Tetra Muon Cooling Ring

S. Kahn, R. Fernow

Brookhaven National Laboratory, Upton, NY 11973

V. Balbekov, R. Raja, Z. Usubov

Fermi National Accelerator Laboratory, Batavia, IL 60510

**Abstract.** Recent simulations have shown that muon cooling rings can
effectively reduce both longitudinal and transverse emittance. The muon
collaboration is investigating several varieties of muon cooling rings.
This study looks at the first of these ring cooling scenarios that was
proposed by V. Balbekov. This simulation of this ring shows significant
cooling in the hardedge field approximation. We discuss the status of
using realistic fields in the tetra simulation.

# INtroduction

Muon cooling rings have shown promise in simulation studies as a
technique for reducing muon beam emittance. In muon ring coolers, the
muons pass through absorber material followed by acceleration repeatedly
for a number of turns. This can produce cooling if there is sufficient
focusing at the absorber. Currently there are three different ring
coolers that are being investigated by the collaboration: 1) The Tetra
four sided solenoid cooling ring, 2) the RFoFo cooling ring, and 3)
dipole edge focused cooling rings. This paper will primarily discuss the
simulations that have been performed for the Tetra ring. The Tetra ring,
which was originally proposed by Valeri Balbekov \[1\], was the first of
the muon cooling ring designs to show significant phase space cooling.
The Tetra ring consists of long straight sections with a large solenoid
magnet surrounding RF cavities and a liquid hydrogen absorber to provide
transverse cooling. The field in the long solenoid varies from 2 T at
the ends to 5 T in the center where the absorber is placed, so as to
reduce the beta function at the absorber. The RF cavities are adjusted
to replace the energy lost in the absorber. In the corner region between
the two straight sections there is a short straight section with two
opposite field solenoids sandwiched between two 45° dipole bend magnets.
The bend magnets are combined function dipoles with a field index of 0.5
to provide focusing in both planes. Figure 1 shows the layout of the
cooling ring, along with the principle parameters that describe it. The
solenoid field flips sign in each of the solenoid channels, as indicated
in the figure. The field flip occurs in the center of the short solenoid
channel. A LiH wedge absorber is positioned near the field flip
position.

![](media/image1.jpeg){width="5.797916666666667in"
height="2.6631944444444446in"}

**Figure 1.** Layout of the Tetra muon cooling ring and a table of the
principle parameters describing the ring.

Simulations of the Tetra ring have been carried out by Balbekov using
his own simulation program \[2\], by Fernow using ICOOL \[3\], and by S.
Kahn, R. Raja and Z. Usubov using GEANT. The ICOOL simulation study
shows significant cooling with merit factors up to 103 after 15 turns.
(The merit factor, which is the muon transmission divided by the
emittance reduction, is a measure used to compare various cooling
schemes.) As promising as these studies have been, there are problems
with the implementation. First of all, the studies that have been
performed have used hardedge field descriptions, which ignore fringing
fields. Secondly, there is no space between the ring elements, which
could cause the solenoid and dipole fields to couple and would make it
difficult to bring in RF and cryogenic services. Also there is no
reasonable approach yet for injection to or extraction from the ring.

# Hardedge Model

Figure 2 shows the transmission, transverse emittance, and 6D emittance
as a function of turn number for this ring simulated in ICOOL using a
hardedge description of the field. The figure also shows the cooling
merit factor. The cooling ring lattice is highly optimized to provide
this performance. In an attempt to add a 15 cm gap on each side of the
wedge bend magnet, matching coils were added symmetrically to the ends
of the adjacent solenoid magnets to compensate and to match into the
bending magnets such that α=0. The matching coil current was varied to
maximize the transmission and minimize the final emittance while keeping
the focusing of the solenoids,$\int_{}^{}B_{s}^{2}\text{ds}$, and the
value of *B~s~* at the absorber unchanged. Inserting this 15 cm gap
reduces the transmission by 80% and increases the transverse emittance
by a factor of 1.5. The longitudinal emittance is unchanged. The
orientation angle of the

![](media/image2.wmf){width="2.8736220472440945in"
height="2.089763779527559in"}![](media/image3.wmf){width="2.9133858267716537in"
height="2.089763779527559in"}![](media/image4.wmf){width="2.9255905511811022in"
height="2.001968503937008in"}![](media/image5.wmf){width="2.858267716535433in"
height="1.9964566929133858in"}

**Figure 2:** Transmission, transverse emittance, 6D emittance and
figure of merit for the Tetra cooling ring with hardedge fields.
Emittances are in meters and the transmission is given in parts per mil.

![](media/image6.jpeg){width="5.790972222222222in"
height="3.251388888888889in"}

**Figure 3:** A comparison of a realistic field description (dotted) to
the hardedge description (solid). The figure shows the axial component
(left) and the vertical component (right) of the field for one quadrant
of the ring.

wedge absorber should be adjusted when varying the fields, however the
results are not strongly sensitive to this.

# Realistic Fields

The hardedge field description does not satisfy Maxwell's equations and
there is a concern that fringing fields may alter the cooling efficiency
of the ring. An attempt to make a realizable design of a magnet system
for the Tetra cooling ring was reported at the PAC01 Accelerator
Conference \[4\]. In this report the 45° bend magnets are 1.45 T iron C
magnets with the poles shaped to give the field index of 0.5. The pole
region is highly saturated, but the magnet does provide the proper field
index. Field clamps on the ends of the magnet are provided to reduce the
fringing field from the dipole magnet. In order to minimize the fringe
field from the large aperture of the long solenoid magnet, iron plates
(with extra coils at the inner radius) are placed on the ends of the
solenoid to return the flux. A similar arrangement is made for the short
field flip solenoid. The resulting field from this more realistic magnet
system is compared to the hardedge field in figure 3. The figure shows
both the axial and the vertical component of the field for one quadrant
of the tetra lattice. This field is currently being implemented in the
GEANT model of this ring.

As another step toward simulating the tetra ring with realistic fields,
the hard edge solenoids (with fictitious mirror plates) in the ICOOL
model were replaced with an elliptic integral representation that
produces a fringe field that extends into the dipole magnet region. The
bend magnet in this study is still represented by the hard edge
description. The solenoid fringe field in this approximation does not
follow the curved reference path in the bend magnet. The emittances,
transmission and merit factors obtained by replacing the hard edge
solenoid by this more realistic representation remain essentially
unchanged.

# Acknowledgments

This work was performed with the support of the US DOE under Contract
No. DE-AC02-98CH10886.

# References

1\. V. Balbekov et al., *Muon Ring Cooler for the MUCOOL experiment*,
Proceedings 2001 Particle Accelerator Conference, Chicago, IL, 2001, pp.
3867-9.

2\. V. Balbekov, *Ring Cooler Progress,* Muon Collaboration Note
MUC-NOTE-246 (2002).

3\. R. Fernow, *Hardedge ICOOL Model of the Balbekov Square Cooling
Ring,* Muon Collaboration Note MUC-NOTE-258 (2002).

4\. S. Kahn et al., *Design of a Magnet System for a Muon Cooling Ring
Experiment,* Proceedings 2001 Particle Accelerator Conference, Chicago,
IL, 2001, pp. 3239-41.