Longitudinal Dynamics in High-Frequency FFAG
Recirculating Accelerators
J. Scott Berg
Brookhaven National Laboratory; Building 901 A; PO Box 5000; Upton, NY 11961-5000
Abstract. A recirculating accelerator accelerates the beam by passing through accelerating cavities multiple
times. An FFAG recirculating accelerator uses a single arc to connect the linacs together, as opposed to multiple
arcs for the different energies. For most scenarios using high-frequency RF, it is impractical to change the phase
of the RF on each pass, at least for lower energy accelerators. Ideally, therefore, the FFAG arc will be isochronous,
so that the particles come back to the same phase (on-crest) on each linac pass. However, it is not possible to make
the FFAG arcs isochronous (compared to the RF period) over a large energy range. This paper demonstrates that
one can nonetheless make an FFAG recirculating accelerator work. Given the arc's path length as a function
of energy and the number of turns to accelerate for, one can find the minimum voltage (and corresponding
initial conditions) required to accelerate a reference particle to the desired energy. I also briefly examine how
the longitudinal acceptance varies with the number of turns that one accelerates.
1.5
LATTICE DESCRIPTION
For the purposes of this paper, a recirculating accelerator
consists of an alternating sequence of identical linacs and
arcs. The arcs are identical in the sense that each has the
same path length as a function of energy. The linacs are
identical in the sense that they all have the same voltage
and the same phase. By "the same phase," I first mean
that the phase of the RF does not change from one turn to
the next. Second, if there are M linacs in the recirculating
accelerator, the phase of one linac differs from that of the
previous linac by 2nk/M for some fixed integer k. If one
has an FFAG arc with a particular characteristic variation
of path length with energy, one can cause the path length
at one particular energy to give the same phase in the
linac following the arc as in the linac preceding the arc
by a combination of adding an additional length to the
arc and varying the aforementioned k. Changing k allows
one to keep the required changes to the dynamics of the
arc lattice small.
There are two extremes in this design: one is the racetrack design, where there are two long parallel linacs
connected by arcs; the opposite extreme is a distributed
RF system, where one has a sequence of short arcs with a
single RF cavity between them. The racetrack design allows one to attempt to suppress dispersion in the linacs,
eliminating longitudinal-transverse coupling. It is generally difficult to suppress dispersion over a large energy range, and in addition, the dispersion suppression
may reduce the dynamic aperture of the system. The distributed RF system allows longitudinal-transverse coupling, but maintains a high degree of symmetry in the
system, in principle giving a good dynamic aperture. The
longitudinal-transverse coupling may not be so impor-
o
x 0.5
-0.5
10
12
14
Energy (GeV)
16
20
FIGURE 1. Path length as a function of energy for a single
6.5 m FFAG cell [1], calculated using COSY INFINITY [2].
Path length is given as a fraction of a 200 MHz RF period. The
solid line is the exact path length variation, the dashed line is a
quadratic approximation.
tant, however, since we are on-crest, and the energy gain
does not vary so strongly with time-of-flight (that variation is what causes the longitudinal-transverse coupling).
It turns out that the path length in an FFAG arc is
generally well approximated as a quadratic function of
energy (see Fig. 1 for an example). One would expect
that it is desirable to minimize the total variation in the
path length over the desired energy range, and so one
generally adjusts the lattice design to place the minimum
of the parabola in the center of the energy range of the
arc. Thus, the path length as a function of energy takes
the form
7
—T
-
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
213
18
(1)
1.5
We expect Ar to vary very little as TQ is adjusted over a
small range. Adjusting T0 changes the energies for which
the particle will see the same phase betweeen subsequent
linacs.
Time-of-flight variation with energy is ignored in the
linacs. For the purposes of this study, and considering the
relatively large energies that these recirculating accelerators are designed for, it is a very good approximation
to distribute any path length variation with energy in the
linacs into the adjacent arcs.
.**•**"». '
1
0.5
t
?
cd
&
°
-0.5
-1
EQUATIONS OF MOTION
,
10
The equations giving the energy and time-of-flight at the
entrance and exit of the linacs are
20
30
Linac Pass
T*f*f*T
40
,
50
FIGURE 2. Phase as a function of linac pass when A0 = 1
and N = 50.
(2)
discrete case. In particular, since the continuous approximation has a solution for all N, we might expect that
the discrete system does as well. In the case of the distributed RF system, A0 is small and N is very large, and
the discrete equations are in fact a very good approximation to the continuous ones. For a racetrack system, however, it is far from clear that the approximation is good.
Since A0 is relatively large, the change of x in one step
can be large, making it questionable whether the contuous approximation is really very good. However, we will
subsequently see that for a large number of turns, a large
fraction of the steps occur at points where the change in
x is small, and xn is large and therefore the change in p is
also small. The continuous approximation thus turns out
to give the correct results for large numbers of turns both
qualitatively and nearly quantitatively as well.
(3)
Here En is the energy after the nth pass through a linac
and Tn is the time-of-flight relative to the crest in the nth
linac. It is useful to change coordinates to xn = corn and
—^
F
(4)
Pn =
Then the equations become
(5)
>o
(6)
where v = 2V/(Emax -£min), A0 = coAr, and 00 = coro.
Say we want to accelerate from £max to Emin in TV
turns. Then /?0 = — 1 and pN = 1. The problem that
we wish to solve is given these endpoint conditions,
minimize v by varying JCQ, the phase at which you enter
the first linac, and 00. This solution will depend only on
A0 and N.
Note that this problem can be formulated for any voltage profile (e.g., something different from cos0) as well
as a different relationship of time-of-flight to energy.
Changing the voltage profile will only change the results quantitatively, not qualitatively [3]. Changing the
relationship of time-of-flight to energy will be discussed
later.
EXAMPLE
We now find the minimum-v solution of Eqs. (5-6) for
A0 = 1. This is a relatively large phase swing: remember that the phase errors accumulate, and so after only 4
steps with this phase error, one would certainly be decelerating. This example is more appropriate for a racetrack
configuration: the phase swing per arc would be orders of
magnitude smaller for a distributed arc system (but there
would be correspondingly more linac passes required).
If one makes 50 linac passes for this system and performs the aforementioned optimization, the phase as a
function of turn number is shown in Fig. 2. Note that
the reference particle crosses the crest three times; this is
related to the parabolic shape of the path length. The particle spends most of its time at the two turning points in
phase (turns 8-18 and 32-42); these are the points where
the path length error is near zero. Due to the large phase
at this point, the particle is not gaining very much energy,
and so remains at the point where the path length is near
Continuous Approximation
It is useful to make a continuous approximation to
Eqs. (5-6) [3]. The continuous approximation indicates
that for AfA0 large, v —> A0/12, 00 —> A0/4, and jt0 —>
—7T/2. The reason that these results are interesting is that
they give a good approximation to what will occur in the
214
0.1
0.5
0.5
0.05
2V/∆E
2V/∆E
(E-Ei)/(∆E/2)
0.4
0.4
| Q3
0.3
£> 0.3
(N
0.2
0.2
-0.05
0.1
0.1
000,
00
0
55
10
10
10
15 20
20 25
20
30
15
25 30
15
Half-Turns
Half-Turns
35
35
40
40
45
-0.1
50
FIGURE 3.
3. Linac
Linac voltage
voltage as
as aa function
function of the number of
FIGURE
linac passes.
passes.
linac
-60
-40
-20
-20
00
20
20
20
Phase
(deg)
Phase (deg)
40
40
40
60
60
60
FIGURE 5. Initial phase space that
linac
that is
is accepted
accepted in
in aa 88 linac
linac
pass system. Each color
represents aa band
band of
of ±0.01
±0.01
color represents
±0.01 in
inpppNNN...
1.5
1.5
0.1
0.1
0.05
0.05
-ωτ
-ωτ00
(E-E
(E-Eii)/(∆E/2)
)/(∆E/2)
11
0.5
0.5
0
-0.05
-0.05
000
0
55
10
10
10
15 20
20 25
25 30
30
20
30
15
25
15
Half-Turns
Half-Turns
35
35
40
40
45
45
50
50
-0.1
-0.1
FIGURE4.
4. Initial
Initial phase
phase as
as aa function
function of
of the
the number
number of
of linac
linac
FIGURE
passes.
passes.
passes.
-60
-60
-40
-40
-20
-20
00
20
20
Phase
Phase (deg)
(deg)
40
40
60
60
60
FIGURE
accepted
in
24
linac
FIGURE 6.
6. Initial
Initial phase
phase space
space that
that isis accepted
accepted in
in aa 24
24linac
linac
pass
pass system.
system.
long time.
time. This
This is
is the
the mechanism
mechanism by
zero for
for aa long
which the
the
zero
by which
number of
of turns
turns in
in this
this
particle can
can spend
spend an
an arbitrary
arbitrary number
particle
system.
system.
Figure333shows
shows the
the voltage
voltage as
as aaa function
function of
of the
the number
number
Figure
shows
the
voltage
as
function
of
the
number
Figure
of
linac
passes,
and
Fig.
4
shows
x
.
For
large
numbers
of linac passes, and Fig. 4 shows x000. For large numbers
of turns,
turns, these
these (including
(including φφ00,,, not
not shown)
shown) do
do
appear
not
shown)
do appear
appear to
to
of
turns,
these
(including
to
of
00
be
approching
the
large
N
limits
given
above.
Note
that
be
approching
the
large
N
limits
given
above.
Note that
that
be approching the large N limits given above. Note
while itit
it is
is not
not clear
clear that
that xXxQ0 →
—> −
—ππ
n/2
in
Fig.
4,
from
while
is
not
clear
that
→
−
/2 in
in Fig.
Fig. 4,
4, from
from
while
/2
0
Fig.
2,
one
can
see
that
the
maximum
phase
swing
is
Fig.
2,
one
can
see
that
the
maximum
phase
swing
is
Fig. 2, one can see that the maximum phase swing is
slightly
larger
than
—
jc
.
The
fact
that
A0
is
large
causes
0
slightly
larger
than
−x
.
The
fact
that
∆
φ
is
large
causes
slightly larger than −x00. The fact that ∆φ is large causes
this difference
difference from
from the
the continuous
continuous approximation.
approximation. It
It
is
this
difference
from
the
continuous
approximation.
It is
is
this
really
the
regime
near
the
turning
points
in
phase
that
really
the
regime
near
the
turning
points
in
phase
that
really the regime near the turning points in phase that
approaches the
the continuous
continuous approximation.
approximation. It
It
turns
out
approaches
the
continuous
approximation.
It turns
turns out
out
approaches
that
the
voltage
limit
as
N
—»
°°
is
very
slightly
less
than
that the
the voltage
voltage limit
limit as
as N
N→
→∞
∞ is
is very
very slightly
slightly less
less than
than
that
what is
is found
found in
in the
the continuous
continuous approximation,
approximation, and
and the
the
what
what
is found
in the
continuous approximation,
and the
difference
is
also
due
to
the
the
finite
A0.
difference is
is also
also due
due to
to the
the the
the finite
finite ∆
∆φφ ..
difference
Finally, Figs.
Figs. 5 and
and 66 demonstrate
demonstrate how
how the
acceptance
Finally,
the acceptance
acceptance
Finally,
Figs. 55 and
6 demonstrate how
the
varies with
with the
the number
number of
of turns.
turns. A
A smaller
smaller number
of
varies
number of
of
varies with the number of turns. A smaller number
turns seems
seems to
to caus
caus aa large
large region
of phase
space to
be
turns
region
of
phase
space
to
be
turns seems to caus a large region of phase space to be
accepted; what
what one
one sees
sees at
at 88 linac
linac passes
is
very
close
to
accepted;
passes is
is very
very close
close to
to
accepted;
what one
sees at
8 linac passes
what
from on-crest
acceleration.
For
what one
one expects
expects from
on-crest acceleration.
acceleration. For
For aa large
large
number
of
linac
passes
(24
here),
one
notes
number of linac passes (24 here), one notes that
that most
most
everything
within
small
band
everything that
that is
is accepted
accepted ends
ends up
up within
within aa small
small band
band
of
reference
momentum
the
final
energy.
In
fact,
of the
the reference
reference momentum
momentum at
at the
the final
final energy.
energy. In
In fact,
fact,
while
it
appears
that
there
is
a
smaller
total
acceptance
while
it
appears
that
there
is
a
smaller
total
acceptance
while it appears that there is a smaller total acceptance
for
for more
more turns,
turns,
the
phase
space
area
ending
up
within
for
more
turns, the
the phase
phase space
space area
area ending
endingup
upwithin
withinaaa
small
energy
band
at
the
end
is
mmuch
larger
for
small energy
energy band
more
small
band at
at the
the end
end is
is mmuch
mmuch larger
larger for
for more
more
turns.
The
analysis
of
acceptance
is
still
very
preliminary,
turns. The
The analysis
turns.
analysis of
of acceptance
acceptance is
is still
still very
very preliminary,
preliminary,
and
and must
must be
be
studied
more
thoroughly.
and
must
be studied
studied more
more thoroughly.
thoroughly.
REFERENCES
REFERENCES
REFERENCES
1.
1.
1.
2.
2.
2.
Carol
Private communication.
Carol Johnstone.
Johnstone.
Carol
Johnstone. Private
Private communication.
communication.
K.
Makino
and
M.
Berz. COSY
INFINITY version
8.
K.
Makino
and
M.
K. Makino and M. Berz.
Berz. COSY
COSY INFINITY
INFINITY version
version 8.
8.
Nuclear
Instruments
and
Methods,
A427,
pp.
338-343
Nuclear
Instruments
and
Methods,
A427,
pp.
338–343
Nuclear Instruments and Methods, A427, pp. 338–343
(1999).
(1999).
(1999).
3.
Proceedings of
Snowmass 2001.
3. J.
J. Scott
Scott Berg.
Berg.
3.
J.
Scott
Berg. Proceedings
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of Snowmass
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215