N.B. Try the XML version of this page for better math typesetting.

Isochronous and Scaling FFAGs

Isochronous and Scaling FFAGs







R. Baartman


TRIUMF







April, 2002





Abstract: 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 1 MW beam power), TRIUMF (500 MeV H-, up to 0.1 MW beam power). How do the beam dynamics change if an FFAG is isochronous? E.g. can we still achieve large acceptance?

Isochronism

Let us define q to be the angle of the reference particle momentum w.r.t. the lab frame. Orbit length L is given by speed and orbit period T:


L=
()

ds=
()

rdq = bcT.
The local curvature r = r(s) can vary and for reversed-field bends even changes sign. (Along an orbit, ds=rdq > 0 so dq is also negative in reversed-field bends.) Of course on one orbit, we always have



()

dq = 2p.
What is the magnetic field averaged over the orbit?



B
 
=

()

Bds


()

ds
=

()

Br dq

bcT
.
But Br is constant and in fact is bgm0c/q. Therefore



B
 
= 2p
T
m0
q
 g     Bc g = Bc



1-b2
.
Remember, b is related to the orbit length: b = L/(cT)=2pR/(cT) R/R. So



B
 
= Bc



1-(R/R)2
.

iso.png
Of course, this means the field index is k=R/B[dB/dR]=[(b)/(g)][(dg)/(db)] = b2g2 constant.

So isochronism non-scaling.

Focusing (flat field)

For the moment, let us imagine that there are no sectors, no azimuthal field variation, just radial variation with field index k=b2g2.

We know the transfer matrix in such a dipole, and can write directly:

nr2
=
1+k
                               
nz2
=
-k
=
1+b2g2
                               
=
-b2g2
=
g2
                               
<
0
nr
=
g
                               
nz
=
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, so the rf voltage was made as large as possible to achieve the final energy before an accumulated phase slip of p/2. In addition, some vertical focusing was achieved by accelerating on the falling side of the rf voltage. In this way, such cyclotrons achieved maximum energies of 20 to 30 MeV (protons).

To reach higher energy, it was necessary to release isochronism, ramp the rf frequency, and pulse the machine. Unfortunately, this meant that the maximum intensity was 2 or 3 orders of magnitude lower than for the (cw) cyclotron.

Focusing (AVF or FFAG, regular lattice)

Strong focusing was invented/developed in the 50s. This had implications for

The application was by far simplest for synchrotrons. For FF machines, the extended nature of the field was a calculational headache. That is why synchrotrons developed rapidly in the 50s (when large computing power was unavailable), while synchro-cyclotrons of the FFAG type did not. Nowadays, large computing power is freely available.

For Azimuthally-Varying Field (AVF) or the special case of Alternating Gradient (FFAG), let us use Mathematica to calculate the tunes. To make it transparent, let us consider all identical dipoles and drifts; no reverse bends. We have drifts d, dipoles with index k, radius r, bend angle f, and edge angles f-q:

ffagsector.png

In addition, imagine that the edges are inclined by an extra angle x. This is called the ``spiral angle'' (hard to draw).

In this hard-edged case, the ``flutter'' F2 (B-[`B])2/[`B]2=R/r-1.

Aside: Notice that the particle trajectory (blue curve) does not coincide with a contour of constant B (dashed curves). This has large implications for using existing transport codes to describe FFAGs.

ffagtrig3.png

These expressions are identical to those originally derived by Symon et al. in the original 1956 Phys. Rev. paper about FFAGs.

But beware! Since there is now a distinction between local curvature (r) and global (R), the definition of field index is ambiguous. The local index, used in the dipole transfer matrix, is


k= r
B
dB
dr
,
while the Symon formula uses


k = R
B
dB
dR
R
r
As we proved, it is in fact this latter quantity which must be equal to b2g2 for isochronism. We therefore still have


nr=g                               (isochronous)
But


nz2=-b2g2+F2(1+2tan2x)        (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.

Example: TRIUMF cyclotron

cyclotron-1972sm.png

EnergyRbgx1+2tan2xF2nz
100 MeV175 in.0.4700.00.30 0.28
250 MeV251 in.0.78473.30.20 0.24
505 MeV311 in.1.177220.00.07 0.24

Irregular FFAG cyclotrons

By adjusting the flutter F and spiral angle x as functions of R, we can arrange to make nz constant.

But what about nr? Can we change it by departing from the regular N-cell lattice? Perhaps, but no one has ever tried it.

In synchrotron language, isochronism means g = gt. But we know that for a ``regular'' lattice, gt=nr. Hence, nr=g. For an irregular lattice,


1
gt2
= nr3
R


n 
|an|2
nr2-n2
where an is the Fourier transform of bx3/2/r. For N identical cells where N >> nr, this gives gt=nr because 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 ~ nr, gt can be moved away from nr:


1
gt2
= 1
nr2
+ 2|aS|2
R
nr3
nr2-S2
This is the approach used to make gt imaginary in e.g. the JHF main ring. Whether this can be used to fix nr in a cyclotron where necessarily gt=g, is not clear, since it prescribes a peculiar variation of aS with R.

Conclusions:

The chief virtue that FFAG scaling machines have over non-scaling isochronous machines is the constancy of the tunes. This results in huge acceptance ( > 1000p mm-mrad). 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 nr=g, many resonances must be traversed to reach high energy. This reduces acceptance to only a few p mm-mrad.




File translated from TEX by TTH, version 2.87.
On 25 Apr 2002, 16:26.