV.

These detailed measurements allowed a series of cross-checks against values expected from a combination of the OPERA field computations, theoretical modeling of the spiral sector field  [2, 3, 15], the \( r \)-dependence of the field and the integrated field, the field index \( K \) and the spiral angle along the entrance and exit field boundaries. The measurements were considered sufficiently comprehensive to be used to validate the magnet design and fabrication; details can be found in Refs.  [13, 14].

3.14.4 Validation by beam dynamics

The field measurements validate the design to the extent that fields and parameters derived from the fields are in accord with OPERA design simulations, or, where the agreement is not satisfactory, the reasons can be understood. The latter included effects such as saturation observed at the maximum experimental current, \( I=225 \) A, which yields 1.933 T at \( R=3.46 \) m, a value 0.2 T beyond the nominal design value. Nevertheless it was decided that further tests of a beam dynamics nature were necessary for completeness:

  • - The design is expected to yield constant tunes over the good field region, and at the various operating energies, as a result of the variable chamfer and clamp plate methods. The results are illustrated in Fig. 11,

  • - From the OPERA design studies, the dynamic aperture of the ring is expected to be of the order of \( 1\,\pi \)mm horizontally and \( 0.2\,\pi \)mm vertically, normalized.(Note that this is well beyond what medical proton beams actually require). This is illustrated in Fig. 12 and in Table 4.

These beam dynamics simulations have been performed using measured 2-D mid-plane field maps. They are summarized here, but more detail can be found in Ref.  [14].


(a) Horizontal tune \( Q_r \) derived from OPERA data, with currents in the 120-200 A range. At 200 A the OPERA model field is 1.7 T at R=3.46 m.



(b) Vertical tune \( Q_y \) derived from measured fields at 135, 180 and 225 A. At 225 A the measured field is 1.933 T at R=3.46 m.

Figure 11:  Horizontal and vertical tunes from tracking in 2-D field maps. The horizontal motion is found to be unstable at 225 A in the R=330 cm region. This is a conse- quence of \( K\approx 4.6 \) (compared to the \( K=5 \) design value) due to saturation  [13, 14]. However, this happens at a field regime far off the nominal design.


(a) Horizontal



(b) Vertical

Figure 12: Maximum stable motion invariant in the \( R=3125\, \)mm region. Results from tracking in measured 2-D field maps (a) horizontal, either in the absence of vertical motion, or with small initial vertical motion (inner invariant); (b) corresponding results in the vertical plane.

Table 4:  Dynamic apertures (DAx, DAy, horizontal and vertical, respectively) from 1000-turn tracking, using field maps. DAx (respectively DAy) is the area delimited by the largest horizontal (resp. vertical) invariant as shown in Fig. 12a (resp. Fig. 12b).

From measured From OPERA
field maps 3D field maps
\( R \) region E DAx DAy DAx DAy
(mm) (MeV) (\( \pi \mu \)m)
Maximal current (\( B_0=1.933 \) T)
2900 38.0 1800 900
3125 86.5 2600 800
3300 156 5500 1500
\( 80\%I_{max} \) (\( B_0=1.606 \) T) (\( B_0=1.7 \) T)
2900 15 4000 1500 2500 900
3125 35.9 1500 1200 2900 1000
3300 67.3 1700 1400 3500 950
\( 60\%I_{max} \) (\( B_0=1.227 \) T)
2900 15 1200 900
3125 35.9 1200 900
3300 67.3 2200 900

Table 4 shows that the vertical DA from the measured field maps (\( 1200\sim 1500\,\pi \mu \)m, vertical) is larger than expected from the OPERA field map (\( 900\sim 1000\,\pi \mu \)m). This indicates that a greater \( K \) could be a considered (which would decrease the vertical DA), so yielding smaller radial excursion over the acceleration range (Eq. (5)) and therefore a smaller magnet.


The RACCAM project opted for a gap shaping method to design a spiral sector, scaling FFA field. The magnet was prototyped and measured. It was the first of its kind and delivered the expected performance.

This technology has the potential for compactness because it allows a high packing factor, which is a consequence of the short drifts, a characteristic of the strong focusing feature of FFA optics. A more compact magnet than the RACCAM prototype, with similar 230 MeV reach, could be achieved based on different technologies (permanent magnets, super-ferric magnets)  [16], and possibly with greater field index \( K \) for smaller radial beam excursions.

  • [1]  S. Antoine et al., “Principal design of a protontherapy, rapid-cycling, variable energy spiral FFAG”, NIM A 602 (2009) 293-305.

  • [2]  J. Fourrier, “Les accélérateurs à champ fixe et gradient alterné FFAG et leur application médicale en protonthérapie”, PhD Thesis, IN2P3/LPSC and J. Fourier University, Grenoble, Oct. 2008.

  • [3]  T. Planche et al., “Design of a prototype gap shaping spiral dipole for a variable energy protontherapy FFAG”, NIM A 604 (2009) 435–442.

  • [4]  F.T. Cole, “A memoir of the MURA years”, April, 1994.

  • [5]  FFAG Accelerators, ICFA Beam Dynamic Newsletter, No. 43, Eds. C.R. Prior, W. Chou (2007), pp. 15-156.

  • [6]  A. Osanai et al., “Study of integer betatron resonance crossing in scaling FFAG accelerator”, TH6PFP079, Proceedings of PAC09, Vancouver, BC, Canada;

  • [7]  A link to the FFAG workshops series :

  • [8]  F. Méot, “The ray-tracing code Zgoubi - Status”, NIM A 767 (2013).

  • [9]  Ch. Mazzara, “Etude medico-economique d’un centre de protonthérapie equipé d’un accélérateur FFAG à extraction multiple”, Rapport de stage Master2 de physique médicale, IN2P3/LPSC and J. Fourier University, Grenoble (2010).

  • [10]  See for instance, K.R. Symon et al., “Fixed-Field Alternating Gradient Particle Accelerators”, Phys. Rev. 103 6 (Sept. 15, 1956).

  • [11]  D. Neuvéglise and F. Méot, “An Alternative Design for the RACCAM Magnet with Distributed conductors”, FR5REP095, Proceedings of PAC09, Vancouver, BC, Canada;

  • [12]  F.T. Cole, “MARK V expanded equations of motion”, MURA/FTC-3, 19 Jan. 1956.

  • [13]  M.-J. Leray et al., “Magnetic measurements of the RACCAM prototype FFAG dipole”, Proc. PAC 09 Conf., Vancouver (2009);

  • [14]  S. Antoine et al., et al., “Tracking periodic parameters in the measured magnetic field maps of a spiral FFAG”, Proc. PAC 09 Conf., Vancouver (2009);

  • [15]  J. Fourrier, F. Martinache, F. Méot, J. Pasternak, “Spiral FFAG lattice design tools, application to 6-D tracking in a proton-therapy class lattice”, NIM A 589 (2008) 133-142.

  • [16]  B. Qin, Y. Mori, “Compact superferric FFAG accelerators for medium energy hadron applications”, NIM A, Volume 648, Issue 1, 21 August 2011, Pages 28-34.