Issue 76

Beam Dynamics Newsletter

3.6 FFA-based ISIS Upgrade Options

Shinji Machida1, STFC Rutherford Appleton Laboratory, United Kingdom.

1 On behalf of the FFA ISIS accelerator physics design team


Among the various accelerator upgrade options being studied by the ISIS neutron and muon source group at RAL, the design of a Fixed Field alternating gradient (FFA) accelerator  [1–3] is being developed. An FFA has several advantages for the proton driver of a spallation neutron source mainly because of the constant field of the main lattice magnets. First, it can deliver high average beam current. This is possible because the acceleration pattern is solely determined by the RF voltage and frequency, not by the kind of magnetic field ramping found in synchrotrons. Secondly, its energy efficiency can be high. DC magnets can easily be designed with superconducting technology or permanent magnets might even be used. Thirdly, an FFA allows a flexible time structure for neutron users. Depending on the type of experiment, the range of the neutron energy spectrum and therefore the time period for measurements are different. Some users prefer 10 Hz operation and others require 50 Hz. The possibility of a high repetition rate from an FFA accelerator could be used to advantage by delivering pulsed beams to multiple target stations at different rates so that neutrons could be generated with a variety of time structures.

The idea of using an FFA as a proton driver for spallation neutrons is not new  [4–8]. In the 1980s, there was a proposal by Argonne National Laboratory (ANL) called ASPUN. A similar machine was considered as an afterburner of KENS at KEK. In the 1990s, as an option for a short pulse version of the European Spallation Source, an FFA was designed by Jülich. Unfortunately none of these proposals went further than paper studies.

A major revival of the FFA accelerator concept took place in 1999 when the first proton FFA was constructed in Japan and 1 ms acceleration was demonstrated  [9]. This had a big impact on the high current proton accelerator community. Many advantages were foreseen for FFAs over conventional accelerator architectures for a range of projects, including future spallation neutron sources, and these were studied in detail in the projects that followed  [10–12]. Traditionally, ever since the first generation of spallation facilities like IPNS at ANL, KENS at KEK, ISIS at RAL and LANSCE at Los Alamos, driving accelerators for neutron production have been based either on rapid cycling synchrotrons (RCS) or full energy linacs plus an accumulator ring (AR) to compress the long linac beam pulse to a microsecond. No one can deny that a great deal of R&D is needed if FFAs are to be used for these purposes, but there is certainly the potential for FFAs to make a breakthrough in this area of application, and in particular for an ISIS upgrade.

3.6.1 Accelerator choice

Discussions with the neutron user community indicate a requirement for a pulsed neutron source in the future to have a similar time structure to ISIS (10 and 50 Hz) but with higher beam power. Table 1 shows the minimum requirements requested.

Table 1: Neutron users’ requirements.

Beam power 1.25 MW
Repetition rate 10 to 40 Hz
Number of target stations preferably 3

The main parameters of the proton driver meeting these requirements are those shown in Table 2

Table 2: Main accelerator parameters.

Kinetic energy 1.2 GeV
Average beam current 1.04 mA
Repetition rate 90 Hz

Our first attempt is to design a new proton driver to fit into the existing ISIS tunnel even though it constrains the radius to around 24 m. The tunnel was initially constructed for the weak focusing synchrotron NIMROD, and there is plenty of width. As an injector, a 0.4-0.5 GeV superconducting linac is assumed. With 0.4 GeV injection energy, the momentum ratio, injection to extraction, will be a factor of 2. On the other hand, a beam energy of 0.5 GeV is known to be the optimum for muon production. A beamline from the injector to the muon target is likely therefore be considered if there is sufficient demand.

We have considered two options for the ring lattice: one is a horizontal orbit excursion FFA and the other is an FFA in which the orbit moves vertically (vFFA). The former is based on the so-called DF spiral design proposed by the author  [13]. The latter is more challenging because this kind of FFA has never been built before although some proposals exist  [14, 15], most recently by Brooks  [16]. Below we give details of both designs.

DF spiral

Using edge angles with respect to the beam orbits for focusing in one plane is a promising idea  [17]. Designing a lattice with spiral FFA magnets with constant edge angle independent of the beam momentum eliminates reverse bending magnets and results in a smaller circumference. This means, however, that there is no means of adjusting focusing in the transverse plane. The field gradient of the main magnets could be adjusted if trim (correction) coils are attached, but this is not enough to explore the whole tune space. This could be a serious issue because it is hard to fix the transverse tune precisely at the design stage of an accelerator lattice, especially for high current proton accelerators where the tune depends on the beam current.


Figure 1: FFA-based ISIS upgrade lattice with 25 cells. Parameters are given in Table 3.

A DF spiral lattice incorporates normal as well as reverse bending magnets with a spiral edge angle. Compared with a conventional spiral sector FFA, the circumference tends to be larger due to the reverse bend magnets. However, the flexibility introduced by reverse bends cannot be overlooked. It is essential for the initial commissioning to find the best working point in tune space for day-to-day operation afterwards; this is a well known fact in synchrotron operation. More details on the design and dynamics are discussed in  [13].

Table 3 shows the parameters identified for an upgrade lattice for ISIS.

Table 3: Parameters of DF spiral 1.2 GeV FFA.

Kinetic energy 0.4 (0.5) - 1.2 GeV
Reference radius 24 m
Number of cell 25
Magnet longitudinal length (\( Bd \), \( Bf \)) (0.60, 1.21) m
Packing factor 0.35
Straight section 3.58 m
Spiral angle 62 degrees
\( k \) index 20.6
Ratio \( Bd \)/\( Bf \) strength -0.47
Orbit excursion 0.8 m
Cell tune (H, V) (0.2076, 0.2096)
Ring tune (H, V) (5.19, 5.24)
Transition gamma 4.6

The number of cells was chosen as 25 so that a systematic 5th order resonance coincides with an integer. Space charge driven resonances at a quarter integer prohibit an operating tune just above a quarter integer. A cell number that is a multiple of 5 gives the largest resonance-free space between an integer and a quarter integer. As for cell tune, almost equal horizontal and vertical tunes are chosen because this is the choice of the high current accelerators built most recently, e.g. SNS and J-PARC. The spiral angle around 60 degrees approximates to the upper limit from an engineering point of view.

Figure 1 shows the top view of the DF spiral lattice. Figure 2 shows the beta function at the injection momentum and Fig. 3 shows the vertical magnetic field strength along the orbit at the injection and extraction momenta.


Figure 2: Beta functions at injection energy (0.4 GeV).



Figure 3: Vertical magnetic field along the orbit at injection and extraction mo- menta. The horizontal axis shifts depending on its momentum due to the spiral angle.


With main magnets whose vertical field strength increases (or decreases) in a vertical direction with proper restoring forces in transverse directions, the equilibrium orbit for different beam momenta shifts vertically. One of the nice features of this arrangement is that the orbit radius is constant, like synchrotrons. Not only that, the horizontal dispersion function is zero and the momentum compaction factor becomes zero, meaning the transition energy is infinite. In that respect, it is like a linac where path length is independent of beam momentum. There is another advantage in the vFFA concept, and that is the separation of the scaling property from the geometrical arrangement of the lattice footprint. In principle, the ring could have any shape and it would still be possible to maintain a scaling property as long as the vertical magnetic field satisfies the design shape of scaling magnets, (q.v.). A simple rectangular shape for the main magnets and the coil geometry is another advantage. A negative curved coil, which is inevitable in the spiral magnet, can be avoided.

We have first examined a test ring lattice as a prototype of a larger vFFA ring designed for the ISIS upgrade. The test ring will take the 3 MeV beams from RAL’s R&D injector, FETS,  [18] and accelerate them to around 30 MeV. Our aim is to demonstrate the vFFA concept experimentally for the first time and establish a design procedure. The rest of the main parameters are given in Table 4.

Table 4: Parameters of test ring vFFA.

Kinetic energy 3 - 27 MeV
Reference radius 3.9789 m
Number of cells 10
Magnet longitudinal length (\( Bd \), \( Bf \)) (0.5, 0.5) m
Packing factor 0.40
Straight section 0.75 m
\( m \) index 2.1 \( \mathrm {m}^{-1} \)
Ratio \( Bd \)/\( Bf \) -0.24
Orbit excursion 0.6 m
Cell tune (H, V) (0.17, 0.19)
Ring tune (H, V) (1.70, 1.90)
Transition gamma infinite

Magnetic fields are expanded from the ideal mid-plane field (the mid-plane for a vFFA is a zero-displaced plane in the horizontal direction) so that the fields satisfy Maxwell’s equations. The only assumption made here is that the shape of the fringe field and the length of fall-off are of the form \( g(x) \) as in the equations below:

\{begin}{align*} B_z\left (z,x,y\right )&=B_0\sum _{i=0}^{\infty }B_{zi}\left (z,x\right
)y^i\,, \\ B_x\left (z,x,y\right )&=B_0\sum _{i=0}^{\infty }B_{xi}\left (z,x\right )y^i\,, \\ B_y\left
(z,x,y\right )&=B_0\sum _{i=0}^{\infty }B_{yi}\left (z,x\right )y^i\,, \{end}{align*}


\{begin}{align*} B_{z0}(z,x)&=\exp (mz)g(x)\nonumber ,\\ B_{x0}(z,x)&=\frac 1m\exp
(mz)\frac {\mathrm {d}g}{\mathrm {d}x}\,,\\ B_{y0}(z,x)&=0\,, \{end}{align*}

where \( x \) is longitudinal, \( y \) is horizontal and \( z \) is vertical. \( m \) is the field index, which is about equal to the field index \( k \) of a conventional FFA divided by the reference radius. \( B_0 \) is the reference field strength.

Figure 4 shows the magnetic fields obtained from the equations above. These correspond to a position slightly off mid-plane along the longitudinal direction to show the horizontal field.


(a) Vertical field \( B_z \)


(b) Longitudinal field \( B_x \)



(c) Horizontal field \( B_y \)

Figure 4: Magnetic fields in the vFFA magnet. Values are given slightly off mid- plane (\( y\neq 0 \)) to show the non-zero horizontal field.

Figures 5 and 6 show the closed orbits found for different momenta. As expected, the beam orbit is fixed in the horizontal plane, while it moves vertically with momentum. For a given momentum, the orbit shifts slightly along the magnets in the vertical direction because of the horizontal magnetic field which is non-zero off midplane. The vertical field along the orbit is shown in Fig. 7 which shows relatively higher fields compared with a conventional FFA.


Figure 5: The orbit, seen from the top for a single cell, is independent of beam momentum. The rectangular blocks represent the \( Bd \) magnet (left), which bends the beams outward, and the \( Bf \) magnet (right), which bends the beams inward.



Figure 6: The orbit seen from the side for a single cell. As beams are accelerated, the orbit moves up. There is a slight shift in the vertical direction along the magnets due to the horizontal field.


Figure 7: Vertical magnetic field along the orbit as a function of momentum.

For use as an accelerator at a user facility, easy operation with sufficient tuning knobs must be guaranteed. The transverse tune can be adjusted by the field index \( m \) and the magnetic field ratio, \( Bd/Bf \). Additional control from the relative distance in the radial direction of \( Bd \) and \( Bf \), together with \( m \) and \( Bd \)/\( Bf \), give coverage of the whole range of cell tune-space from 0 to 0.5 in both decoupled tune spaces, as shown in Fig. 8.


Figure 8: Tune-ability of the focusing in terms of cell tune. The field index \( m \) and magnetic field ratio \( Bd/Bf \) are the main parameters available for changing tune. Occasionally adjusting the displacement between \( Bd \) and \( Bf \) in the radial direction is necessary to explore the whole cell tune space from 0 to 0.5 in both directions. The colour scale indicates a rough idea of dynamic aperture, from 0 (the worst) to 5 (best).

Dynamic aperture is a particular concern in a vFFA because of intrinsic, relatively stronger nonlinearity in the lattice magnets. The optics are coupled in the two transverse planes, which creates another complexity in the dynamics. The dynamic aperture was investigated, first, by looking at surviving particles from a random 4D distribution, as shown in Fig. 9, top row; and, secondly, by tracking particles positioned equally in one of the decoupled coordinates, as shown in Fig. 9, bottom row. Both results show a normalised dynamic aperture of around 30 \( \pi \) mm.mrad for the test ring, which is more than enough considering the actual limitation from physical aperture.







Figure 9: Dynamic aperture in the test ring. The survival of particles in 4D phase space is shown in the top row: horizontal in coupled coordinates on the top left, vertical in coupled coordinates on the top right. Decoupled, orthogonal space coordinates are used in the bottom row.


Although any major upgrade to ISIS accelerators will not happen in the immediate future, the team has started looking at detailed options for accelerator designs. A FFA has several advantages over a conventional proton driver, such as an RCS or linac+AR. However, no large-scale facility using an FFA as part of the accelerator infrastructure exists and extensive physics design and hardware R&D in many areas are necessary before making a final choice. Accelerator designs based on DF spiral magnets and a vFFA have been explored. The vFFA in particular seems an interesting and promising option.

  • [1]  T. Ohkawa, Proc. Annual meeting of JPS (1953).

  • [2]  K.R. Symon, D.W. Kerst, L.W. Jones, L.J. Laslett, and K.M. Terwillinger, Phys. Rev. 103 (1956) 1837.

  • [3]  A.A. Kolomensky and A.N. Lebedev, Theory of Cyclic Accelerators (North-Holland, Amsterdam, 1966), p. 332.

  • [4]  Y. Ishikawa, et. al., Proc. of the 8th meeting of the Int. Collaboration on Advanced Neutron Sources (1985) 17.

  • [5]  R.L. Kustom, T.K. Khoe and E.A. Crosbie, IEEE Trans on Nuclear Science, Vol. NS-32, No. 5 (1985) 2672.

  • [6]  P.F. Meads, Jr. and G. Wüstefeld, IEEE Trans on Nuclear Science, Vol. NS-32, No. 5 (1985) 2697.

  • [7]  S.A. Martin, E. Zaplatin, P.F. Meads, Jr., G. Wüstefeld and K. Ziegler, Proc. of the 13th Int. Conf. on Cyclotron and their Applications, Vancouver (1992) 701.

  • [8]  H. Jungwirth, R.L. Kustom, S. Martin, P.F. Meads, Jr., E. Zaplatine and K. Ziegler, Proc. of the \( 14^{th} \) Int. Conf. on Cyclotron and their Applications, Cape Town (1995) 625.

  • [9]  M. Aiba, K. Koba, S. Machida, Y. Mori, R. Muramatsu, C. Ohmori, I. Sakai, Y. Sato, A. Takagi, R. Ueno, T. Yokoi, M. Yoshimoto and Y. Yuasa, Proc. of European Particle Accelerator Conference 2000 (EPS-AG, 2000) 581.

  • [10]  T. Adachi, M. Aiba, K. Koba, S. Machida, Y. Mori, A. Mutoh, J. Nakano, C. Ohmori, I. Sakai, Y. Sato, M. Sugaya, A. Takagi, R. Ueno, T. Uesugi, T. Yokoi, M. Yoshii, M. Yoshimoto and Y. Yuasa, Proc. of Particle Accelerator Conference 2001 (2001) 3254.

  • [11]  M. Tanigaki, Y. Mori, M. Inoue, K. Mishima, S. Shiyoya, Y. Ishi, S. Fukumoto, and S. Machida, Proc. of European Particle Accelerator Conference 2006 (2006) 2367.

  • [12]  S. Machida, et. al., Nature Physics, Vol. 8, No. 3 (2012) 243, DOI: 10.1038/NPHYS2179 (2012).

  • [13]  S. Machida, Phys. Rev. Lett. 119 (2017) 064802.

  • [14]  T. Ohkawa, Bull. Amer. Phys. Soc., 30, (1955) 20.

  • [15]  J. Teichmann, Sov. J. At. Energ. 12 (1963) 507.

  • [16]  S. Brooks, Phys. Rev. ST Accel and Beams 16 (2013) 084001.

  • [17]  D.W. Kerst, E.A. Day, H.J. Hausman, R.O. Haxby, L.J. Laslett, F.E. Mills, T. Ohkawa, F.L. Peterson, E.M. Rowe, A.M. Sessler, J.N. Synder and W.A. Wallenmeyer, Rev. of Sci. Instrum. 31 (1960) 1076.

  • [18]  A. Letchford, et. al., Proc. of 6th International Particle Accelerator Conference 2015 (2015) 3959.