Abstract This paper explores the possibility of reaching high energies in isochronous ring cyclotrons with radial sectors by using negative valley fields to increase the magnetic flutter. A simple model was used to generate field maps for 4-GeV and 13-GeV proton rings, whose orbit properties were then studied using the CYCLOPS equilibrium orbit code. Field maps were also generated for two FFAG designs (both non-scaling, one isochronous and one not) and their orbit properties evaluated with CYCLOPS -the first time that a cyclotron code has been used on FFAGs

M K Craddock

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_____________________________________________________________________________________________________________________ 
*Work supported by the National Sciences and Engineering Research 
Council of Canada through grant SRO 328338-05. 
#TRIUMF receives federal funding under a contribution agreement 
through the National Research Council of Canada. 
HIGH-ENERGY CYCLOTRONS WITHOUT SPIRAL  
M.K. Craddock*, University of British Columbia and TRIUMF#,  
Y.-N. Rao, TRIUMF, Vancouver, B.C., Canada 
Abstract 
This paper explores the possibility of reaching high 
energies in isochronous ring cyclotrons with radial sectors 
by using negative valley fields to increase the magnetic 
flutter. A simple model was used to generate field maps 
for 4-GeV and 13-GeV proton rings, whose orbit propert-
ies were then studied using the CYCLOPS equilibrium orbit 
code. Field maps were also generated for two FFAG 
designs (both non-scaling, one isochronous and one not) 
and their orbit properties evaluated with CYCLOPS – the 
first time that a cyclotron code has been used on FFAGs. 
INTRODUCTION 
In the past, designs have been presented for isochron-
ous ring cyclotrons to accelerate protons to energies of 
10-15 GeV[1,2], as these would allow cw, and therefore 
very-high-intensity, operation. These would have been 
three-stage devices, using the 500-MeV TRIUMF 
cyclotron as injector, followed by a 3.5-GeV booster ring. 
The fundamental problem in designing a high-energy 
cyclotron is, of course, how to counter the strong axial 
defocusing term -β2γ2 created by the steeply rising aver-
age magnetic field Bav = γBc that is needed to keep the 
particles (of mass m0) isochronous at a kinetic energy of 
(γ - 1)m0c
2. For this, these designs relied on edge focusing 
by spiral magnet sectors to supplement the magnetic 
flutter F2 ≡ 〈(B(θ)/Bav – 1)2〉: 
  νz
2 ≈ -β2γ2 + F2(1 + 2tan2ε).   (1) 
That axial stability could be achieved in this way was 
confirmed by tracking orbits through simulated magnetic 
fields using the equilibrium orbit code CYCLOPS[3]. The 
feasibility of extracting the beam efficiently by exciting a 
radial resonance was also confirmed [4] using the general 
orbit code GOBLIN.  
The high spiral angles ε required, however, lead to 
various practical problems: strong distorting forces on the 
magnet coils (particularly if these are superconducting), 
restricted space for rf cavities and injection and extraction 
equipment, and strong radial kicks during acceleration.  
More recently, Rees[5] has proposed an isochronous 
radial-sector FFAG design (IFFAG) for accelerating 
muons from 8 to 20 GeV. This employs a novel five-
magnet 0doFoDoFod0 lattice cell, termed a “pumplet” 
from the Welsh word pump (pronounced pimp) for five, 
where the d magnets (and Fs at low energy) are reverse 
bending, and the d, F and D magnets each have special 
field profiles B(r). With long drift spaces between the d 
magnets, and N = 123 cells, the circumference is 1255 m.  
Méot et al.[6] have used the ray-tracing code ZGOUBI to 
follow muons through a simulated field grid and confirm 
the orbit properties Rees predicts: good isochronism, a 
radial tune that drops with energy from 0.38N to 0.20N 
(rather than rising ≈ γ as is usual in isochronous fields), 
and an axial tune dropping slightly from 0.14N to 0.08N. 
The physical mechanism by which the lattice achieves 
axial stability up to γ = 190 is unclear. The reverse fields 
certainly provide relatively high flutter, but even so, F2 is 
only ≈200 at 8 GeV and drops to ≈30 at 20 GeV, far 
below the ≈6,000 and ≈36,000 respectively, which (1) 
would require in the absence of spiral. 
In view of these intriguing results, and the practical 
difficulties presented by superconducting spiral magnets, 
it seemed interesting to explore how far the energies of 
radial-sector cyclotrons could be raised by inserting 
reverse-bend magnets to increase the flutter. The next 
section shows how the parameters for such devices may 
be determined, and compares the theoretical predictions 
with the results from a cyclotron ray-tracing code. 
Finally, we present results for some other FFAG lattices 
(including IFFAG) obtained with the CYCLOPS code. 
RADIAL-SECTOR CYCLOTRONS WITH 
REVERSE BENDS 
As this is an exploratory study, we make the simplest 
possible assumptions: N radial sectors, hard-edge 
magnets, no drift spaces, and equal but opposite hill and 
valley fields: 
   Bh = –Bv = B(r) = γB0 .   (2) 
Denoting the angular widths of the hills and valleys as 
2πh/N and 2π(1 - h)/N respectively, and neglecting 
scalloping effects on the orbit length: 
Bav = 2(h – ½)B .   (3)  
The magnetic flutter is determined entirely by h, and so is 
the same at all energies : 
F2 = ¼(h – ½)–2 – 1 .  (4)  
For the axial focusing to remain positive up to some 
maximum energy γm , but no further, (1) tells us that: 
h – ½ = 1/2γm.   (5) 
Assuming that the maximum magnetic field available, Bm, 
is applied at maximum energy γm , then the “central field” 
Bc and “cyclotron radius” Rc are given by: 
    Bc = Bm/γm
2    (6) 
Rc = (m0c/e)γm
2/Bm .   (7) 
From (7) we see that the ring radius required increases 
as the square of the desired maximum energy, making 
such designs impractical for very high energies. The 
recipe for hill and valley field strength is:  
    B(r) = (Bm/γm)/ √{1 – (r/Rc)2}.  (8) 
Note that Symon's circumference factor[7], the ratio of 
the actual circumference to that obtainable with the same 
maximum field, but no reverse bends: 
  C = γm .   (9) 
Cyclotrons and Their Applications 2007, Eighteenth International Conference
370
We choose the number of sectors N ≈ 3γm so that νr ≈ γ 
remains well below the N/2 resonance. 
As examples we have studied two cases with specif-
ications similar to those of the spiral-sector ring cyclo-
trons mentioned above: one accelerating protons from 1 
to 4 GeV, and the other from 3 to 13 GeV. We assume a 
maximum magnetic field Bm = 5 T. In the first case, 
γm = 5 leads to N = 15, h = 0.6, F
2 = 24, Bc = 0.2 T, and 
Rc = 15.65 m. CYCLOPS was then run on a simulated 
magnetic field grid with B(r) calculated from (8). The 
CYCLOPS results showed that both the flutter and the 
axial tune (Fig. 1) were lower than predicted, νz becoming 
imaginary above 3.4 GeV, where F2 had dropped to 10.9. 
This occurs because the orbit scalloping causes the field 
seen by a proton to vary, rather than being piecewise 
constant, as the above equations assume. To remedy this, 
we have attempted to produce a new field map with the B 
contours shaped to match the orbit arcs: 
    B(r,θ) = (Bm/γm))/ √{1 – (Rhv/Rc)
2} (10) 
where Rhv(r,θ) is the radius at which the orbit through 
(r,θ) crosses the hill-valley boundary. A fully satisfactory 
field map remains to be completed, but some preliminary 
results for low energies show that the full theoretical 
flutter can be obtained. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
In the second case, γm = 15 leads to N = 45, h = 0.533˙, 
F2 = 224, Bc = 0.022˙ T, and Rc = 140.83 m. The tracking 
results (Fig. 2) showed similar behaviour to those for the 
low-energy ring, with νz dropping to zero at 10.8 GeV. In 
both rings the radial tune νr increases ≈ γ. 
The radii for both rings are of course, considerably 
larger than those for their spiral-sector counterparts (10 m 
and 41 m respectively), and in practice would be enlarged 
further by the inclusion of drift spaces for the rf cavities.  
FFAG STUDIES USING CYCLOPS 
F0D0-2:  
Our first test of CYCLOPS on an FFAG lattice was 
made with F0D0-2, designed by J.S. Berg[8] for 
accelerating muons from 10 to 20 GeV. Both the positive-
bending D and negative-bending F are sector magnets, in 
which the field magnitudes decrease outwards with 
constant gradient. The CYCLOPS results agreed very 
closely with Berg's for all the parameters examined (orbit 
radius, beta functions, tunes, and orbit time – the latter 
two being shown in Figures 3 & 4). Note that this linear 
non-scaling design does not aim to be perfectly iso-
chronous. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Figure 2: Tunes νr  νz in a 13 GeV radial-sector cyclotron. 
 
Figure 1: Tunes νr, νz in a 4 GeV radial-sector cyclotron. Figure 4: Orbit time deviation in the F0D0-2 FFAG: 
JSB – Berg; CYC – CYCLOPS. 
Figure 3: Tunes in the F0D0-2 FFAG: JSB – Berg; 
CYC – CYCLOPS. 
Cyclotrons and Their Applications 2007, Eighteenth International Conference
371
IFFAG:  
Because of its greater complexity, this lattice (described 
in the Introduction above) presented a greater challenge to 
producing an adequately detailed field map. To represent 
the hard-edge magnets with sufficient accuracy probably 
requires a finer mesh with more grid points than the 
current version of CYCLOPS is able to digest. The 
symptoms of this were a strong dependence of tune on 
mesh size at some energies, and poorer than usual closure 
of the orbits.  Nevertheless,  the tune results (see Figure 5)  
show general agreement with Rees's values, confirming 
his success in creating a magnet arrangement that can 
maintain positive axial focusing in an isochronous field at 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
very high γ values. They also confirm Teng’s suggestion 
in 1956 that high-order terms in the series expansions for 
the tunes in isochronous machines may drive νr away 
from γ [9]. 
ACKNOWLEDGEMENT 
The authors are grateful to Rick Baartman for helpful 
discussions on various aspects of this work. 
REFERENCES 
[1] M.K. Craddock, C.J. Kost, J.R. Richardson, 
Proc.CYC'78, IEEE Trans. NS-26, 2065 (1979). 
[2] J.I.M. Botman, M.K.Craddock et al., Proc. PAC'83, 
IEEE Trans. NS-30, 2007 (1983). 
[3] M.M. Gordon, Part. Accel. 16, 39 (1984). 
[4] R. Baartman, G.H. Mackenzie, R. Laxdal, R. Lee, 
ibid., 2010. 
[5] G.H. Rees, FFAG04 Workshop, KEK (2004) 
http://hadron.kek.jp/FFAG/FFAG04_HP/plenary/14O
ct04_rees.ppt ; G.H. Rees (this conference). 
[6] F. Lemuet, F. Méot, G. Rees, Proc. PAC'05, 2693 
(2005). 
[7] K.R. Symon, D.W. Kerst, et al., Phys. Rev. 103, 1837 
(1956) 
[8] J.S. Berg, FFAG Workshop, BNL, October 2003, 
 http://www.cap.bnl.gov/mumu/conf/ffag-
031013/Berg3.pdf 
[9] L.C. Teng, Rev. Sci. Instr. 27, 1054 (1956). 
 
 
 
 
 
 
 
 
 
Figure 5: Betatron tunes in the isochronous IFFAG: 
GHR – Rees; CYC – CYCLOPS. 
Cyclotrons and Their Applications 2007, Eighteenth International Conference
372


HIGH-ENERGY CYCLOTRONS WITHOUT SPIRAL

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HIGH-ENERGY CYCLOTRONS WITHOUT SPIRAL

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