**Rebunching In The Debuncher for μ-e Conversion Experiments**

David Neuffer

Fermilab

September 27, 2006

**Abstract**

Some scenarios for bunching beam in the Fermilab Debuncher for future
μ-e experiments are discussed. In these scenarios the Debuncher ring is
filled by transfer of debunched beam from the Accumulator, which had
been filled by 1 to 4 Booster batches. A barrier bucket or multiharmonic
rf system can compress the stored proton beam to a relatively short
bunch length (\<200ns) suitable for a μ-e conversion experiment like
MECO.\[1\] A higher-harmonic rf system (such as h=12) can be used to
obtain multiple short bunches that can be used for a
PRISM/PRIME-like\[2\] μ-e conversion experiment.

### Introduction

In this note, we discuss a plan to use 8-GeV proton beam from the
Fermilab Booster, stacking it for multiple turns in the Accumulator,
where the beam is debunched, transferring the beam into the Debuncher,
and then rebunching it to a single bunch, which will then be
slow-extracted toward target which will produce secondary π's, that
decay to μ's and are stopped by a target where μ-e conversion events can
be detected, similar to the proposed MECO experiment.\[1\]

This scenario is designed to be compatible with a future Fermilab proton
plan, presented by D. McGinnis,\[3\] which may be implemented when the
Recycler and Accumulator are not needed to supply antiprotons to the
Tevatron. In that plan, multiple Booster bunchers are stacked in the
Accumulator for transfer into the Recycler for boxcar stacking in the
Recycler, to be then transferred into the Main Injector and accelerated
to supply neutrino experiment. The plan would be modified by diverting
some Accumulator cycles into the Debuncher for μ-e conversion
experiments.

The complete scenario is discussed in more detail in ref. \[4\]. In the
present note we discuss the rebunching in the Debuncher aspect of the
μ-e Conversion Experiment at Fermilab.

After 1 to 4 booster batches are transferred into the Accumulator and
debunched, the beam is extracted into the Debuncher. In the Debuncher
the beam is rebunched to a single bunch with full width less than
\~200ns. The rebunched beam is slowly extracted over the remainder of
the \~1.5s cycle time, with the extracted proton beam sent to the
external target that produces π's for the μ-e conversion experiment. In
this section we discuss the bunch formation

The proton beam in the Accumulator is debunched to a relatively small
energy spread, and fills the circumference (474m or 1600ns). After
accumulation of 4 Booster batches the full emittance is estimated to
84×0.38 = 32 eV-s, which implies an energy spread of 20MeV (full width)
in the fully debunched beam with length of \~1600ns. An extraction gap
of \>45ns is introduced into the circumference-length bunch (by a
barrier bucket or harmonic rf) and a fast kicker extracts the long bunch
into the Debuncher. The Debuncher has an \~30m greater circumference,
which increases the initial gap in the circulating beam to \>145ns.

The μ-e experiment requires that the beam be bunched to a small fraction
of the ring circumference. This means the rf must increase that initial
gap and compress the bunch to a short length. We have chosen a
full-length Δτ~B~ of Δτ~B~\<200ns as a goal in the bunching, which
should meet the μ-e experiment requirements. The bunch must be held to
that length throughout the extraction cycle, with minimal leakage into
the interbunch gap that could lead to mistimed extracted beam. The
experiment goal is to reduce leakage of mistimed protons into the
extraction region to \< 10^-9^ of the primary beam intensity.

**Rf Bunching**

Scenarios for rf bunching in the debuncher have been explored, initially
by considering only the longitudinal motion of the beam. The equations
of motion are:

![](media/image1.wmf)

![](media/image2.wmf)![](media/image3.wmf)

where n is the number of turns, φ is the particle phase (longitudinal
position) and ΔE is the particle energy offset from the reference
energy. V~rf~(φ) is the rf voltage, *m* is the proton mass, β=v/c and γ
= (1-β^2^)^-1/2^ are the usual kinematic factors, and α~p~ = 1/γ^2^ --
1/γ~t~^2^ is the momentum compaction factor. In the Debuncher, γ=9.52
and γ~t~=7.6, so α~p~ =-0.006. The small value of α~p~ means the motion
is relatively isochronous, which implies that the longitudinal motion is
relatively slow. That means that only a small amount of V~rf~ will be
needed in the rebunching, but that bunching may be relatively slow.

**Barrier Bucket rf**

A simple model for bunching would be to follow the bunching procedure
used in the Recycler, which uses barrier bucket rf waveforms to bunch
and debunch antiproton beams. In an initial example we can grow a square
wave potential at two ends of the bunch and then move the phases of that
square wave together until the two ends of the potential meet. In the
present example we considered square waves of + and -- 20 kV amplitude,
with 40° phase widths (\~180ns). The bunching must be slow enough to
avoid phase space dilution; in the initial example the compression
occurs over 0.17s (\~10000 turns). At that point a single harmonic h=4
(2.38MHz) rf system is imposed at 25kV, with amplitude increased to 75kV
over 0.03s is imposed, and that rf is used to maintain the beam in a
short bunch structure over the remainder of the 1.5s cycle. (\~1.3s)

The h=4 rf system was used since it is similar in voltage and frequency
to an rf system that is currently used in the Main Injector for Tevatron
bunch coalescing. (That rf will not be needed after 2010.) This system
bunches the beam to an rms bunch length of \~40ns, with the entire beam
held within a \~200ns full width. The energy full width of the beam is
increased to \~±100MeV. Phase-space dilution is limited to \< \~20% in
initial simulation. 1-D simulation results of the model system are
presented in figure 2.

**Multi-harmonic rf buncher**

The barrier bucket approach required use of an inefficient low-Q rf
system, that may be somewhat expensive, and the rf waveforms will be
less ideal than our initial model. We may also consider a bunching
system consisting of a combination of low-harmonic rf systems, and these
could be obtained from high-Q fixed-frequency cavities. In an initial
example, we consider a combination of h=1, 2, 3, 4 rf cavities with
maximum voltages of 30, 15, 10 and 7.5kV respectively, and combined in
order to obtain an approximately linear rf wave form. In an initial
simulation these cavities are ramped from 0 to full strength over
\~0.055s. As in the above example the 4^th^-harmonic rf is ramped up to
75kV to hold the beam for the remainder of the 1.5s extraction cycle.
Simulation results are dispayed in figure 3.

The final beam state is similar to the above example, with an rms bunch
length of \~40 ns and an energy full width \< \~±100MeV. The finite
number of harmonics resulted in a slightly more dilute phase space than
a more idealized linear or barrier bucket rf would obtain; however, a
more complete optimization and a more accurate evaluation of barrier
bucket nonlinearity may remove the small difference. The bunching did
occur in a significantly shorter time than the above barrier bucket
example, however.

**Discussion**

Further study of instability intensity limitations is needed. The
compressed beam has an enhanced space charge tune shift. A formula for
that tune shift is:

> ![](media/image4.wmf),

where B~F~ is the bunching factor. With B~F~ = 0.06, N= 1.5×10^13^,
ε~Fermi~ = 6πε~N,rms~ = 20πmm-mr, we obtain δν ≅ 0.1, which should be
acceptable.

The energy spread of ±100MeV corresponds to δp/p = ±1%, which is less
than the full acceptance of the Debuncher, which was designed to be ±2%.
It is therefore possible to compress the beam to shorter lengths, with
more rf and bunching time, but the additional reduction would be less
than a factor of 2. The minimum bunch length is thus \> 100ns and this
is somewhat greater than the MECO-BNL design microbunch length of
\<30ns. Some study will be needed to determine whether this is adequate.

While the simulations show bunch compression, a complete assessment
would show some small amount of beam leaking into the interbunch gap,
and the desired extinction factor of 10^-9^ cannot be obtained by the
bunching rf alone. The bunching rf can provide an extinction of at best
\~10^-3^. Additional extinction procedures will be needed, including
additional gap-clearing rf and kickers, adjustment of the slow
extraction dynamics and kickers to improve extinction, and extracted
beam transport clearing kickers. Improved event discrimination in the
μ-e decay detector will help.

**PRISM/PRIME compatible bunching mode**

While this initial study is for a BNL-MECO style μ-e conversion
experiment, a different approach is being developed by the PRISM/PRIME
collaboration. In that approach, the proton beam is formed into short
bunches, and single-bunch extracted onto a target. for to produce short
π→μ bunches. The short μ bunches are phase-energy rotated in the PRISM
ring to small energy-spread bunches which are then single bunch
extracted onto the PRIME target/detector. The phase-energy rotation
requires bunches with σ~τ~ \< 5ns, roughly an order of magnitude shorter
than the single intense bunch discussed above

However, the Debuncher proton beam could instead be bunched into a
number of short bunches, which would then be single turn extracted in
PRISM/PRIME mode. As an example we bunch on harmonic h=12 (\~7.14 MHz).
The beam is initially adiabatically bunched, with an rf that increases
to 10 kV over \~0.04s. The rf voltage is then increased to 200kV, and
the bunches are phase-energy rotated in 0.001s to short δτ (σ~τ~ = 4ns).
Simulation results are shown in fig. 4. (This is, qualitatively, the
inverse of the present Debuncher cycle which takes a string of short
antiproton bunches (small δτ, large δE), fast-rotates them to small δE,
and adiabatically debunches to small δE, continuous δτ for transfer to
the accumulator.) At this point a bunch or bunches could be sent to the
proton target for the PRISM φ-E rotation ring.

To maintain the short bunch length for later bunch extractions a
higher-harmonic, high voltage rf system would be needed. An h=36 or 48
system with V~rf~ ≅ 1MV would be needed. Alternatively, the beam could
be rotated to long bunches (\~0.001s), held at 10kV, and rerotated at
200kV for short bunch extractions.

In the baseline SNUMI-compatible mode, this produces \~12 bunches of
protons, with \~1.25×10^12^ p/bunch every 1.5s, or \~10^20^
protons/Snowmass-year. This is an order of magnitude less that desired
by PRISM/PRIME. If not shared with SNUMI, this could be increased by
\~4×, which may still not be enough. An eventual "proton driver" booster
upgrade would reach that level, however.

**References**

\[1\] W. Molzon, Spokesperson, BNL P940 proposal "A search from μ^-^N →
e^-^N with Sensitivity Below 10^-16^" (Muon-Electron COnversion),
September 1997.

\[2\] The PRIME working group, "An Experimental Search for the μ^-^-e^-^
Conversion Process at an Untimate Sensitivity of 10^-18^ with PRISM",
Letter of Intent to the JPARC 50-GeV Proton Synchrotron Experiment,
January 1, 2003.

\[3\] D. McGinnis, "A 2 Megawatt Multi-Stage Proton Accumulator",
Beams-Doc-1782, 2005.

\[4\] C. Ankenrandt et al., "Using the Fermilab Proton source for a μ-e
Conversion Experiment", Fermilab TN in preparation, October 2006.

  ----------------------------------- --------------- -------------------
  **Debuncher Ring Parameters**                       

  Parameter                           Symbol          Value

  Circumference                       C=2πR           504m

  Beam Momentum                       P               8.89 GeV/c

  Transition                          γ~t~            7.52

  Betatron functions (max)            β~x~, β~y~, η   19.8, 17, 2.2m

  Tunes                               ν~x~, ν~y~      9.66, 9.76

  Period                              C/βc            1690ns

                                                      

                                                      
  ----------------------------------- --------------- -------------------

![](media/image5.png){width="5.15in" height="4.278346456692914in"} Fig.
1: Overview of Booster-Accumulator-Debuncher system.

Figure 2. 1-D simulation of bunch compression.

A: Longitudinal distribution after 0.169s of barrier bucket compression

B: Longitudinal distribution after further h=4 bunching, beam can be

(Emittance numbers on plots are rms emittances; full emittances are a
factor of 6π larger)

A:![](media/image6.png){width="3.8333333333333335in"
height="2.420138888888889in"}

B:![](media/image7.png){width="3.848611111111111in"
height="2.720833333333333in"}

**Figure 3.**

Beam after simulation of h=1, 2, 3, 4 adiabatic bunching with h=4 bunch
compression; beam properties are similar to the previous barrier-bucket
example but with somewhat more phase-space dilution.

![](media/image8.png){width="5.125in" height="3.542361111111111in"}

**Figure 4**. Simulations of h=12 bunching to obtain short proton
bunches for PRISM compatible extraction, A: initial debunched beam. B:
after adiabatic bunching to moderate length. C: after fast bunching
(20kV/turn rf) to obtain short bunches (\~4ns rms length.

A:![](media/image9.png){width="3.0in" height="2.0729166666666665in"}

B:![](media/image10.png){width="3.0055555555555555in"
height="1.9041666666666666in"}

C:![](media/image11.png){width="3.0in" height="2.0729166666666665in"}