Isochronous and Scaling FFAGs
R. Baartman
TRIUMF4004 WesbrookMall, Vancouver, BC, Canada V6T2A3
Abstract. We explore the differences between isochronous FFAGs (i.e. cyclotrons) and non-isochronous FFAGs
(i.e. where the beam is necessarily pulsed). The general formula for tunes is derived. This is used to understand
the acceptances and also the suitability of such machines as high energy and/or high power accelerators.
INTRODUCTION
If an FFAG can be made isochronous, acceleration can
occur without ramping the rf frequency. But this is of
course simply a cyclotron, of which many examples exist
worldwide: 2 of the largest are PSI (580 MeV p, up to
2mA beam current[l]), TRIUMF (500MeV H~, up to
0.2mA beam current). We would like to know how the
beam dynamics change if an FFAG is isochronous. In
particular, can we still achieve large acceptance?
ISOCHRONISM
Let us define 6 to be the angle of the reference particle
momentum with respect to the lab frame. Orbit length L
is given by speed and orbit period T:
The local curvature p = p(s) can vary and for reversedfield bends even changes sign. (Along an orbit, ds =
pdO > 0 so dO is also negative in reversed-field bends.)
Of course on one orbit, we always have
0.2 0.4 0.6 0.8
scaled radius
1.0
FIGURE 1. Magnetic fi eld law for an isochronous macine.
FOCUSING
What is the magnetic field averaged over the orbit?
-
§Bds
fBpdO
Flat Field
For the moment, let us imagine that there are no sectors, no azimuthal field variation, just radial variation
with field index k = jS2/2.
We know the transfer matrix in such a dipole, and can
write directly:
L/2 _
v? = -k
But Bp is constant and in fact is ^jm^c/q. Therefore
Bc
Remember, /3 is related to the orbit length: /3 = L/(cT)
2nR/(cT)=R/R00.So
B=
2
= r
Bc
*
= -/3V
< o
vr = 7
vz = imaginary
So why do such cyclotrons work at all? Lawrence's
first cyclotrons, and those built before the early 50s, had
no sector focusing. To obtain vertical focusing, k was
made slightly negative. This resulted in phase slippage,
Of course, this means the field index is k = f ^
^ 0 = jS2/2 ^ constant.
So isochronism implies non-scaling.
CP642, High Intensity and High Brightness Hadron Beams: 20th ICFA Advanced Beam Dynamics Workshop on
High Intensity and High Brightness Hadron Beams, edited by W. Chou, Y. Mori, D. Neuffer, and J.-F. Ostiguy
© 2002 American Institute of Physics 0-7354-0097-0/02/$ 19.00
200
In
this hard-edged
hard-edged case,
case, the
the "flutter"
“flutter” FF22 ≡
≡ h(B
h(B−
In this
the
“flutter”
In
=
((B
-−
22
2
22i/B
2 = R/ρ − 1. Aside: Notice that the particle traB)
B) )/B
i/B = R/p
R/ρ -− 1. Aside: Notice that the
the particle traB)
jectory
(blue curve)
curve) does
does not
not coincide
coincide with
with aacontour
contourof
of
jectory (blue
jectory
constant
B
(dashed
curves).
This
has
large
implications
constant BB (dashed curves). This has large implications
for using
using existing
transport codes
codesto
todescribe
describeFFAGs.
FFAGs.The
The
existing transport
transport
for
so
sothe
therfrfrfvoltage
voltagewas
wasmade
madeas
aslarge
largeas
possible to
so
the
voltage
was
made
as
large
as possible
possible
to achieve
achieve
the
of
the final
final energy
energy before
before an
an accumulated
accumulated phase
phase slip
slip of
of
the
final
energy
before
an
accumulated
phase
slip
π7T/2.
by
π/2.
/2.In
Inaddition,
addition,some
somevertical
verticalfocusing
focusingwas
was achieved
achieved by
by
In
addition,
some
vertical
focusing
was
achieved
accelerating
In
this
acceleratingon
onthe
thefalling
falling side
side of
of the
the rf
voltage. In
In this
this
accelerating
on
the
falling
side
of
the
rfrf voltage.
voltage.
way,
such
cyclotrons
achieved
maximum
energies
of
20
way, such
such cyclotrons
cyclotrons achieved
achieved maximum
maximum energies
energies of
of 20
way,
to
30
MeV
(protons).
to 30
30 MeV
MeV (protons).
(protons).
to
To
To reach
reach higher
higher energy
energy with
with simple
simple magnet
magnet poles,
poles,
To
reach
higher
energy
with
simple
magnet
poles,
ititit was
necessary
to
release
isochronism,
was
necessary
to
release
isochronism,
ramp the
the rf
was necessary to release isochronism, ramp
frequency,
frequency, and
and pulse
pulse the
the machine.
machine. Unfortunately,
Unfortunately, this
this
frequency,
and
pulse
the
machine.
meant
meantthat
thatthe
themaximum
maximumintensity
intensity was
was 22 or
or 33 orders
orders of
of
meant
that
the
maximum
intensity
magnitude
magnitudelower
lowerthan
thanfor
forthe
the(cw)
(cw)cyclotron.
cyclotron.
magnitude
lower
than
for
s=
Ρ * Φ;
s
= p*0;
RL
Sin@ΦDD;
eΘ =
= ArcSin@HΡ
ArcSin [ (p/ RL
R)Sin@ΦDD;
Sin [0] ] ;
f11 =
= pΡ /TTan@Φ
Θ+++ fΞD;
ΞD;
fi
a n [ 0 -- 0Θ
];
Θ
ΞD;
Ρ /TTan@Φ
ff22 =
=p
a n [ 0-- Θ
0 ---£ΞD;
];
d == 22 R*
ΘD;
R *S
Sin@Φ
d
i n [ 0- 0ΘD;
];
RR == -Ρ J1 + FN;
k=
* pΡ 
R;
k
= KΚ *
/ R;
kxx =
= Sqrt[l
Ρ;
Sqrt@1 +
kD 
k
+ k]
/pΡ;
;
kyy =
= Sqrt[k]
Ρ;
Sqrt@kD 
/ pΡ;
;
Cosh@k
sD
Sinh@k
Cosh@kyyy s]
sD Sinhfky
Sinh@kyy sD
sD
i Cosh[k
s]
/kkyyy
j
N.
N.
Mzz :=
M
:= j
jJ
Sinh@k
sD
Cosh@k
Sinh@kyyy s]
sD
Cosh@kyy sD
sD
Coshfky
s]
k kkyy Sinh[k
2
Azimuthally-Varying
Azimuthally-Varying Field
Field
Azimuthally-Varying
Strong
Strong focusing
focusing was
was invented/developed
invented/developed in
in the
the 50s.
50s.
Strong
focusing
was
invented/developed
Thishad
hadimplications
implicationsfor
for
This
This
had
implications
for
1
0 y 11 dd ii 11 00yy
i
z
z
j
z
j
zz
j
j
z..
N.
N.j
1
11
jj
j - €€€€€
z.J
j - €€€€€
1z
11zz
f11
{ 00 11 kk - f€€€€€
{{
k f2
Cosh@kyyy s]
sD
Sinh@k
sD
sD Sinh[k
Sinh@kyyy s]
sD
y
Cosh[k
/kkyyy y
z
NNz
J
z
zz
z
Cosh@k
sD
Cosh@kyyy sD
sD
kkyy SSinh@k
i n h f k yyySsD
]
Cosh[k
s]
{{
cyclotrons: they
they could
could now
now be
be isochronous
isochronous
•••cyclotrons:
cyclotrons:
they
could
now
be
AND
AND
vertically
focusing
vertically
focusing
vertically focusing
synchro-cyclotrons:tunes
tunes need
need be
be no
no longer
longer near
near
•••synchro-cyclotrons:
synchro-cyclotrons:
tunes
need
1,so
sobeams
beamswere
weresmaller,
smaller,space
spacecharge
chargelimits
limits higher
higher
1,1,
so
beams
were
smaller,
space
andsynchrotrons:
for synchrosynchro•••and
and
synchrotrons:same
same advantages
advantages as
as for
for
cyclotrons
cyclotrons
cyclotrons
2
Ν
= FullSimplify[
FullSimplify@
v
z =
FullSimplify@
zz
HArcCos@Series@
(ArcCos[Series[
HArcCos@Series@
HM
1DD
MMzZz@@2,
( Mzz@@1,
[ [ l , 1]]
[@@2,
[ 2 , 22DDL
] ] ) 2,
HM
1DD+++M
2DDL
2,
z@@1,
8Φ,
0,
3<DDL
^^2
22
{0, 0,
}]]) 
/ H2
( 2 ΦLD
0ΦLD
)]A
8Φ,
0, 3
3<DDL
H2
2
Theapplication
application was
was by
by far
far simplest
simplest for
for synchrotrons.
synchrotrons.
The
application
was
by
far
The
synchrotrons.
ForFF
FFmachines,
machines,the
the extended
extended nature
nature of
of the
the field
field was
was
For
FF
machines,
the
extended
field
For
calculationalheadache.
headache. That
That is
is why
synchrotrons dedecalculational
headache.
That
is
why synchrotrons
aaacalculational
velopedrapidly
rapidlyin
inthe
the50s
50s(when
(whenlarge
large computing
computing power
power
veloped
rapidly
in
the
50s
(when
large
veloped
wasunavailable),
unavailable),while
whilesynchro-cyclotrons
synchro-cyclotrons of
of the
the FFAG
FFAG
was
unavailable),
while
synchro-cyclotrons
was
typedid
didnot.
not.Nowadays,
Nowadays,large
largecomputing
computingpower
power is
is freely
freely
type
did
not.
Nowadays,
large
computing
freely
type
available.
available.
available.
ForAzimuthally-Varying
Azimuthally-Varying Field
Field (AVF)
(AVF) or
or the
the special
special
For
Azimuthally-Varying
Field
(AVF)
For
case
of
Alternating
Gradient
(FFAG),
let
us
use
Mathcase of
of Alternating
Alternating Gradient
Gradient (FFAG),
(FFAG), let us use Mathcase
ematica
to
calculate
the
tunes.
To
make
it
transparent,
ematica to
to calculate
calculate the
the tunes. To
To make it transparent,
ematica
letus
usconsider
considerall
allidentical
identicaldipoles
dipoles and
and drifts;
drifts; no
no reverse
reverse
let
us
consider
all
identical
dipoles
let
bends.
We
have
drifts
d,
dipoles
with
index
k,
radius
bends.
We
have
drifts
d,
dipoles
with
index
&,
radius
p,
bends. We have drifts d, dipoles with
k,
ρρ,,
bend
angle
φ
,
and
edge
angles
φ
−
θ
:
bend
angle
0,
and
edge
angles
0
—
0:
bend angle φ , and edge angles φ − θ :
2
2
2
J-Κ
+ FF
+
Tan@ΞD
O@ΦD
(-K ++
(1 +
a n [ ^ ] 22)LN
1+
++O
[ 0 ] 22
J-Κ
F H1
H1
+ 22
2T
Tan@ΞD
LN
O@ΦD
2
Cos@k
sD
Sin@k
sD
i
Sin[k
/kkkxxx N.
Cos@kxxx s]
sD
Sin@kxx s]
sD
iJ Cos[k
j
j
Mrrr :=
:=
M
:= j
M
N.
J -kx Sin@kx sD
j
Cos@k
sD
-k Sin[k s]
Cos[k
Cos@kxxx s]
sD
k
k -kxx Sin@kxx sD
0
1
i 1
y
ii 1
yy
1 0
0z
1 00z
j
i
y.J 11 dd N.j
z
zz..
j
1
1
z
jj
j
j
.J 0 1 N.j
1
1
j €€€€€
z
j
z
1
€€€€€
11 zz
€€€€€
1
€€€€€
f2
f
0
1
k
{
k
{{
1
k f2
{
k f1
Cos@k
sD
Sin@k
yy
Cos@kxxx s]
sD
Sin@kxx sD
sDkkxx Nz
J Cos[k
zz
z
Nz
J
Cos@k
--k
k xxx SSin@k
i n [ kxxx ssD
]
C
o s [ kxxx sD
ssD
]
-k
Sin@k
sD
Cos@k
{z
{
2
−θ
φφ−θ
d/2
d/2
ρρ
φφ
θ
θ
R
R
R
2 r = FullSimplify@
Ν
v
r =
Ν
= FullSimplify[
FullSimplify@
r
ArcCos@Series@
ArcCos[Series[
ArcCos@Series@
HM
1DD ++M
M @@2,
( Mrr @@1,
[ [ l , 1]]
[ 2 , 22DDL
] ] ) 2,
r@@1,
HM
1DD + Mrrr [@@2,
2DDL
2,
8Φ,
0, 3}]]
3<DD 
2
{0, 0,
/ H2
( 2 ΦLD
0ΦLD
) ] A^^2
8Φ,
0,
3<DD
H2
2
2
2
sin(φ)
sin(θ)
sin(0)
sin(φ)
sin(θ)
=
=
R
R
ρρ
R
H1
+xΚL
+ 0O@ΦD
( l ++
) +
[0] 2
H1
ΚL
+ O@ΦD
FIGURE 3. Mathemat
Mathematica
code to
to generate FFAG
FFAG tunes.
FIGURE
ica code
FIGURE 3.
3. Mathematica
code to generate
generate FFAGtunes.
tunes.
resulting expressions (see
(see Fig. 3)
3) are identical
identical to those
resulting
resulting expressions
expressions (see Fig.
Fig. 3) are
are identical toto those
those
originally
derived
by
Symon
et
al.
in
the original
original 1956
originally
originally derived
derived by
by Symon
Symon etet al.
al. in
in the
the original 1956
1956
Phys. Rev. paper
paper about FFAGs[2].
FFAGs[2].
Phys.
Phys. Rev.
Rev. paper about
about FFAGs[2].
But beware! Since
Since there is
is now aa distinction
distinction between
But
But beware!
beware! Since there
there is now
now a distinctionbetween
between
local
curvature
(
ρ
)
and
global
(R),
the
definition of
of field
local
local curvature
curvature (p)
(ρ ) and
and global
global (/?),
(R),the
thedefinition
definition offield
field
index is
is ambiguous.
ambiguous. The
The local
local index,
index, used
used in
in the
the dipole
dipole
index
index
is
ambiguous.
The
local
index,
used
in
the
dipole
transfer matrix, is
is
transfer
transfer matrix,
matrix, is
ρ dB
k = p_d_B_
ρ dB,
B dρ ,
k = Bdp'
B dρ
d/2=R sin((t>-9)
sin(φ−θ)
d/2=R
d/2=R
sin(φ−θ)
FIGURE 2.
2. FFAG
FFAG sector
sector magnet
magnet with
with no
no spiral.
spiral. Notice
Notice that
FIGURE
FIGURE
2.orbitFFAG
sector
magnet
with
no spiral.
Notice that
that
the
(blue)
does
not
follow
a
line
of
constant
field.
the (blue)
(blue) orbit
orbit does
does not
not follow
follow aa line
line of
of constant
constant field.
fi eld.
the
In addition,
addition, imagine
imagine that
that the
the edges
edges are
are inclined
inclined by
by an
an
In
In
addition,
imagine
thatinthe
edges
are
inclined
by the
an
ξ
,
not
shown
the
figure.
This
is
called
extra
angle
extra angle
angle ξ%,, not
not shown in the figure. This is called the
extra
“spiral angle".
angle”. shown in the figure. This is called the
"spiral
“spiral angle”.
201
while the Symon formula uses
RdB
BdR
1C — — __
,R
p
~ If —
As we proved, it is in fact this latter quantity which must
be equal to jS2/2 for isochronism. We therefore still have
(isochronous)
But
v2 = -/3272 + F2 (I + 2tan2 £)
(isochronous)
Aside: We could have Mathematica print out the
next higher order, but it would be in error because the
trajectories are not circular arcs.
FIGURE 4. TRIUMF cyclotron. This is the lower pole,
shown during construction (1972).
Example: TRIUMF cyclotron
the n = 0 term gives by far the largest contribution to the
sum.
If the lattice is arranged to have a superperiodicity
n = ±S where S ~ vr, yt can be moved away from vr:
See for example [3] for some details of the magnetic
field configuration.
TABLE 1. Vertical tune at a few radii in the TRIUMF
cyclotron.
Energy
R
100 MeV
250 MeV
505 MeV
175 in.
251 in.
311 in.
1
l+2tan 2 £ I F2
0°
47°
72°
0.0
3.3
20.0
0.30
0.20
0.07
r2
0.28
0.24
0.24
=
1 . 2\«s
R v2-s2
This is the approach used to make yt imaginary in e.g. the
JHF main ring [5]. Whether this can be used to fix vr in
a cyclotron where necessarily yt = y, is not clear, since it
prescribes a peculiar variation of as with R.
A peculiar feature of the TRIUMF cyclotron is that
the peak field is only 0.5 T. The reason for this is that
the accelerated particle is H~, and higher fields at the
500 MeV energy needed for efficient meson production
would strip the loosely bound extra electron. This makes
the flutter at large radii very small and forces reliance
on spiral angle instead. The extreme spiral angle used
was considered quite daring at the time the design was
conceived.
ACCEPTANCE
The chief virtue that FFAG scaling machines have over
non-scaling isochronous machines is the constancy of the
tunes. This results in huge acceptance (> lOOOTrmmmrad[6]), especially in the bending plane. But because
they cannot be isochronous, they must be pulsed.
The chief virtue of isochronous FFAGs is that they can
run cw and reach very high intensity. However, because
vr = y, many resonances must be traversed to reach high
energy. This reduces acceptance to zero: all beam would
be lost if energy gain per turn were as low as it is in
non-isochronous pulsed FFAGs. However, since the rf
frequency is fixed, one can use high Q cavities and very
high energy gain per turn. For example, a 9 GeV proton
cyclotron with 30 sectors was studied[7] and found to
have an acceptance of less than 1 Trmm-mrad normalized
even with an energy gain of 25 MeV per turn. This was
due to the intrinsic 30/3 resonance crossed at y = 10.
Irregular FFAG cyclotrons
By adjusting the flutter F and spiral angle £ as functions of/?, we can arrange to make vz constant.
But what about vr? Can we change it by departing
from the regular //-cell lattice? Perhaps, but no one has
ever tried it.
In synchrotron language, isochronism means y = yt.
But we know that for a "regular" lattice, yt = vr. Hence,
vr = y. For any lattice[4],
CONCLUSIONS
where an is the Fourier transform of j8£/2/p. For N
identical cells where N ^> vr, this gives % = vr because
FFAG synchrocyclotrons can be designed with very large
acceptance and repetition rate and so can be used for ac-
202
celerating very large time-averaged current or very large
phase space populations as is the case for example with
uncooled secondary beams. If an FFAG is isochronous,
then the rf can have constant frequency (and the beam
can be cw), but the acceptance advantage over synchrotrons is lost because isochronism implies that the
horizontal tune cannot be constant (is actually equal to
7). Isochronous FFAGs already exist, accelerating protons up to 7 = 1.6. It seems possible to extend such designs to energy of a few GeV.
ACKNOWLEDGMENTS
TRIUMF receives federal funding via a contributionagreement through the National Research Council of
Canada.
REFERENCES
1. Schmelzbach, P., Stammbach, T., Adam, S., Mezger, A.,,
and Sigg, P., these proceedings (FFAG) (2002).
2. Symon, K., Kerst, D., Jones, L., Laslett, L., and Terwilliger,
K., Phys. Rev., 103, 1837-1859 (1956).
3. Craddock, M., and Richardson, J., (PAC1969) IEEE Trans.
Nucl. Sc., 16, 415^20 (1969).
4. Courant, E., and Snyder, H., Annals of Physics, 3, 1^8
(1958).
5. Mori, Y., these proceedings (JHF) (2002).
6. Mori, Y., these proceedings (FFAG) (2002).
7. Baartman, R., Mackenzie, G., and Lee, R., (PAC 1983)
IEEE Trans. Nucl. Sc., 30, 2010-2012 (1983).
203