Phase II Proposal Project Summary/Abstract

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Phase II Proposal


Project Summary/Abstract


Company Name: Particle Beam Lasers, Inc., Northridge, CA 91324

Project Title:

Development of a 6
-
Dimensional Muon Cooling System Using Achromat
Bends and Design, Fabrication and Test of a Prototype H
igh Temperature


Superconducting (HTS) Solenoid for the System

Principal Investigator: Dr. Alper A. Garren

Technical Topic 31:


Advanced Concepts and Technology for High Energy Accelerators

Subtopic Letter b):


Novel Device and Instrumentation Developmen
t


The use of muon particle beams have possible commercial applications in the fields of
biotechnology, medicine, and nanotechnology as well as in basic “physics science” engaged
in the study and commercialization of advanced accelerator technologies, in t
his case, the
study of acceleration, containment and storage of muon beams for both science and
commercial applications. A major obstacle for building a case for future muon colliders or
neutrino factories has been the lack of an experimental demonstratio
n of the principle of
ionization cooling of muons and in particular 6
-
D cooling and emittance exchange. Past
work by our company in this area has focused on lattice design, simulation studies and
magnet design for a compact gas
-
filled storage ring for 6
-
D

cooling of muon beams. Based
on this previous work, we considered in Phase I an extension of those design results to allow
credible injection and extraction of the beam. The basic new idea is the incorporation of
extended straight sections between achro
mats in the lattice. Both open and closed systems
(i.e. ring) were studied. Although more work needs to be done, the Phase I feasibility studies
made advancements as some beam cooling was observed under two of the five cases studied.

The Phase II project

will continue the refinement and optimization of the preferred lattice of a
6
-
D muon cooling system using achromat bends. Work will continue to define and develop a
credible beam injection/extraction scheme. A high temperature superconducting (HTS)
solen
oid, a crucial sub
-
system of the 6
-
D muon cooling machine, will be designed, built, and
tested during this phase. Further investigation of applications for cooled muon beams outside
the high energy physics community will also be made.


Commercial Applicati
ons and Other Benefits: Cooled beams of muon particles for use in
elementary particle physics experiments are needed to advance mankind’s understanding of
the fundamental nature of energy, the elementary constituents of matter and the forces that
control
them. A robust, simple and economical cooling system to cool ion and particle beams
have use in ion lasers, biotech, medical, and nanotechology applications. Development of
HTS magnet technology may revolutionize future medical and accelerator facilities
. Various
magnets in muon colliders, hadron colliders, facilities for rare isotope beams (FRIB) benefit
significantly from the ability of HTS to produce high fields and ability to handle and
economically remove large energy depositions.


Key Words: 6
-
Dime
nsional Cooling of Muon Beams


Summary for Members of Congress: The Phase II project proposed is a continuation of
initial work to develop a 6
-
Dimensional muon cooling system using achromat bends. The
ultimate goal is to design, build, test and operate s
uch a system at a national accelerator site

1

or commercial ion laser facility. High Temperature Superconducting magnet technology is
expected to make future accelerator facilities highly energy efficient.




Phase II Project Narrative


Cover Page



Particl
e Beam Lasers, Inc.

18925 Dearborn Street

Northridge CA 91324
-
2807



Principal Investigator:


Alper A. Garren



Project Title:

Development of a 6
-
Dimensional Muon Cooling System


Using Achromat Bends and the Design, Fabrication and


Test of a Prototype Hig
h Temperature Superconducting


(HTS) Solenoid for the System



Topic Number:


31) Advanced Concepts and Technology for






High Energy Accelerators



Subtopic Number:

b) Novel Device and Instrumentation Development



DOE Grant Number:

DE
-
FG02
-
07ER84855












2

Significance, Background Information, and Technical Approach



a) Identification and Significance of the Problem or Opportunity and Technical
Approach




The U.S. Department of Energy is highly interested in the development of nov
el devices and
instrumentation for use in producing intense low energy muon beams suitable for precision
muon experiments, and intense high energy muon beams suitable for neutrino factories and/or
muon colliders. The DOE is particularly interested in the
development of concepts or devices
for ionization cooling, including emittance exchange processes, and injection and extraction
schemes for muon cooling rings. During Phase II we will continue to develop a 6
-
D muon
cooling system using achromat bends. In

particular the preferred lattice will be refined and
optimized and a prototype high temperature superconducting solenoid (a critical sub
-
component of the system) will be designed, fabricated, and tested. A test demonstration
system will be studied and th
e possibility of constructing the system by a Phase III SBIR will
be considered. The work proposed builds on the current Phase I effort and our earlier study
of a compact gas
-
filled storage ring for 6
-
D cooling of muon beams that was supported by an
earl
ier Phase I SBIR under DOE Grant No. DE
-
FG02
-
04ER84037. Illustrations of the gas
-
filled storage ring are shown in Figures 1 and 2 below.






1.6 m



Figure 1 Possible Gas
-
Filled Muon Storage Ring with Iron Magnets




and RF Cavities for Cooling Demons
tration









3




Figure 2 Cut Through of the Beam Pipe, RF Cavity and Magnet


Two significant technical challenges are presented when one considers the development of an
intense muon beam. The first is the production and collection of the muons and th
e second is
the reduction of the phase space (cooling) of the muon beam in order to facilitate the ultimate
application of the muon beam for physics research [1]. In order to optimally cool the muon
beam it is desired to collapse its extent in 6
-
D phase s
pace, i.e. in each of the three space and
three momentum dimensions [2]. A principle technique for muon beam cooling is ionization
cooling in which the magnitudes of 3
-
dimensional momentum vectors of the muon particles
are reduced via energy loss in an ion
izing media followed by the subsequent restoration of
only the longitudinal momentum component with rf power.


Whereas 4D transverse cooling can be achieved in a linear channel, it is necessary for the
beam to have dispersion so that longitudinal cooling c
an also be realized. This is because
dispersion gives the beam a correlation between energy and transverse displacement. The
placement of absorbing wedges in the beam creates a corresponding correlation between
particle energy and energy loss, and this a
llows longitudinal cooling. Thus it is natural to
consider rings where dispersion arises naturally from the bending in the dipole magnets. It is
possible to construct rings with wedge absorbers installed [3]. An important consideration in
material selec
tion is that in addition to cooling the beam via energy loss, the material will also
"heat" the beam through the mechanism of multiple Coulomb scattering. The most favorable
material for cooling is hydrogen since this imparts the least harmful heating of t
he known
materials. Hydrogen based wedge absorbers are proposed in which liquid hydrogen is
employed. This is a technical challenge due to the need for the implementation of cryogenic
systems and the power requirements associated with heat removal. We p
ropose an alternative
to the wedge absorbers in which the whole system is loaded with high pressure hydrogen gas
so the complexity associated with cryogenic systems can be avoided. In the system, the path
lengths of the high momentum particles are natural
ly greater that low momentum particles
when the particles circumscribe the ring. This will lead to the longitudinal phase space of the
beam being cooled as the dE/E of the beam is reduced through differential reduction of
particle energy. We also propose

to study a discrete system where LiH absorbers are inserted
for the cooling.



4

The significant technical issues to be addressed in our scheme is to 1) identify appropriate
lattices which can achieve the cooling we seek; 2) establish that the magnets can b
e
reasonably constructed and operated; 3) insure that the rf system can be implemented into the
system; 4) determine the best injection/ejection system for introducing and extracting the
beam for the cooling system. Our Phase I work demonstrated 6
-
D cooli
ng under two
scenarios and a plausible muon beam injection/ejection system has been identified.


Achromatic System

Incorporation of Achromats into the Cooling System


In an earlier proposal, we considered a ring system using energy loss by allowing the
ci
rculating muons to pass through high
-
pressure hydrogen gas, in contrast to previous rings
studied by ourselves and others which incorporated short wedge
-
shaped liquid hydrogen
(LH
2
) absorbers. Our rings featured weak
-
focusing scaling lattices with four wed
ge
-
shaped
dipoles separated by drifts, and acceleration by rf cavities to recover the energy loss. In both
cases


LH
2

and high
-
pressure gas
-

the dispersion of the ring produces a dependence of
energy loss on momentum. A closed ring tends to be more eff
icient than an open system
because of the reuse of the magnets and rf cavities. Ring systems of this type are shown in
Figures 3 and 4. Other ring systems designed for the cooling of muon beams have been
proposed by Balbekov [4] and Palmer[5].


Unfortuna
tely, it has not yet been possible to establish a convincing injection/extraction
scenario for such a closed ring. This situation, we believe, can be rectified by the introduction
of dispersion
-
free solenoid
-
focused straight sections into the ring lattice
at points separated by
a 180
o
phase advance in each transverse plane. When this is done, the 4
-
cell lattice described
above becomes a racetrack with two straight section cells, each with a 45
o
phase advance,
separating the two 180
o
achromat bends (See Fig
. 5). The tune of the modified ring becomes
1.25 compared to near 1.00 for the original 4
-
cell ring. We will incorporate solenoids within
the straight sections which will serve several functions. First, they will form the basis for
beam matching into and
out of the achromat bends. Second, they will provide for the
necessary focusing which allow the beam to remain in the straight channel. Third, the
solenoids will allow for a rotation of the beam such than the x and y dimensions are coupled.
This will lea
d to a uniform cooling rate for both the x,x' and y,y' planes. The solenoid cells
can have arbitrary lengths tailored to the requirements of the injection/extraction kickers. Of
course, the longer these straight sections are, the less efficient the coolin
g will be.


Another solution to the injection/extraction problem is to abandon closed ring systems in
favor of open ones, as have been considered in previous linear designs of Palmer [6]. These
designs promise efficient transverse but no longitudinal cooli
ng. We propose to take
advantage of the unique flexibility offered by our achromatic bends approach to explore a
simple topological change that converts a set of racetrack rings into S
-
bend structures, as
suggested by Figure 6. This structure should also g
ive 6
-
D cooling. Although this will forego
the cost efficiency advantage inherent in rings, we can now taper the cooling segments to fit
the decreasing beam
-
envelope radii as the cooling progresses, thus raising the possibility of
more efficient cooling.
Many variations can be contemplated, including excursions into the
vertical direction which will allow for XZ plane coupled cooling to be alternated with YZ
plane coupled cooling.


5








Figure 3. A dipole
-
only 4
-
cell ring lattice. The required focusing
is accomplished by
adjusting the dipole entrance and exit pole
-
face angles.



Figure 4. A Gas
-
Filled Weak Focusing Dipole
-
only Ring.





6



Figure 5 Racetrack ring with dipoles and solenoids. Each arc has two 90 degree wedge
-
shaped dipoles and each stra
ight section has two focusing solenoids (rectangular boxes).





7









Figure 6 S
-
Bend Structures of an Open System using dipoles and solenoids.






8


Previous Design of a Magnet Suitable for Achromatic Bends


Dipole magnets have been designed for the p
urpose of circulating large emittance muon
beams in small compact rings similar to what we envision. These magnets have large vertical
apertures in order to minimize the losses. The pole edges are rotated with respect to the
incident closed orbit path so
as to provide horizontal focusing. Table 1 shows the
parameters that describe one particular ring for which this magnet would be suitable. The
fields generated by the magnets are used to track muons in the ring. Because of the small size
of the ring,
the local variation of the field along the ring is important.



These dipole magnets were designed to bend a 172 MeV/c reference muon on the
closed orbit by 90°. The magnet pole faces are rotated by 22.5° in order to provide edge
focusing in the horizonta
l plane. Figure 7 shows a diagram of the mid
-
plane geometry of one
cell of the ring.




Parameter

Value

Dipole Field

1.8 T

Number of Cells

4

Reference Momentum

172.12 MeV/c

Ring Circumference

3.81 m

X Aperture

±
20 cm

Y Aperture

±
15 cm


P
z

Acceptance

±
10 MeV/c

Minimum β
X


38 cm

Maximum β
X


92 cm

Minimum β
Y


54 cm

Maximum β
Y


66 cm

Hydrogen Gas Pressure

40 Atm @ 300º K

RF Gradient

14 MV/m

Total RF Length

1.2 m

Total Turns

100




Table 1: Parameters that describe the muon cooling ring.






9



Figure 7: Sketch of the cooling ring magnet.


It is desirable to have a horizontal aperture of ±20 cm about the closed orbit. The field should
be as uniform as possible in that region. The vertical aperture needs to be ±15 cm in order to
contain the ful
l beam. The dipole field is chosen to be 1.8 Tesla so that the iron yoke does
not significantly saturate. The coils are placed around the poles and because of space
limitations they will likely be superconducting. This is the design that was used for th
e field
calculations. The horizontal dimensions shown in the diagram are measured from the center
of the ring. The amount of iron chosen was to insure that the return yoke would not saturate.
The poles are shaped to provide a more uniform field in order
to improve the horizontal
dynamic aperture. The field needs to be reasonably uniform over a region ±20 cm about the
reference orbit.


The field from this magnet was computed using the finite element code TOSCA. A single
magnet was modeled and the presenc
e of the other magnets was taken into account with the
boundary conditions. TOSCA provides the ability to track particles within the program. By
launching muons at various start positions on a symmetry plane, a closed orbit can be found.
Using the mid
-
p
lane field map from TOSCA, the field and its harmonics can be calculated
along the closed orbit path. The harmonics are determined from the variation of the mid
-
plane field in the plane transverse to the closed orbit path at each point along the path.


Th
e ICOOL simulation program can accept the field as a fourier decomposition of each of
these harmonics along the closed orbit path. The field can be reconstructed by an expansion
in variables of the local coordinate system defined by the closed orbit path.

The errors in the
field calculated in this manner are expected to grow with distance from the closed orbit.
TOSCA also provides a field map where the field errors are more uniformly distributed in
space. This field map can also used for inclusion into
GEANT for the purpose of particle
tracking.






10

Beam Dynamics


We will adopt the successful strategy incorporated in our earlier work, namely exploring first
the linear lattice solutions to various cells and then transposing the parameters into the
tracki
ng code ICOOL. Important issues already accessible at this stage will include the
chromatic acceptance of the ring. This will be crucial for the success operation of a 6D
cooling system for circulating muon beams, in that these beams will have a substant
ial dp/p
component. We will then be able to examine beam dynamic issues for the interesting cases of
included rf cavities and evaluate the impact of installing absorbers at various points in the
ring. We will consider both H
2

gas filled rings with various

pressures as well as solutions in
which lithium hydride absorbers are place at key points in the lattice where the beta function
is minimal in the ring. Once satisfactory solutions are located, we will then exploit the
advantageous achromatic structure
of the lattices by appending chains of cells together.
These chains need not take only the form of rings but may instead take the form of S curves
which will present one with the advantage of avoiding the need for elaborate injection and
extraction system
s in order to transport beam through the structure. Giving this S structure,
one can also contemplate a transport system in which the lattice cells are tapered to allow for
maximal cooling performance as the beam traverses the structure. Finally, it is a
pparent that
one need not remain exclusively within a horizontal plane but we can also envision lattice
segments which will bend the beam in the vertical plane thus allowing for coupled cooling in
the vertical as well as the horizontal plane.



A Possible

Model to be Constructed to Test the Achromatic System


As we progress on the study of the lattice for the achromat structure, we will study and
consider the possibility to construct a small test model with small magnets appropriate to the
energy of the in
jected particles. To test injection and extraction, it seems best to use an
electron beam. Such beams exist at BNL and UCLA and could be available for such a test.
While we do not expect to be able to cool such a beam, we can study the efficiency of
inj
ection and extraction as well as the lattice properties and tune of the machine and compare
with our simulations. While not part of this proposal, it may be possible to later upgrade this
system to actually inject and cool low energy muons. We would also

search for existing rings
or magnets that could be used in our test to save resources.



b) Anticipated Public Benefits


The mystery of the neutrino is of key interest to segments of the public population and the
U.S. government. The fact that different
flavors of neutrinos can transform into each other
has also caught the public interest. A µ
±

Neutrino Factory sending a beam over 2000 km
would clearly excite people. An achromat system with either a high pressure gas or LiH
cooler could be a key componen
t of a Neutrino Factory and a Muon Collider. A further
possibility is to develop a muon collider to study a certain form of scaler particle that could
hold the key to the origin of CP violation in Nature and the existence of matter over
antimatter in the
early universe. The Director of Fermilab has stated there is interest in a
muon collider in the long term future of Fermilab. We believe this would generate public

11

interest. In a more general sense, a robust/simple system to cool beams could have other

applications such as ion lasers and even medical applications both having possible public
benefit.


Industrial and Commercial Applications of Cold Muon Beams from a High Pressure
Gas Ring Cooler


(a)

Possible element selection by muon radiography in general


Cosmic ray muons were used years ago to study the pyramids in Egypt by L. Alveraz.
There could be new commercial uses of very cold energetic muon beams that have been
cooled by the gas ring cooler. These beams would likely have to be accelerated to greater

than 600 MeV energy in some cases and would need an energy spread of less than 100
KeV and a very small spot size. Examples of objects that could be studied at the required
energy:


(i)

Human head: 60 MeV

(ii)

Homeland security search for fissile ~600 MeV material
s in trucks (with oil for
example)


Other applications of very cold intense beams could be muon catalized fusion. Current studies
of this process show low efficiency using intense cold muons with a clean deabration to low
energy could boost the efficiency.


(b)

Use of intense sources of muons in condensed matter studies and nanotechnology and
other technology


A 6
-
D ring cooler as described here could help collect very large numbers of cold
muons. In principle these muons could be decelerated to low energy by d
E/dX
systems or other means with low energy on electrostatic device and dE/dX combined.
In this case we would have a new source of cold muons to use for condensed matter
studies. There are two key reasons why cold muons are useful:

a.

The range of the muon ca
n be very small, allowing the muons to stop inside of
nanostructures (the range of a 1 KeV muon is 8nm).

b.

The polarization of the muons can be used to test the magnetic fields inside the
structure.

Both of these methods are in use today around the world, bu
t the muon intensities are
rather small (i.e. for PSI of order 10
6



10
7
). We quote from a recent talk at Nufact04
in Osaka Japanon the advantage of higher muon fluxes: “High quality muon beams
(flux, emittance, brilliance) would have great impact on the a
pplication of muons in
nanoscience (e.g. microbeam, possibility of lateral resolution on

m scale,
investigation of ~100

m x 100

m samples”.


Several of these schemes to decelerate muons have been pioneered by K.
Nagamini as collaborators at RIKEN.


We l
ist several results from this type of research at the end of this section. This
could be a unique tool for the study of nanostructures since no other particles can have

12

a nanometer range and then decay probing various features of the nanostructures. For
ba
sic method see E. Morenzoni et al, NIMB
192
(2002), 254.


For the ring cooler the beam could be extracted and then sent through an RFQ
system to reduce the energy to the desired level. It may be possible to increase the
flux of muons for this research by u
p to three orders of magnitude.

That would greatly enhance this field of research. In Phase II we would attempt a
conceptual design of such a system of decebration for muon technology science.


i) Study of DNA with a cold polarized muon beam



In the work

of E. Torikai
1

et al, the use of labeled electrons with muons, which was
first applied to conducting polymers [sentence fragment]. The possibility of the use of muon
spin relaction can measure the electron transfer in the DNA strands. High intensity, co
ld
polarized muons made possible by ring muon coolers could carry out more important studies.



ii) Nanotechnology studies with cold polarized muon beams



In the work pf T.J. Jackson
2

cold polarized muons are used to measure magnetic fields
below the surf
ace of superconducting material. This is a new technique for nanotechnology
that can have major implications for the industrial use of 6
-
D muon coolers such as the one
proposed here.


Some references to some applications to this section


a.

A Suter et al, Phy
s. Rev. Lett.
92

(2004), 087001 uses low energy polarized
muons to probe beneath the surface of a superconductor to ~ 200nm depth.

b.

R. Khasann et al, Phys. Rev. Lett.
92

(2004), 05602 uses low energy muons to
determine the
16
O/
18
O ratio at a few 100 nm befo
re the surface of YBa
2
Cu
3
O
7
-
8
(a high temperature superconductor).

c.

K. Nagamina, Uses of slow muons in life sciences, J. Phys. G Nucl. Phys.
29
,
1507 (2003).


(c)

Bunch collection in the high pressure gas ring for a

+

-

collider



There is an exciting poss
ibility that one of the key problems for developing a

+

-

collider can be partially solved in the ring cooler. The problem is that a collider
requires single bunches of

+
and

-

. However the number of bunches from the target
may be in the 100s. At the 2
001 Snowmass meeting we proposed one possible
solution using “parking rings” to store the primary bunches and then to extract them
and overlap. This scheme will increase the final emittance.


The ring cooler discussed here introduces two new components in
the ring:


1)

The particles entering the ring at higher energy and spiral as the energy is
lost




1

E. Torikai et al, Hyperfine Interactions
138
, 509 (2001).

2

T.J. Jackson et al, Phys. Rev Letters,
84
, 4958 (2000).


13

2)

The high gradient RF produces “deep buckets” that will capture the
particles.

There is a likelihood that each incoming bunch will be captured into the three bucket
s.
The following injected bunch will follow the same process. If we consider the
ultimate process of a very high pressure gas and very high RF energy gradient it is
easy to see how this bunch capture (or compression) occurs.


One goal of the Phase II progr
am would be to carry out a full simulation of
this process. Already in the current simulation studies we can see some evidence that
such a process can occur in a strongly dissipative/high gradient RF ring.


(d)

The SUSY Higgs Factory

+

-

collider


In the ear
ly days of the muon collider (~1992
-
95) the focus was on the study of the
Higgs Boson in a “Higgs factory”

+

-

collider; see the attached invited paper for the
Nufact 04 meeting for references. This collider required a very small energy spread (~
few MeV
) and a center of mass energy of 120


150 GeV. This in turn requires
extremely good 6
-
D cooling.


Within the model of supersymmetry there will be other key Higgs bosons [A,
H] that are opposites CP parity. These are expected to be nearly mass degenerate,
split
by GeVs (not MeVs). with mass in the few hundred GeV/c
2
range. Experiments at the
LHC are likely to detect the combined [A, H] state, but unlikely to separate the two
states. A e
+
e
-

linear collider will also have difficulty. The

+

-

collider can mor
e easily
scan through these states and separate them.


In addition the 6
-
D cooling required now is much less. The current simulation
of a full scale high pressure gas ring cooler indicates that the 6
-
D cooling could be
adequate for such a collider. Additio
nal transverse cooling could still be needed, but
the feasibility of the 6
-
D ring cooler could lead to the development of a new scheme
(Figure 1) for such a SUSY Higgs factory. A small test ring studied in Phase II and
constructed with 6
-
D cooling observed

would be a big step towards the development
of such a cooler in the USA.


c) Degree to which Phase I has Demonstrated Technical Feasibility


SBIR Phase 1 Lattices


The basic strategy that we follow is to examine rings with achromatic bends. These lattic
es
allow for dispersion free straight section before and after the bends. Depending on the
orientation of the magnetic field in each of the bends, we can have either closed rings or open
structures. The benefit of closed rings is that we can more effici
ently re
-
use the installed rf
cavities thus making for cheaper construction and operations costs. The benefit of open cell
structures is that we can greatly relieve the injection/extraction issues with regard to
introducing the beam into the system and th
en taking the beam out of the system. In addition,
we will have the option to taper each subsequent lattice so as to take advantage of a beam
which is shrinking in size as it cools.




14

We have explored a number of different ring lattices for this study;

for each there is a
corresponding open or S
-
configuration. Cooling simulations were done on many of these
lattices with variations of the design momentum and corresponding magnet strengths.


All of the ring lattices have a racetrack configuration with two

180 deg arcs and two straight
sections. The magnets are dipoles and solenoids. The arcs are achromatic horizontally and
nearly so vertically. The result is that the dispersion is zero in the straight sections. This
enables the S
-
configuration.


The arcs c
ontain 4 dipoles separated by solenoids (a 3
-
dipole arc lattice was also designed).
The straight sections contain either 2 or 3 solenoids. In order to reduce the horizontal and
vertical coupling, the fields of successive solenoids alternate in direction
.


We have chosen to concentrate on two cases, case 2 and case 4. Case 2 has 3 solenoids per
straight section: one at each end and one in the center. Case 4 has 2 solenoids per straight
section: one at each end and none in the center. This feature has th
e disadvantage of leading
to increased beta functions in the lattice and hence a smaller dynamic aperture. It does,
however, result in straight sections between the bends which can more easily accommodate
injection/ejection kickers.


The strategy envision
ed is to cool the beam in two stages. The first stage would use the case 2
lattice in the open, repetitive S
-
configuration. In this configuration we will take advantage of
the ability to taper the lattices so that each section can scale down in dimensions

while the
fields scale up with the same scale factor. This will lead to each section having identical tunes
but result in diminishing beta functions and hence greater cooling. The second stage would
use the case 4 lattice in the closed ring configuration

where tapering is not an option.


The arguments for this strategy are that in the first stage injection and extraction is relatively
trivial with the open lattice, the case 2 lattice has smaller beta function values and larger
acceptance than the case 4 l
attice and the S
-
configuration periods can be tapered, that is they
have successively smaller dimensions and higher magnet strengths.



The second stage ring injection
-
extraction problem is made easier by the longer open space in
the straight section and
the smaller emittance beam provided by the cooling in the first stage.


Figure 8 is an illustration of the Case 2 ring. Figure 9 showed te Case 2 beta functions and
dispersion. Figure 10 illustrates the Case 2 design with an open configuration. Figure 11 i
s an
illustration of the Case 4 ring. Figure 12 shows the Case 4 beta functions and dispersion.




15


Figure 8: Case 2 ring.


Figure 9: Case 2 beta functions (green and red curves) and dispersion (blue curve).


16


Figure 10 Case 2 Design with Open Configura
tion


17




Figure 11 Case 4 Ring with dipoles and solenoids without center solenoids


18


Figure 12 Case 4 beta functions (green and red curve) and dispersion (blue curve)



Details of the Lattice Design


The arc dipole edges lie along radii extending from
a common point, the arc center. The two
edges of each dipole subtend an angle 1/2 that of the dipole bend angle (45 deg for a 4
-
dipole
arc) and the edge angle to the reference orbit is 1/4 of the bend angle, or 11.25 deg. The
dipoles have zero gradient and

they contribute half of the total arc phase advance of 180 deg.
The other half is provided by the solenoids. The reasons for this arc design are to make the
arcs achromatic horizontally and nearly so vertically, and to make the beta functions nearly
equal

in both planes. As a result, the dispersion is zero in the straight sections and their
solenoids are able to transmit the beta functions between the two arcs. The global fractional
tunes and the amplitude of the beta function oscillations in the ring are

determined by the
strength and spacing of the straight section solenoids.


Demonstration for Cooling


We can see clearly the results of 6
-
D cooling for the case of 145 MeV/c circulating muon
beams within the ring using the Case 2 lattice (See figures 13,
14, and 15 below). Here we
use a set of four 2 degree liquid hydrogen wedges per full turn placed at the minimal beta
points of the lattice. We propagate two sets of muon beams through the lattice, one a warm
beam with initial normalized emittances of 5.
3mm
-
rad in the horizontal plane; 6mm
-
rad in

19

the vertical plane; and 8mm
-
rad in the longitudinal plane. As a second example we begin
with a cold beam, i.e 0mm
-
rad in all three planes. Note that the warm beam is cooling toward
the equilibrium emittance whi
le the cold beam is warming. The equilibrium emittances for
this example are: 3mm
-
rad in X, 2mm
-
rad in Y; and 4.5mm
-
rad in Z.



Figure 13 Emittance evolution in the horizontal plane




Figure 14 Emittance evolution in the vertical plane





20



Figure
15 Emittance evolution in the longitudinal plane


Closed Orbits


We use a code which finds closed orbits by repeatedly running ICOOL. In the process, we
also obtain approximate eigenvalues (i.e., tunes). These results are shown in Figures 16 and
17. N
otice the two stable regions separated by a stopband. One may be able to operate in the
stop band as long as one spends enough time out of it and has enough cooling. Since in the
middle of the stop band, there is over a 10% growth in one direction, this
would seem
difficult. The bounds in the range are caused by the inability to find closed orbits

.

To have longitudinal focusing, one wants wants the machine to be non
-
isochronous. This
design becomes isochronous at a momentum around 170

MeV/c. One there
fore will run in
the lower
-
energy passband.


Next, particles were launched at various amplitudes about their energy
-
dependent closed
orbits to determine the energy
-
dependent dynamic aperture of the machine. To do this, one
uses the eigenvectors of the tra
nsfer matrix at each energy to determine the transformation
from normalized variables to the real phase space coordinates. Particles are then launched
with various values of their normalized amplitude (in an uncoupled system, this normalized
amplitude wou
ld be
)
/(
2
mc
p
a

, where
a

is the maximum displacement from the closed orbit
for betatron oscillations with this amplitude,
p

is the momentum,


is the Courant
-
Snyder beta
function,
m

is the particle mass, and
c

is the speed of light) to determ
ine whether they survive.
The results are shown in Figure 18. One sees poor dynamic aperture for low energies and
better dynamic aperture at high energies. Dynamic apertures are shown only for the chosen
passband because they are impossible to compute i
n the stopband. These results are
indicative of a positive tune shift with amplitude, in that lower energy particles move toward
the low energy/high tune stopband as their amplitude increases, whereas the high energy
particles move away from the high
-
ener
gy stopband as their amplitude increases. Thus, in
choosing an operating region for this machine, we will choose momenta close to the


21

maximum (we choose 152

MeV/c) while staying somewhat away from the minimum
momentum (we choose 136

MeV/c).



Figure 16:

Time of flight on the closed orbit as a function of total momentum







22



Figure 17: Real part of the eigenvalues as a function of total momentum.The machine is
stable if the absolute value of all eigenvalues is less than 1. There is instability betwee
n about
153 and 159 MeV/c (the unstable

eigenvalues don't appear on the plot). For stable
eigenvalues, the value plotted is the cosine of 2


times the tune.


RF Voltage


The equations of motion can be approximated by


)
(
)
(
0
E
T
E
T
dn
d




E
V
dn
dE




)
sin(



(
1
)

Here
n

is the number of cells,


is the time of flight relative to a reference particle,
E

is the
particle energy,
V

is the maximum voltage gain per cell,


is the angular RF frequency,


is
the RF phase,

E

is

the energy lost per cell due to cooling, and
T
(
E
) is the time of flight per
cell as a function of energy. This results in a Hamiltonian


E
V
E
E
E
T
E
S











)
cos(
)
)(
(
)
(
min
0

(
2
)

where




E
E
E
d
E
T
E
S
min
)
(
)
(

(
3
)






23


Figure 18: Horizontal (left) and vertical (right) amplitude and energy of particles that
survived for 120 cells. If a point is plotted, the particle survived.


For a periodic system,
)
(
0
E
T

would need to be some multiple of


/
2
, but that constraint
will be ignored here, and
0
E

will be treated as a variable to be adjusted.

There are 5 steps in performing this computation:

1.

Create
)
(
E
T
by making an interpolating function from t
he ICOOL data. I use B
-
Splines with 10 basis functions and order 4.

2.

Compute
)
(
E
S
. This is trivial from the spline representation.

3.

Compute
)
(
0
E
T
. This is found from the condition that the value of the Hamiltonian
on th
e top and bottom branch of the separatrix is equal. We choose two energies

E

and

E

for the separatrix bounds. I used energies corresponding to the momenta
136

MeV/c and 152

MeV/c. Then


.
)
(
)
(
)
(
0







E
E
E
S
E
S
E
T

(
4
)

The resulting time is 27.412

ns.

4.

Invert
)
(
E
T

to find
0
E
. This is straightforward using a Newton iteration on the
spline representation. The momentum corresponding to this energy is
143.520

MeV/c. Note this is
not

the reference momentum in ICOOL.

5.

Find the voltage and phase that gives this maximum separatrix. The voltage can be
found from the equation


)],
)(
(
)
(
)
(
[
2
)]
/
(
sin
2
/
[
)
(
0
0
0
1
2
2
E
E
E
T
E
S
E
S
V
E
E
E
V
















(
5
)

and the phase from


.
sin
E
V




(
6
)



24


Figure 19: RF Voltage and phase as a function of the energy loss in the absorber for the
parameters given in the text.

For our parameters, the RF phase and voltage as a function of the energy loss per c
ell in the
absorber is shown in Figure 19. Note that for large phases, the acceptance in time becomes
reduced significantly. Thus one would like to run with relatively modest energy losses, say
around 1

MeV per cell.


To ensure that particles with energy

E
0

are synchronized with the RF, one must set the
ICOOL reference momentum properly. However, the ICOOL reference particle does not
follow the closed orbit at
E
0
, but instead follows a fixed trajectory with a velocity determined
by the “reference momentu
m.” So, one does a run with an arbitrary reference momentum,
finds the time that the reference particle takes, then computes the momentum that a particle
would need to have, following that same trajectory, to have a time
T
(
E
0
). For the values
above, that

value is 143.955

MeV/c.


We next examine the machine acceptance without any absorbers. The results are shown in
Figure 20. Note the energies transmitted above the stopband. This is likely due to two
effects: first, the positive tune shift with amplitu
de described above; second, the fact that
large amplitude particles have a longer time of flight, meaning that to remain synchronized to
the RF, they will need to have a higher energy.

Cooling

First, one must determine the desired energy loss. For one 6.6
25

m long cell, there is a decay
of 0.74%. To have a merit greater than unity, one would need an energy loss of at least

25

0.5

MeV in the absorber. Since the energy acceptance is only around 15

MeV, one does no
want to lose too much energy, otherwise beam
will be lost due to the energy




Figure 20: Longitudinal (left) and transverse (right, blue for horizontal and red for vertical)
initial conditions for particles that survived for 120 cells with RF and no absorbers.


variation (although this could pote
ntially be corrected by adjusting magnet currents to the
variation in the particle energy down the channel).


Based on the arguments seen given so far, about 1

MeV of energy loss is desirable. An initial
run was done with 6° liquid hydrogen wedges which w
ere adjusted to have the proper energy
loss, the results of which are shown in Figure 21. Note that for now, multiple scattering,
energy straggling, and decay loss have been turned off, since this simplifies the process of
optimizing the lattice. After an

initial mismatch, the longitudinal and vertical emittances fall,
while the horizontal emittance grows. This is indicative of excessive coupling from the
wedge. Furthermore, note the continuous loss of particles, even after the energy mismatch.
This may

be partially due to

the growth in the horizontal emittance, but it appears to also be
coming from longitudinal losses. Particles with large longitudinal amplitudes are not
damping fast enough, and are eventually getting lost. This appears to be related
to the strong
horizontal
-
longitudinal coupling. This may be corrected by injecting a more properly
matched beam with an emittance appropriate to the acceptance of the system, which we will
do at a later stage in this study.


The emittances are computed as

follows: first, the covariance matrix for each plane is
computed. There is a transformation matrix which would transform a circular distribution to
the actual elliptical distribution; if the covariance matrix is


,






pp
xp
xp
xx





(
7
)

then the transformation matrix is


26


,
/
/
0
/








xx
xx
xp
xx








(
8
)

where


2
xp
pp
xx







(
9
)

is the emittance. We then multiply the coordinates for each particle by the inv
erse of this
transformation for each plane. We then take the sum of the squares of the two resulting
coordinates. If this number is greater than
k
2

times the emittance for any of the planes, the
particle is thrown out (this is referred to as a
k


cut).
This process is repeated until no more
particles are cut. The purpose of this process is to thrown out particles which probably
should be considered “lost,” but which contribute disproportionately to the emittance due to
their large amplitudes.

One can pe
rform an exponential fit to the central region of the curves in Figure 21 to obtain
the approximate cooling rates for the three planes. The results were 60



10
-
5

m
-
1

in the
horizontal plane,
-
88



10
-
5

m
-
1

in the vertical plane, and
-
114



10
-
5

m
-
1

in t
he longitudinal
plane. Call the inherent (without coupling) rate for the transverse
k

, and for the longitudinal
k
||
. Then the observed cooling rates
k
x
,
k
y
, and
k
z

are related by


C
x
k
k
k






k
k
y

,
||
C
z
k
k
k



(
10
)

where
k
C

is a coupling factor which is approximately proportional to the wedge angle. From
these equations,
k
C

is approximately 148



10
-
5

m
-
1
. Say that we want
k
x

=

k
z
. In




Figure 21: Initial cooling run with 6° wed
ge angle. Solid lines are with a 3


cut, and dashed
lines are with a 4


cut.


27


that case, we would want
k
C

=

61



10
-
5

m
-
1
. To achieve this, the wedge angle should be

reduced from 6º to around 2.5º. From a run with 2.5º, one finds two effects: first, tha
t while
there is now damping in all planes, we have not achieved
k
x

=

k
z
; second, that there are still
losses after the initial mismatch. Repeating the exponential fits that were done for the 6º
case, one estimates a corrected wedge angle of 1.7º. As for

the losses, it appears that in fact
the particles leaving the bucket are the source of the later losses. There appears to be very
strong longitudinal
-
horizontal coupling which is contributing to these losses (this is likely
what one is seeing in the osci
llations of the longitudinal emittance in Figure 21. The effect
may be remedied by choosing an initial distribution with a sufficiently small longitudinal
emittance that these high
-
amplitude particles don’t appear.


Phase I accomplishments:


We have const
ructed a lattice, achromatic at the reference energy which gives cooling in all
planes when far from equilibrium.


We have analyzed the lattice, finding the energy
-
dependent behavior of that lattice, in
particular tunes and time of flight. We have analyze
d the longitudinal dynamics in the lattice.
We have found the transverse dynamic aperture as a function of energy.


We have identified aspects of the beam dynamics in the system that are in need of further
analysis. In particular, to improve the transmis
sion in the lattice, we should modify the lattice
to make the dynamic aperture further from the equilibrium emittance. We should also reduce
the negative effects of transverse
-
longitudinal coupling.


Injection into and Extraction from the Ring


Injection



It is convenient to divide this process into two parts. Here we will assume that the
beam pulse is shorter than the revolution time of the particles around this ring. In this case the
usual technique is to use a septum magnet in a field
-
free region to be
nd the (matched)
injection beam parallel to the local reference orbit displaced from the reference orbit by the
beam width plus the septum thickness. One quarter of a betatron oscillation downstream from
the septum magnet, the injected beam crosses the ref
erence orbit at an angle. A fast kicker
magnet is located (also in a field
-
free region) which deflects the beam so that it is parallel to
the reference orbit. As soon as the beam has passed through the kicker, the field in the kicker
is reduced to zero. Si
nce the beam is matched, this results in no emittance growth by the
process.


Another solution is to inject the beam directly into the aperture using a channel that
passes through the bending magnet. This is possible if the magnetic field can be shielded.
The
use of a superconducting pipe to exclude the field was first developed at SLAC.
3





3

F. Martin et al, SLAC Pub. 1040,
May 1972, A Four Meter Long Superconducting Magnet
Flux Exclusion Tube for Particle Physics Experiments. M. Perl, Private communication.


28


A sketch of this scheme is shown in Figure 22. This device was constructed at SLAC
and we intend to follow up the construction and results of the test and the existence
of the
remaining components at SLAC on future work. An illustration of the injection
superconductor pipe is shown in Figure 23. The design was made to shield a 2T magnetic
field which is larger than the needs for injection in our case.



Figure 22: Inject
ion scheme through bending magnet





29


Figure 23: Injection superconductor pipe



We illustrate more on this flux tube here in quotes from the paper in the abstract: “The
design and construction of a field
-
free channel through the transverse central magnet
ic field
of a 54
-
inch aperture magnet is described. This flux
-
free path is an essential part of the
experiment at the Stanford Linear Accelerator Center to measure the electron production of
neutral rho mesons. The flux exclusion tube itself is made of Nb
3
Sn tape bonded with lead
-
in
solder to form a rigid tube of laminated superconducting material, approximately four meters
long and from 5 to 25 mm in diameter. The transverse magnetic field shielded exceeds 1.5 T.
The performance of the tube and its associa
ted cryostat is discussed. In order to bring the

30

injected beam into the correct cooling ring aperture a small kick will be required. We believe
an induction kicker is the best possibility to be used.


Renderings of a 6
-
D Muon Cooling System




Figures 24
and 25. Figure 24: View of the ring with injection flux tube shown and the
injection/extraction kicker as well as the other components of the ring. Figure 25: Same as
Figure 24 but a different perspective. The wedges of LiH are shown here.


Figure 26:
Flat view of the ring showing the various components.


31



Extraction of the Cold Beam



As an example of this, we might wish to kick a 250 MeV/c meam of
-
.3 m width into
the ring with a kicker. The amplitude implied is .15 m. The beta function of the ring is

about
.5 m, so the kick angle is about 0.3 rad, implying a strength of 0.25 Tm. The distance between
magnets is about .25 m, so the kicker field is about 1 T, which much be switched off in
several nanosec. This is not beyond current practice. (Kickers usu
ally employ fields of .01
-

.04 T with 50


100 nsec fall times). An alternate strategy is to employ the dissipative forces
to gain effective strength. Absorbers may reduce the energy of the beam sufficiently to cause
the beam radius to shrink sufficiently

to miss the absorbers on subsequent revolutions as the
cooling takes place.


For the extraction system we propose on induction kickers with a rise time of 50 ns as
studied by Don Summers and Bob Palmer and reported in a UCLA meeting on ring coolers in
200
2.



Figure 27: Induction Kicker for Injection






32

Table 2 is taken from the Palmer and Summers report and can be found in the UCLA Ring
Cooler Meeting Report.


Table 2.




Cooling



CERN

p


Ind Linac

f


1.05



f


1.05



f


3



m


V

1.05 108



c m/s

3 108




n



mm

10




x

m

1.0



f Bdl

.43

.088


L m

1.0



5

5.0

t
rise

ns

50

90

40

t ns

100

500

100





2


B T

.42



0.018

0,6

X m

.42

.08


Y m

.63

.25


J
magnetic

J

8200



13

8000

I kA

150


73

V
1turn

kV

5,700

800


n
units

30

10

50

V
p.s.
kV

190

80

190


Parameters of an induction kicker for injection worked out by B. Palmer and D. Summer.



The estimated position of the induction kicker is shown in Figures 24, 2
5, and 26. This
means it can be used for injection and extraction. Further study of this kicker will be needed
to match its timing and properties to the injected and extracted beam.


RF System for the Ring Cooler


33


A frequency near 200 Mhz has been chosen

to capture and cool the muon beam. Since the
cavity will be immersed in high pressure hydrogen gas, and the rf pulse length is short, we
might be able to operate the cavities with rather high peak surface fields. The most obvious
choice of cavity is the q
uarter wave drift resonator. The transmit time factor is


.



This means the energy gain varies by about 15% from the axis of the cavity to te edge of the
useful region (Beta, Gamma = 2, a = 0.25m). We assume now that this is acc
eptable, but
needs more study. Since superconducting magnets can use only about 60% of the coil radius,
there is a space between the coil and the inner surface to store the rf magnetic energy. On the
other hand, there is a potential problem with sparking i
n the magnetic field. This also needs
more study. The cavities are short, only about 37 cm in length. The presence of dipole (m=0)
modes need study to see if their suppression is necessary. In order to keep all cavities in the
correct phase, it might be wi
se to investigate the use of coupling in pi mode to simplify the
power source. In this way it might be possible to drive the whole system with a single power
source.


The Phase II Project


a) Technical Objectives


We wish to demonstrate a viable system for

6
-
D cooling which will ultimately entail
circulating a muon beam in a ring to allow for multiple passes through accelerating cavities.
This will lead to smaller foot
-
print cooling systems, less capital investment in cooling
components and reduced operati
ng costs.


Key objectives will include: a) design of open cooling structures to allow for easy injection
and tapered cooling; b) a credible injection scenario which will allow for injection into a
compact ring; c) a robust ring design for final cooling; d
) an ejection system from the cooling
ring.


In order to allow for a 3
-
fold factor in lattice tapering, we anticipate the need for solenoids
capable of operations with axial B
-
fields up to 10T. We propose to accomplish this using
HTS (High Temperature Sup
erconducting) conducting material in order to allow for higher
operating temperatures (30 to 40 degrees K) and hence simpler operating conditions and low
operating costs.



34

In keeping with the desire for simpler operations, we will incorporate lithium hydri
de
absorbers into the cooling strategy, thus obviating the need for LH2 in the cooling system.


b) The Work Plan


Plans for simulations in Phase II


We have evidence that the equilibrium emittance is closer to the dynamic aperture than we
would like. This

can be seen from the dynamic losses that occur during cooling. One of the
primary goals of the beam dynamics studies in Phase II would be to improve this situation.


The dispersion and its derivative are zero only at the reference energy for this lattice
. This
leads to significant longitudinal
-
transverse coupling for particles with large longitudinal
oscillation amplitudes, which appears to be the primary limitation in the dynamic aperture in
the lattice. This nonlinear dispersion could be straightforwa
rdly reduced by reducing the
bend angle in each cell.


The equilibrium emittance itself could be improved if we were able to lower the beta
functions (over the entire energy range) at the absorber. We can study making the lattice cell
more compact to acc
omplish this.


We will study other variations of the lattice cell that our Phase I studies have indicated would
improve the performance of the lattice.


Currently the lattice is running in the fourth passband (tunes between 1.5 and 2.0 per cell).
We could

potentially improve the energy acceptance by re
-
designing the lattice to run in a
lower passband. A larger energy acceptance would allow for a larger energy loss in the
absorbers, improving the cooling rate. We can also potentially improve the energy
ac
ceptance by lowering the magnitude of the frequency slip factor.


We could also increase the cooling rate by using a longer absorber, but this would reduce the
phase acceptance of that lattice. Phase acceptance may not be essential, but at some point
ther
e will be a problem since RF cavities are not distributed everywhere. Modest adjustments
to the RF parameters, such as the reference energy, may also give improved performance.
When the lattice if formed into a closed ring, the RF frequency and the ring
circumference
will have to be made commensurate.


For the case where the lattice is not closed, the alternating bending directions will break the
lattice symmetry and give additional resonances near the quarter tunes. If the lattice were
achromatic at the

quarter tune, this resonance would be weak. While the lattice is still
relatively achromatic at the quarter tune, it is not perfectly so, so we will need to study the
effect of this resonance.


Running at a higher energy would be desirable since that wou
ld raise the sum of the damping
partition numbers.



35

All of these studies require that the lattice properties be understood over the entire energy
range of the machine, not just at the reference energy. We have demonstrated in our Phase I
work that we have
the tools to do this effectively.







Applications: Related Research or R&D


Scientific Goals


i) Low Energy µ
+
µ
-

Colliders


In the model of supersymmetry there will likely be one low
-
mass Higgs (h
0
) and two high
-
mass (or supersymmetric) Higgs A and H. F
or the parameter tan ß, larger values lead to a
near mass degenerate system of H and A states, most likely in the 300
-

500 GeV mass range.
Current evidence on SUSY suggests a large value of tan ß. In this case the coupling of H and
A to tt and gauge boson
s is sharply reduced, making them difficult to produce and study at
the Large Hadron Collider or Next Linear Collider. In this sense the muon collider is
complementary to the Ile and the LHC (more on a SUSY higgs factory later).


ii) High Energy µ
+
µ
-

Colli
ders


The FNAL director has approved a long range plan to study a 1.5 TeV µ
+
µ
-

collider! The
cooling methods proposed here could be important for this plan. This collider is
complementary in all ways to the next International Linear Collider (ILC) being
planned by
the international high energy physics community. We quote:


“The U.S. government seeks improved methods to study the fundamental properties of matter
and energy and shed ligt on the conditions that prevailed at early times in the evolution of th
e
universe.. The International Linear Collider (ILC) is one such method but has limited reach in
energy. Muon colliders are another such method, with the potential for substantially higher
energy reach; their technical challenges require substantial R&D. S
BIR grants havebeen an
important means by which innovative approaches to overcoming the challenges of muon
colliders have been worked out. From being merely an idea in the minds of researchers fifteen
years ago, the muon collider concept is now beginning t
o enter the mainstream as evidenced
by the recent establishment by Fermilab Director Oddone of the Muon Collider Task Force,
charged with laying out an R&D path towards the possible future construction of a muon
collider at Fermilab (possibly as an upgrade

to the ILC).



As just mentioned, the potential market is the U.S. government and its National Laboratories


a billion
-
dollar
-
plus market. If a muon collider (or muon
-
based neutrino factory) is built,
muon cooling techniques and apparatus will be a signi
ficant share of that market. There is a
reasonable possibility that the proposed muon
-
cooling approach would be one ultimately

36

usef in a muon collider or neutrino factory. (It is too early in the R&D process to try to
estimate the probability more precise
ly).




c) Performance Schedule


Beam Dynamics Studies/Lattice Design Effort



0


6 months


Finalize performance characteristics of Case 2 lattice. Identify





improvements.



3


12 months


Design improved lattices and incorporate LiH absorbers.



6
-

18 months


Detailed dynamic studies of improved lattices



9
-

18 months


Study tapered S
-
channel



18


24 months

Consolidate results of study and prepare Phase II final report.






Facilities and Equipment


Consultants and Subcontractors



Phase II
Funding Commitments (Commercial)


Phase III Follow
-
On Funding Commitments


Bibliography and References