# Presentation: 248181

## Muon acceleration- amplitude effects in non-scaling FFAG -

- Shinji Machida
- CCLRC/RAL/ASTeC
- 26 April, 2006
- _[http://](http://hadron.kek.jp/~machida/doc/nufact/)__[hadron](http://hadron.kek.jp/~machida/doc/nufact/)__[.](http://hadron.kek.jp/~machida/doc/nufact/)__[kek](http://hadron.kek.jp/~machida/doc/nufact/)__[.](http://hadron.kek.jp/~machida/doc/nufact/)__[jp](http://hadron.kek.jp/~machida/doc/nufact/)__[/](http://hadron.kek.jp/~machida/doc/nufact/)__[~machida](http://hadron.kek.jp/~machida/doc/nufact/)__[/doc/nufact/](http://hadron.kek.jp/~machida/doc/nufact/)_
- _ffag/machida_20060426.ppt & pdf_

**Notes:**
I would like to focus on amplitude effects in non-scaling FFAG today. I should take every efforts to save FFAG since it is in danger now.

## Contents 

- Code benchmarking
- Possible cures of “amplitude problem”
- Matching between two FFAG rings
- Next steps

**Notes:**
First, I will talk about codes and its benchmarking. Secondly, I will propose possible cures of amplitude problem. Then, I will mention a matching issue between two FFAG rings. Finally, I will make some comments on next steps.

## Code benchmark (1)

- Zgoubi and S(hinji’s)-code had discrepancy (?)
- Initial distributions were different.
- Zgoubi assumed 6-D ellipsoid. S-code assumed 2-D ellipse independently in horizontal, vertical, and longitudinal space.

- (5 to 10 GeV: 30  mm in transverse, 0.05 eVs in longitudinal.)

- S-code

- Zgoubi

**Notes:**
So-called “amplitude effects” or “amplitude problem” was known for several month now, but it was not clear quantitatively, at least at the time of last ISS meeting at KEK, because two codes had discrepancy. Zgoubi shows much less effects than S-code. It turns out then two codes use different initial distributions and that is the source of discrepancy. Zgoubi assumed 6-D ellipsoid and S-code did 2-D ellipse independent in two transverse and longitudinal space.

## Code benchmark (2)

- With the same initial condition: independent in each plane.
- Large amplitude particles make a problem also on Zgoubi.
- Another confirmation of the problem by Keil with MAD.

- (10 to 20 GeV: 30  mm in transverse, 0.05 eVs in longitudinal.)

- S-code

- Zgoubi

**Notes:**
If we use the same initial condition, for example, independent distribution in transverse and longitudinal, two code results agree quite nicely. In other words, Zgoubi also shows the amplitude problem. Recently another confirmation was made by Eberhard using MAD.

## Code benchmark (3)

- (10 to 20 GeV: 30  mm in transverse, 0.05 eVs in longitudinal.)

- Correlation in T only
*J**l**<a* and* J**h**+J**v**<a*

- Correlation in L and T
*J**l**+J**h**+J**v**<a*

- No correlation
*J**l**<a*, *J**h**<a*, and* J**v**<a*

- Correlation between transverse and
- longitudinal space eliminates large
- amplitude particles in both planes.

*a* is amplitude in normalized phase space.

- Some particles are not accelerated.

**Notes:**
Those picture shows longitudinal phase space for different initial distributions. If we assume emittance independently in three phase space, the effects looks really bad and that is what S-code assumed before. If there is strong correlation between three phase space, which eliminates large amplitude particles in both planes. Effects are not so bad. This was what Zgoubi assumed before. In real life, the most plausible distribution is something between two extreme like this.

## Code benchmark (4)

- In real life, there will be some correlation between horizontal and vertical, but probably independent of longitudinal.
- The best way is to take particle distribution at the end of linac or RLA. Use it as an initial distribution to FFAG.

**Notes:**
As a working assumption in the following simulation, I say there will be some correlation between horizontal and vertical, but probably independent of longitudinal.

Of course, the best way it to take particle distribution at the end of linac or RLA and use it as an initial distribution to FFAG. Then, the end to end simulation become possible.

## Possible cures (1)

- Longitudinal dynamics is parameterized by

- Relative energy gain
- per phase slip.

- RF frequency relative
- to revolution freq.

- 36  mm

- 25

- 0

- Phase (1/2 pi)

- dp/p (normalized)

- Left figure is slightly different
- for finite amplitude particle.

- Amplitude effects

**Notes:**
We want to cure the problem. Let me look at the longitudinal dynamics once again. It is parameterized by two parameters “a” and “b”. With some combination of “a” and “b”, longitudinal phase space looks like this. Idea is to accelerate a particle using one of those path. Unfortunately, location of those path depends on transverse amplitude. Simulation in 6D phase space shows a large amplitude particles lags behind a small amplitude particle in longitudinal phase. It cannot be accelerated if it is more than 36 pi mm for this example.

## Possible cures (2)

- Cure 1
- Increase *V* to increase “a”.
- Cure 2
- Decrease ** to increase “a”.
- Cure 3
- Flatten crest by introduction of higher harmonics.

**Notes:**
In order to solve the problem, one can easily think of three ways of cures. The first one is to increase voltage. The second one is to decrease RF frequency, and the third one is to introduce higher harmonics in RF wave form.

## Possible cures (RF voltage)

- =0 pi, V=1
- dp/p=0.36%

- =30 pi, V=1
- dp/p=2.8%

- =30 pi, V=1.4
- dp/p=0.88%

- V/V_nominal

- Dp/p_rms

- 1

- 2

- =30 pi

- 0%

- 5%

**Notes:**
Those picture show some examples of 10 to 20 GeV muon ring. This is a reference with zero amplitude. Once we assume finte amplitude such as 30 pi mm, emittance is distorted. However, if we increase voltage by 40%, emittance comes back in good shape although it is not as good as the zero amplitude case. If we draw a curve of dp/p as a function of voltage for 30 pi mm, it looks like this. 

## Possible cures (RF voltage)

- 50% increase of RF voltage makes dp/p around 1% up to 50 pi mm.

**Notes:**
This is another way of looking at the problem and its cure. With nominal RF voltage, momentum spread at the end of 10 to 20 GeV becomes large as emittance increase. However, 50% increase of voltage fairly flattened the momentum spread.it shows that even 50 pi mm can be accelerated with small momentum spread.

## Possible cures (higher harmonics)

- =0 pi, h=1
- dp/p=0.36%

- =0 pi, h=1+2
- dp/p=0.42%

- =30 pi, h=1+2
- dp/p=0.58%

- =30 pi, h=1+3
- dp/p=0.77%

**Notes:**
Introduction of higher harmonics is another way to cure the problem. With second harmonics RF, ellipse is flattened as well as with third harmonics.

## Possible cures (higher harmonics)

- Second harmonics makes dp/p around 1% up to 50 pi mm.
- It requires more RF power because second harmonics reduce peak voltage 25%.

## Possible cures (lower frequency)

- 274 MHz
- =30 pi, V=1
- dp/p=2.8%

- 88 MHz
- =30 pi, V=2x88/200
- dp/p=1.2%

- 44 MHz
- =30 pi, V=2x44/200
- dp/p=1.4%

- 2 MHz
- =30 pi, V=4x2/200
- dp/p=3.7%

**Notes:**
Third way is to use lower frequency RF keeping relative voltage high. I tried to see three frequency and the summary is the next page.

## Possible cures (lower frequency)

- Lower frequency helps.
- However, it requires relatively higher voltage and time to complete acceleration.

**Notes:**
It certainly helps. However, it requires relatively higher voltage and time.

## Possible cures (summary)

- Increase of a few 10% of RF voltage or second harmonics makes dp/p around 1% up to 50 pi mm beam.
- That requires additional RF power.
- Amplitude effects can be cured when there is only one FFAG.

**Notes:**
So as far as single FFAG system is concerned, increase of a few 10% of RF voltage or second harmonics makes dp/p around 1% even up to 50 pi mm beam. Of course, it is not free. We need additional RF power in both cases.

## Matching between two FFAG rings (1)

- Zero transverse emittance beam has no problem of longitudinal matching.

-           10                          20
-                        [GeV]

- 0%

- 3%

- dp/p_rms

- longitudinal phase space

- 1%

- 2%

- 5GeV

- 10GeV

- 20GeV

**Notes:**
In the present base line scenario, there are two FFAG cascade. With zero transverse amplitude, beam has not problem of longitudinal matching. There is slight distortion in the second ring, but it is not big.

## Matching between two FFAG rings (2)

- Second harmonics and 10% increase of RF voltage partially cure the problem.
- It depends decay ring acceptance if it is allowed.

- =30 pi, h=1, V=1

- =30 pi, h=1+2, V=1.1

**Notes:**
Finite amplitude beam mess up the longitudinal bunch. However, with second harmonics and 10% increase of RF voltage partically cure the problem even though it is not 100%.

## Matching between two FFAG rings (3)

- Still many parameters we can play with.
- Injection phase of the second ring (partially done).
- Optical parameters of the first ring at injection (partially done).
- Combination of 2 or more cures (partially done).
- etc.
- Need criterion: how much dp/p is tolerable.

## Next steps 

- Scaling FFAG with “higher” RF frequency
- Does 44 MHz, 88 MHz makes a reasonable bucket in scaling FFAG ?
- How high is the *k*-value supposed to be if the frequency is fixed at 200 MHz ? (Berg)
*k=*1800 (my number), 1100 (mori) vs. 800 (present lattice)
- How much dp/p can be allowed in the decay ring?
- Need a target value for optimization.
- RLA may have as much dp/p as FFAG with amplitude effects: a few percent ?
- dp/p of scaling FFAG with either low and high frequency is unknown.