11/13/23

Higgs factory (HF) report: front end

Chapter 6 Outline

1\. General overview of ionization cooling

.1 transverse cooling

.2 emittance exchange

2\. Summary of status report (SR) front end scenario

.1 rf phase rotation

.2 cooling

.3 simulations

3\. Summary of neutrino factory (NF) front end scenario

.1 induction linac (IL) phase rotation

.2 precooling

.3 bunching

.4 cooling

.5 simulations

4\. Simulation summary

5\. Modified scenarios of NF front end for HF

.1 eliminate it , use SR solution

.2 replace IL with adiabatic buncher

.3 use it with stacking ring

.4 modify it using cooling rings

6\. Final cooling schemes

.1 limit for sFOFO

.2 limit for Li lens

.3 limit for ring coolers

7\. Emittance exchange

.1 schemes

.2 simulations

8\. Assessment

Chapter 6 Higgs Factory Front End

6.1 General overview of ionization cooling

# Cooling for the Muon Collider 

# Introduction

Table X-1 shows some suggested parameters of Higgs-energy μ^+^−μ^−^
colliders. According to these tables, we need bunches of \~10^12^ μ's
within transverse emittances of ε~N,rms~ \~ 10^-4^ m and ε~L,rms~ ≅
0.01m. These emittances are substantially smaller than that of the muon
bunches produced from pion decay, as described in the targetry section.
The transverse and longitudinal emittances of these production beams are
more like ε~N,rms~ \~ 2×10^-2^m and ε~L,rms~ ≅ 1m. Thus we need to cool
in each of the emittances by, roughly, a factor of 100, or in 6-D
emittance by a factor of \~10^6^, and the cooling must be completed
before μ decay. The cooling method which can provide sufficiently fast
cooling is ionization cooling, and the final beam parameters are within
the expected capabilities of ionization cooling.

**Cooling process and requirements**

The cooling process which will be used is ionization cooling. In
ionization cooling (μ-cooling), particles pass through a material medium
and lose energy (momentum) through ionization interactions, and this is
followed by beam reacceleration in rf cavities.(Figure 4-1) The losses
are parallel to the particle motion, and therefore include transverse
and longitudinal momentum losses; the reacceleration restores only
longitudinal momentum. The loss of transverse momentum reduces particle
emittances, cooling the beam. However, the random process of multiple
scattering in the material medium increases the rms beam divergence,
adding a heating term which must be controlled in a complete cooling
system. This cooling method is not very practical for protons, which
would have frequent nuclear interactions, or electrons, which would have
bremstrahlung, but is practical for muons, and cooling rates compatible
with muon lifetimes are possible.

The differential equation for rms transverse cooling is \[1, 2, 3, 4,
5\]:

$\frac{{d\varepsilon}_{N}}{\text{ds}} = - \frac{1}{\beta^{2}E}\frac{\text{dE}}{\text{ds}}\varepsilon_{N} + \frac{\text{βγ}\beta_{}}{2}\frac{d\left\langle \theta_{\text{rms}^{2}} \right\rangle}{\text{ds}} = - \frac{1}{\beta^{2}E}\frac{\text{dE}}{\text{ds}}\varepsilon_{N} + \frac{\beta_{}E_{s}^{2}}{2\beta^{3}m_{\mu}c^{2}L_{R}E}$
(4-1)

where the first term is the energy-loss cooling effect and the second is
the multiple scattering heating term. Here ε~N~ is the normalized
emittance, E is the beam energy, β = v/c and γ are the usual kinematic
factors, dE/ds is the energy loss rate, θ~rms~ is the rms multiple
scattering angle, L~R~ is the material radiation length, β~⊥~ is the
betatron function, and E~s~ is the characteristic scattering energy
(\~13.6 MeV).\[6\] (The normalized emittance is related to the geometric
emittance ε~⊥~ by ε~N~ = ε~⊥~/(βγ), and the beam size is given by σ~x~ =
(ε~⊥~β~⊥~)^½^.)

# Longitudinal Cooling and Emittance Exchange

Cooling to collider intensities requires longitudinal cooling, which is
difficult since ionization cooling does not directly provide
longitudinal cooling. The equation for longitudinal cooling with energy
loss is:

$\frac{{d\sigma}_{E}^{2}}{\text{ds}} = - 2\frac{\partial\frac{\text{dE}}{\text{ds}}}{\partial E}\sigma_{E}^{2} + \frac{d\left\langle {\Delta E}_{\text{rms}^{2}} \right\rangle}{\text{ds}}$
(4-2)

in which the first term is the cooling term and the second is the
heating term caused by random fluctuations in the particle energy. Beam
cooling can occur if the derivative ∂(dE/ds)/∂E \> 0. This energy loss
can be estimated by the Bethe-Bloch equation\[6\]:

$\frac{\text{dE}}{\text{ds}} = 4{\pi N}_{A}r_{e^{2}}m_{e}c^{2}\rho\frac{Z}{A}\left\lbrack \frac{1}{\beta^{2}}\text{ln}\left( \frac{2m_{e}c^{2}\gamma^{2}\beta^{2}}{I} \right) - 1 - \frac{\delta}{2\beta^{2}} \right\rbrack$
(4-3)

where N~A~ is Avogadro's number, ρ, A and Z are the density, atomic
weight and number of the absorbing material, m~e~ and r~e~ are the mass
and classical radius of the electron, (4πN~A~r~e~^2^m~e~c^2^ = 0.3071
MeV cm^2^/gm). The ionization constant I is approximately 16 Z^0.9^ eV,
and δ is the density effect factor which is small for low-energy μ′s.
The energy loss as a function of p~μ~ is shown in Fig. 4-2. The
derivative is negative (or naturally heating) for E~μ~ \< \~ 0.3 GeV,
and is only slightly positive for higher energies. In the
long-pathlength Gaussian-distribution limit, the second term in Eq. 4-3
is given approximately by\[7\]:

$\frac{d\left\langle {\Delta E}_{\text{rms}^{2}} \right\rangle}{\text{ds}} = 4\pi\left( r_{e}m_{e}c^{2} \right)^{2}n_{e}\gamma^{2}\left( 1 - \frac{\beta^{2}}{2} \right) \simeq 0\text{.}\text{157}\rho\frac{Z}{A}\gamma^{2}\left( 1 - \frac{\beta^{2}}{2} \right)\ (\text{MeV})^{2}\frac{\text{cm}^{2}}{\text{gm}}$,
(4-4)

where n~e~ is the electron density in the material (n~e~=N~A~ρZ/A). This
expression increases rapidly with higher energy (larger γ), opposing the
cooling process. After adding this energy straggling, ionization cooling
does not naturally provide adequate longitudinal cooling.

However, the longitudinal cooling term can be enhanced by placing the
absorbers where transverse position depends upon energy (nonzero
dispersion) and where the absorber density or thickness also depends
upon energy, such as in a wedge absorber.\[8\](see fig. 4-3) In that
case the cooling derivative can be rewritten as:

$\frac{\partial\frac{\text{dE}}{\text{ds}}}{\partial E} \Rightarrow \frac{\partial\frac{\text{dE}}{\text{ds}}}{\partial E}|_{0} + \frac{\text{dE}}{\text{ds}}\frac{\eta\rho^{'}}{\beta\text{cp}\rho_{0}} = g_{L}\frac{\frac{\text{dp}}{\text{ds}}}{p}$
(4-5)

where ρ′/ρ~0~ is the relative change in density with respect to
transverse position, ρ~0~ is the reference density associated with
dE/ds, and η is the dispersion (η = d x /d(∆p/p)). We have introduced
the partition number g~L~ to describe the cooling rate related to the
mean momentum loss, and the wedge configuration increases the
longitudinal partition number by ηρ′/ρ~0~. It also decreases the
corresponding transverse partition number by the same amount: g~x~ →
(1-ηρ′/ρ~0~), which decreases the transverse cooling. The sum of the
cooling rates or partition numbers (over x, y, and L) remains constant;
a similar invariant sum of cooling rates, with emittance exchange from
radiation at nonzero dispersion, occurs in radiation damping of
electrons. In ionization cooling, however, there is an energy dependence
of this sum of partition numbers, due to the energy dependence of the
natural energy loss. This sum of partition numbers is \~2 at p~μ~ \> 0.3
GeV/c . Fig. shows this sum of partition numbers as a function of p~μ~;
small values of p~μ~, where the sum becomes less than \~1.5 should be
avoided in cooling.

Eq. 4-3 is the expression for energy spread cooling. The equation for
longitudinal emittance cooling, similar to the transverse cooling
equations, is:

$\frac{{d\varepsilon}_{L}}{\text{ds}} = - \frac{g_{L}\frac{\text{dp}}{\text{ds}}}{p}\varepsilon_{L} + \frac{\beta_{L}}{2}\frac{d\left\langle {\Delta E}_{\text{rms}^{2}} \right\rangle}{\text{ds}}$,

where β~L~ = σ~ct~^2^/ε~L~ is the longitudinal focusing function, which
depends on the rf bunching wavelength and voltage.

## Cooling considerations

Some general considerations on the conditions for cooling, and the
required absorbers and beam transports, can be developed from eq. 4-1 to
4-9. From Eq. 4-1 (with g~x~ from eq. 4-7) we find an equilibrium
emittance from setting the derivative to zero:

$\varepsilon_{N,\text{eq}} = \frac{\beta_{}E_{s}^{2}}{2g_{x}{\beta m}_{\mu}c^{2}L_{R}\frac{\text{dE}}{\text{ds}}}$
(4-11)

This represents the minimal obtainable emittance for a given material
and focusing parameter β~⊥~. From this expression, obtaining small
emittance implies having small β~⊥~ (strong focussing), as well as large
L~R~ dE/ds (small multiple scattering) at the absorber. Table 4-1
displays parameters of typical cooling materials; large L~R~ dE/ds
implies light elements (H, Li, Be, ..) for the absorber material.

  ---------------- ------ ------- -------- ---------- ---------- --------------- ------------- -----------------------
  **Table 4-1:                                                                                 
  Material                                                                                     
  Properties for                                                                               
  Ionization                                                                                   
  Cooling\[6\]**                                                                               

  **Material**            **Z**   **A**    **dE/ds    **L~R~**   **L~R~dE/ds**   **Density**   **g~x~βε~N,eq~/β~⊥~**
                                           (min.)**                                            

                                           MeV/cm     cm         MeV             gm/cm^3^      mm-mrad/cm

  Hydrogen         H~2~   1       1.01     0.292      865        252.6           0.071         37

  Lithium          Li     3       6.94     0.848      155        130.8           0.534         71

  Lith. Hydride    LiH    3+1     7+1      1.34       102        137             0.9           68

  Beryllium        Be     4       9.01     2.98       35.3       105.2           1.848         88

  Carbon           C      6       12.01    4.032      18.8       75.8            2.265         122

  Aluminum         Al     13      26.98    4.37       8.9        38.9            2.70          238

  Copper           Cu     29      63.55    12.90      1.43       18.45           8.96          503

  Tungsten         W      74      183.85   22.1       0.35       7.73            19.3          1200
  ---------------- ------ ------- -------- ---------- ---------- --------------- ------------- -----------------------

#  From consideration of minimum-β~⊥~ focusing conditions (such as in a Li lens, see below), we expect to be able to obtain β~⊥~ ≅ 0.01m, which means the transverse emittances can be cooled to ε~⊥,n~ ≅ 0.0001 m-rad (in hydrogen or lithium). Similarly a cooling rf bucket at 200 MHz can maintain the beam within a longitudinal emittance of \~0.01m, and smaller emittances could be obtained with higher frequency rf systems. Thus, the collider emittance goals are within conceptual reach of ionization cooling. However a complete cooling scenario taking the beam from production emittances to cooling emittances must be developed, and some possible approaches will be discussed below.

#  

**The Status Report Cooling Scenario**

**Figure 4-1**: Concept of ionization cooling.

![](media/image1.wmf)

**Figure 4-2.** (dE/dx)/ρ (MeV/(gm/cm^2^)) as a function of muon
momentum P~μ~ for various atoms. Note that this function is heating
(negative slope) for P~μ~ \< \~0.350 GeV/c and becomes strongly heating
(steep slope) for P~μ~ \< 0.200 GeV/c.

![](media/image2.png){width="5.25in" height="4.325in"}

**Figure 4-3.** The sum of the cooling partition numbers Σ~g~ = (g~x~ +
g~y~ +g~L~) as a function of momentum p (0----500 MeV/c). g~x~ and g~y~
are naturally 1 while g~L~ becomes strongly negative for p \< 200 MeV/c.
Σ~g~ remains greater than 0, which means that ionization loss remains
intrinsically cooling at low momenta.

![](media/image3.wmf)

Momentum P~μ~ (MeV/c)

![](media/image4.wmf)**Fig. 4.4.** Overview of an emittance exchange
section, in which longitudinal emittance is reduced by using a wedge
absorber at nonzero dispersion.

6.2 Summary of status report front end scenario

6.3 Summary of neutrino factory front end scenario

The MC collaboration has undertaken two detailed studies of how we could
build a Neutrino Factory \[FS1,FS2\]. Both studies have a number of
features in common.

A schematic layout of the study 1 scenario is shown in Fig. xxx .
Following the target there is a 50 m long drift, a 100 m long induction
linac for phase rotation, a mini-cooling stage, a 17 m long buncher and
a 140 m long cooling section. The total length of the study 2 front end
is 350 m. The layout of the study 2 scenario is shown in Fig. xxx.
Following the target is an 18 m drift, a 108 m induction linac,
mini-cooling stage, an additional 200 m of induction linac and drift to
complete the pahse rotation, a 55 m buncher and 108 m cooling section.
The total length of the study 2 front end is 541 m. Most of the
following description will concentrate on the more recent and more
detailed results from study 2.

6.3.1 Phase rotation using induction linacs

The energy spread of the pions collected from the target and their muon
decay products is very large. It is in fact much larger than the
acceptance of the following cooling stage and accelerators. The energy
spread must first be reduced using a phase rotation system that
decelerates the high energy portion of the bunch and accelerates the low
energy part. Since a long bunch is acceptable in a neutrino factory,
this can be accomplished by first letting the beam from the target drift
so that the energy spread causes a significant correlation to develop
between the particles' energy and position in the bunch. The drift is
followed by one or more induction linacs. The voltage pulse across the
gaps in the induction linac cells can be tailored to match the shape of
the incoming beam bunch.

By breaking the induction linac up into two or more sections it is
possible to significantly reduce the distortion in the resulting
longitudinal phase space of the muon beam. In Study 2 the fractional
energy spread of the beam after the induction linac is reduced to 3.7 %.
The rms bunch width grows to 27 m. There are 0.49 muons within these
cuts per incident proton on target.

6.3.2 Minicooling

The mini-cooler stage consists of several meters of liquid hydrogen in a
solenoidal field. This is used to provide an \~20% reduction in
transverse normalized emittance prior to bunching.

6.3.3 Buncher

After the energy spread of the beam has been reduced in the induction
linacs

the muons are distributed continuously over a distance of around 100 m.
It is then

necessary to form the muons into a train of bunches prior to cooling and

subsequent acceleration. For the 201 MHz rf frequency this produces
about 70 bunches.

First an 11 m long magnetic lattice section is used to gently transform
the beam from the approximately uniform solenoidal field used in the
induction linacs to the so called \"super-FOFO\", or sFOFO, lattice used
in the remainder of the front end. This is followed by the 55 m long rf
buncher itself, which consists of rf cavity sections interspersed with
drift regions. The two functions of this section are performed
sequentially for design simplicity. There is a significant advantage in
using the same lattice in the buncher section as in the following
cooling region since it avoids adding another complicated 6-dimensional
matching section.

The buncher magnetic lattice is identical to that used in the first
cooling section. It

contains rf cavities in selected lattice cells and no absorbers. The
main rf frequency

was chosen to be 201.25 MHz in the front end, so that the beam would fit
radially

inside the cavity aperture and because power sources and other technical
components are available at this frequency. The 201.25 MHz cavities are
placed at the high beta locations in the lattice, just as in the cooling
section. Harmonic cavities running at 402.5 MHz are placed at beta
minimum locations, where hydrogen absorbers are located in the cooling
section. The buncher encompasses 20 lattice cells, each 2.75 m long.
Maximum bunching efficiency was obtained by breaking the region into
three rf stages separated by drift regions

Second harmonic (402.5 MHz) cavities are used at the entrance and exit
of the

first and second stages to linearize the shape of the rf pulse. All
cavities are assumed to have thin Be windows at each end. They are
modeled in the simulation codes as perfect TM010 pillboxes. The electric
field gradient is gradually raised in the buncher from \~6 to 8 MV/m. A
long drift is provided after the first stage to allow the particles to
begin overlapping in space.

By the end of the buncher, most, but not all, particles are within the
approximately elliptical bucket. About 25 % are outside the bucket and
are lost relatively rapidly, and another 25 % are lost more slowly as
the longitudinal emittance rises from straggling and the negative slope
of the energy loss with energy.

A significant coupling develops in this front end lattice, including the
induction linac, between a particle\'s longitudinal and transverse
motions. This coupling develops because particles with different
transverse displacements or angular divergences take different amounts
of time to move axially down the solenoidal lattice. They thus arrive at
the cavities at different points in the rf cycle, thereby obtaining
different acceleration and different longitudinal and transverse
focusing.

The correlation can be expressed as

$p = p_{o} + \text{CA}^{2}$

where C is the correlation coefficient and the transverse amplitude is

$A^{2} = \frac{r^{2}}{\beta_{}^{2}} + \theta^{2}$

There is little correlation between momentum and angular momentum after
the induction linacs, indicating that the field reversal is correctly
located. The magnitude of the momentum-amplitude correlation is \~ 0.7.
This is a higher value than the correlation of 0.45 that would be
obtained without the mini-cooler. Ideally the correlation should be such
that forward velocity in the following lattice is independent of
transverse amplitude. A value of approximately 1.1 would be required for
this.

6.3.4 Cooling

In the Neutrino Factory the rms transverse emittance of the muon beam
emerging from the induction linac must be reduced to at least \~ 2 mm
rad (normalized) in order to

fit in the downstream accelerators, and be contained in the storage
rings. Ionization cooling is currently our only feasible option \[FS1\].
The cooling channel described below is based on extensive theoretical
studies and computer simulations.

The strength of the solenoidal field should be tapered upwards as the
particles proceed through the cooling stage. The transverse momentum of
each particle will decrease, while

the position of the Larmor center will not, causing the net total
angular momentum of the beam to grow. One must flip the field, while
maintaining good focusing throughout

the beam transport and low β~⊥~ at the absorbers. One of the simplest
solutions, based on the FOFO lattice, is to vary the field sinusoidally.
The transverse motion in such a lattice can be characterized by the
betatron resonances, near which the motion is unstable.

The sFOFO lattice is also based on the use on alternating solenoids, but
is a bit more complicated. A second harmonic is added to the simple
sinusoidal field. As

in the FOFO case, the longitudinal field vanishes at the β~⊥,min~
position,

located at the center of the absorber. This is accomplished by using two
short

\`\`focusing\" coils running in opposite polarity. However, unlike in
the FOFO

case, the field decreases and flattens at β~⊥,max\ ,~ due to a
\`\`coupling\" coil located around the linac. Thus, the transverse beam
dynamics is strongly influenced by the solenoidal field profile on-axis
and by the desired range of momentum acceptance.

The channel operates at a nominal momentum of 200 MeV/c. There are six
sections with steadily decreasing β~⊥,min\ .~ In the first three the
lattice half period length is 2.75 m, and

in the last three lattices this half period length is 1.65 m. The
matching sections between these sections also consist of cooling cells,
which differ from the regular cooling sections only by the current
circulating in the coils, with one exception. A different coil length
must be used in the matching section where the lattice period length
decreases from 2.75 m to 1.65 m, and the absorber has been removed.

The lengths of the linacs are constrained by the lattices themselves, as
the focusing coils have a bore radius smaller than the rf cavities, and
by the rf cell length, which must be optimized to give the high Q
required to reach high gradient. These linacs are always placed in the
middle of the half-period lattice cell. Each cell can be phased
separately. In order to improve the Q of the cavity, the iris of the
cell must be covered with a foil

or a grid.

The baseline design calls for thin, pre-stressed beryllium windows with
thicknesses that increase with radius. This arrangement is justified on
two separate grounds. (i) The power dissipated on the iris or foil goes
like the 4th power of the radius (at relatively small radius). We
therefore need more thickness at higher radius to remove the heat. (ii)
The particles at large radius tend to have large transverse amplitude
and are \`\`warmer\" than the central core. One can therefore afford a
bit more multiple scattering at such large

radius. Note also that the windows on each end of the linacs dissipate
half the power than the windows at the boundary between two adjacent
cells. Thus, these edge windows are thinner than those in the center of
the linac. The use of thin aluminum tubes arranged in a Cartesian grid
has also been considered as an alternative.

The absorber material is liquid hydrogen (LH2). The density is
approximately 0.071 g/cm^3^. The energy loss, as given by the
Bethe-Bloch formula with a mean excitation energy of 21.9 eV, is 4.6 MeV
cm^2^/g. The length of these absorbers is 35 cm for the 2.75 m lattices
and 21 cm for the 1.65 m lattices, respectively. The muons therefore
lose \~12 MeV per lattice cell for the 2.75 m lattices and \~7 MeV for
the 1.65 m lattices. This includes the energy loss in the windows. The
LH2 vessels must also be equipped with thin aluminum windows. Their
thickness is 360 μm (220), with a radius of 18 (11) cm, for the 2.75 m
and 1.65 m lattices, respectively.

6.3.5 Simulation results

The β~⊥,min~ function, derived from the beam second order moments
estimated at the absorber\'s center using Geant4, is shown in Fig. xxx.
The beta function steps down with each new section of the cooling
lattice

![](media/image5.png){width="3.5208333333333335in"
height="2.2395833333333335in"}

> Fig. xxx β~⊥,min~ function, averaged over the relevant momentum bite,
>
> and measured from the second order moments of the beam itself, as the
> cooling progresses, for the entire SFOFO cooling channel. The 5 arrows
> indicate the beginning of new lattice sections.

The transverse and longitudinal emittances through the cooling system
are shown

in Fig. xxx. Emittances are computed using diagonalized covariance
matrices. The emittance values are corrected for correlations between
the variables, including the strong

momentum-transverse amplitude correlation. At equilibrium a transverse
emittance of 2.2 mm rad is reached. The transverse emittance cools from
12 to 2 mm rad. The

longitudinal emittance shows an initial rise as particles not within the
rf bucket are lost, and then an approach to an asymptotic value set by
the bucket size. Naturally, this longitudinal emittance should rise due
to straggling and the negative slope of energy loss with energy.
However, since the rf bucket is already full, instead of an emittance
growth we have a steady loss of particles.

![](media/image6.png){width="3.4270833333333335in"
height="2.1770833333333335in"}

> Fig. xxx The transverse and longitudinal normalized emittances.

Despite the overall particle loss, the numbers of particles within the
accelerator

acceptance increases, as seen in Fig. xxx. The two curves give the
number of particles within the baseline longitudinal and transverse
acceptances. The upper line represents the

values for the accelerator parameters in this study. The lower line,
given for

comparison, gives the values for the acceptances used in Feasibility
Study 1. It is seen that the gain in muons within the accelerator
acceptance due to cooling is 3.1x (or 5x if the study 1 acceptances were
used). If the particle loss from longitudinal emittance growth could be
eliminated, as should be the case if emittance exchange were used, then
these gains would double.

![](media/image7.png){width="3.5541666666666667in"
height="1.9270833333333333in"}

> Fig. xxx The muon to proton yield ratio for the two transverse
> emittance cuts, showing that the channel cools as the density in the
> center of the phase space increases.

Table xxx Beam characteristics summary

  ---------------------- ------- --------- ----------- -------- -----------
  location (end of)      σ~X~    σ~X'~     σ~P~        σ~t~     \<p\>

                         cm      mrad      MeV/c       ns       MeV/c

  induction linac        8.6     95        113                  260

  matching section       5.8     114       113                  260

  buncher                5.3     107       111         0.84     256

  2.75 m cooler          3.0     91        70          0.55     226

  1.65 m cooler          1.8     102       30          0.51     207
  ---------------------- ------- --------- ----------- -------- -----------

The beam characteristics in the buncher and cooler sections are
summarized in

Table xxx. The beam is symmetric in this lattice, so the y properties
are similar to those

in x. We see that the size steadily decreases as we proceed down the
channel.

The angular divergence is kept constant for maximum cooling efficiency.
The

momentum spread of the entire beam is still large after the induction
linac,

but this includes very low and high energy muons that do not get
transmitted

through the subsequent sFOFO lattice. The decrease in energy spread is
due to

particle losses since there is no longitudinal cooling or emittance
exchange in

this lattice. Likewise, the average momentum of the beam decreases until
it

matches the acceptance of the sFOFO lattice. The time spread refers to a

single bunch in the bunch train.

The longitudinal emittance remains more or less stable, at around 30 mm.
This

is in part deceptive. Straggling and imperfections in the longitudinal
to

transverse correlation cause particles to fall out of the rf bucket and
to be scraped away due to the strong π betatron resonances. In fact, the
buncher delivers a full rf bucket to the cooling section and the
longitudinal emittance cannot grow any larger. This scraping occurs on
the combined time scale of a synchrotron period, about 20 m, and on the
growth time of the betatron resonance instability.

The performance of the cooling channel is limited by both multiple
scattering and

the limited momentum acceptance. While the latter is difficult to
estimate

based on computer simulations, it is straightforward to estimate the
former contribution

by simply turning off multiple scattering in the code. If so, the
relative

μ/p~15~ and μ/p~9.75~ yields would increase by 11% and 19%,
respectively.

6.4 Simulation summary

It is worthwhile at this point to summarize how much of the required
cooling effort has been simulated so far. Figure 6.4.1 is a plot of
normalized transverse versus longitudinal phase space.

![](media/image8.wmf){width="5.999305555555556in"
height="4.24375in"}Fig. 6.4.1 Summary of simulation effort

The beam collected from the target is shown at START in the upper right
corner. The contour for a constant 6-dimensional normalized emittance of
0.17 mm^3^ is shown by the dotted line in the lower left corner. A 100
GeV Higgs Factory is indicated by the dot at the end of the dotted line.
The solid connecting the starting point to the Higgs Factory shows the
proposed baseline scenario that emphasizes longitudinal cooling at the
beginning. This shows the preferred cooling path and is not a
simulation.

The neutrino factory goal is the square marked NFPJK. There are three
neutrino factory simulations shown: FS1FO3 is the 3 T FOFO solution and
FS1FL5 is the 5 T single flip solution from Feasibility Study 1, and
FS2SF5 is the 5 T sFOFO solution from Feasibility Study 2. These
simulations all start at a lower longitudinal emittance than is produced
at the target. This is because the neutrino factory can use a bunch
train and the appropriate longitudinal emittance is that of each bunch
in the train. The Higgs factory cannot use a bunch train unless a
stacking ring is available to later recombine the bunches. These
solutions also end at a lower longitudinal emittance than they start
with. There is no real longitudinal cooling. It comes about from
scraping the tails off the longitudinal distributions. Two simulations
are shown that include emittance exchange and thus provide some real
longitudinal cooling. VBHEL5 uses a helical dipole to generate
dispersion in a 5 T single flip solenoid channel. The RING COOLER uses
the dipole fields to generate dispersion and incorporates wedges in the
ring.

The three simulations marked SRAS are alternating solenoid lattice
solutions from the Status Report. These solutions have good transmission
and show the classic behavior of simultaneously cooling transversely and
heating longitudinally. The 31 T solution ends very near the Higgs
Factory requirement. The SRLi10 simulation was of a 10 T surface field
lithium lens. It gets nearby the required values. A 15 T lens could
probably achieve them.

Three points should be obvious from this summary. First, many different
simulations have shown that transverse cooling should be possible over
the required range of transverse emittances. Secondly, emittance
exchange is a critical technology for tying the transverse cooling
sections together in such a way that we follow the baseline scenario
down to the Higgs Factory goal. Lastly, there is still a great deal of
simulation work needed before we have a self-consistent plan, listing
the various cooling sections needed to satisfy the Higgs Factory
requirements.

6.5 Modified neutrino factory front end scenarios

6.6 Final cooling schemes

6.7 Emittance exchange schemes

As we have seen developing a practical method of implementing emittance
exchange is a critical requirement for building a Higgs Factory or any
muon collider for that matter. A number of schemes have been proposed,
some of which are summarized below:

-   standalone lattice sections

-   small dispersion superimposed on transverse cooling lattice

-   ring coolers

-   helical capture of unbunched beam

-   bunch stacking

-   special rf cavity modes

A number of emittance exchange ideas are summarized on a webpage
\[gailweb\] devoted to Emittance Exchange Efforts and in the proceedings
of a workshop \[bnlws\] held in September 2000.

6.7.1 Standalone lattice sections

The idea of the standalone lattice section is that a large amount of
emittance exchange is done in a section of lattice totally isolated from
the transverse cooling. One would cool transversely until the
longitudinal heating became unacceptable. Then a pure emittance exchange
section would be inserted to reduce the longitudinal emittance back down
to an acceptable level. The process would continue through as many
stages as needed to achieve the final emittance requirements.

The first implementation of this idea was adopted for the Status Report
\[sr, rpvan\], where the emittance exchange is achieved using bent
solenoids and wedges. The goal was to achieve a factor of 3 reduction in
momentum spread. Simulations of this scheme showed good exchange of 5-D
emittance (the system did not contain rf cavities and the bunch length
was ignored). Subsequent attempts at designing bent solenoid channels of
this type including rf cavities are not very successful \[rfbs\].
Achieving a large exchange in a single step is not currently favored
because it requires creating a large dispersion and it is difficult to
get the beam out of the section without developing strong correlations
among the phase space variables.

A modified version of this scheme uses separated, nearly isochronous
regions of the lattice to introduce the dispersion \[cw\]. There is no
rf in the dispersive region. This scheme uses smaller dispersion and
aims for a smaller amount of exchange in each stage.

6.7.2 Small dispersion superimposed on transverse cooling lattice

The idea here is to take a successful transverse cooling lattice and
superimpose a small amount of dispersion on the lattice. This idea has
been considered with the dispersion coming from dipoles, bent solenoids
or helical dipoles. The dispersion is assumed to be small enough so that
it doesn't greatly disturb the transverse cooling behavior.

Calculations of the expected performance from adding dipole fields to a
super FOFO transverse cooling lattice were done by Palmer \[rpsf\]. He
puts a dipole over the rf cavity in the middle of the lattice cell. A
gradient field is used to get equal focusing in both transverse planes.
The bend angles are 45^o^, which give a dispersion of \~40 cm. In order
to get emittance exchange in both transverse planes, an 18^o^ helical
twist is introduced into the lattice. Variations of this idea have been
studied by a number of people \[bnlws\].

Another scheme \[vbhel\] uses a rotating dipole to generate dispersion
in a single flip transverse cooling channel. A simulation was made of a
72 m long channel with a 0.3 T rotating dipole in a 5 T solenoid. LiH
wedge absorbers were spaced periodically down the channel. The beam was
reaccelerated using 201 MHz rf cavities. Including nonlinearities and an
initial momentum-transverse amplitude correlation, the 6-D emittance was
reduced from 5300 mm^3^ to 1350 mm^3^. The beam transmission was 81%.
Another simulation of the same scheme, using Geant4 \[elvira\], obtained
a similar answer.

6.7.3 Ring coolers

The ring cooler uses some of the dispersion from the bending dipoles
together with wedges to incorporate emittance exchange. A number of ring
designs have been proposed by Valeri Balbekov.

\(1\) The first design \[vbaip\] attempted to design a ring with a λ I
transfer matrix per turn, where λ is the cooling factor. In this case
the variables in 6-D phase space are independent. It used two bending
sections with wedge absorbers. The dipoles had an n=0.5 quadrupole
gradient superimposed. The straight sections had rf cavities and LiH
absorbers for transverse cooling. Skew quadrupoles were used to control
dispersion in the straight sections. The beam was injected at 225 MeV/c
and circulated at a 9.3 MHz revolution frequency. Cooling takes place
primarily through a reduction in transverse size and bunch length. The
simulation consisted of a mix of tracking in the absorbers and matrix
transport. The 6-D emittance was reduced from 11x10^4^ mm^3^ to 24
mm^3^. Approximately 25% of the muons were lost because of aperture
restrictions and 25% from decays.

\(2\) In the second design \[vbpac\] alternating direction solenoids
were incorporated in a racetrack ring. The solenoids focus the beam
leaving the LiH absorbers into the rf cavities. The arcs contain a bent
solenoid superimposed on the gradient dipoles. The ring has a
circumference of 42.5 m. The muon momentum is \~330 MeV/c and the
revolution frequency is 6.2 MHz. Transverse and longitudinal
nonlinearities significantly reduced the predicted linear cooling
performance. Including nonlinearities the 6-D emittance was reduced from
4.6x10^4^ mm^3^ to 1.2x10^3^ mm^3^. Approximately 50% of the muons were
lost because of aperture restrictions and decays.

\(3\) A higher frequency ring has also been developed \[vbmuc\]. This is
a 32.6 m circumference ring with 201 MHz rf and liquid hydrogen
absorbers. There are 8 dipoles in the arcs with superimposed solenoid
fields and LiH wedges. The simulations still lack realistic fringe
fields around the dipole magnets and any method of injection or
extraction. The 6-D emittance of a pre-bunched beam was reduced from
2800 mm^3^ to 200 mm^3^ in 10 turns. The transmission was 60%.

6.7.4 Helical capture

Y. Derbenev \[derb\] has proposed a "sweeping" method to reduce the
energy spread of an initially unbunched muon beam. Dispersion is created
using a helically rotating dipole field. Wedges must be placed
periodically along the channel. Preliminary simulations \[rfhel\] show
that the method works in principle with ideal beams, but does not work
with the enormous emittance muon beam collected from the production
target

6.7.5 Bunch stacking

If trains of bunches are used in the cooling sections, some form of
bunch stacking must be provided before the beam reaches the collider
ring. One scheme \[ck\] proposed synchronizing the bunches in time by
separating them with a transverse deflector into individual time delay
lines. The bunches are then recombined by merging them in transverse
momentum space. One important issue is the emittance dilution caused by
the stacking process.

Preliminary simulation work on this idea has begun \[yfstack\]. A kicker
magnet inside a solenoid was used to create different temporal paths for
10 bunches. A sector magnet channel was used for stacking the
time-synchronized bunches.

6.7.6 rf cavity modes

There have proposals to use rf cavity modes to reduce the energy spread
in the beam \[km\]. The cavity must be placed in a dispersion region.
The beam can be sent through a normal accelerating cavity off-axis, such
that there is a transverse variation of accelerating field. Any such
exchange that does occur is believed to take place through non-linear
processes only \[courant\].

6.8 Assessment

References

\[bnlws\] http://www.cap.bnl.gov/mumu/exchange/

\[ck\] C. Kim, An emittance exchange idea using transverse bunch
stacking, NuMu note 70, 1999.

\[courant\] E. Courant, Conservation of phase space in Hamiltonian
systems and particle beams, in R. Marshak (ed), Perspectives on Modern
Physics, Wiley, 1966, p. 257-60.

\[cw\] Chun-xi Wang, Solution to emittance exchange?: design principles
and a possible solution, BNL Emittance Exchange Workshop, 2000;
http://www.cap.bnl.gov/mumu/exchange/Wang1.pdf

\[derb\] Y. Derbenev, Conceptual studies on ionization cooling of muon
beam, NuMu note 185, 2000; Sweeping method to cool a large energy spread
of initial muon beam, NuMu note 135, 2000; Resume on sweeping cooling
and cooling agenda in general, NuMu note 133, 2000.

\[elvira\] V. Elvira, P. Lebrun & P. Spentzouris, Simulation of helical
cooling channel using Geant4, NuMu note 193, 2001.

\[fs1\] Feasibility study 1

\[fs2\] Feasibility study 2

\[gailweb\]
http://needmore.physics.indiana.edu/\~gail/emittance_exchange.html

\[km\] Kirk McDonald, Compression of beam energy via off-axis traversal
of an rf cavity, Princeton note /μμ/97-1.

\[rfbs\] R. Fernow, Emittance exchange using thick bent solenoid
elements, NuMu note 166, 2000.

\[rfhel\] R. Fernow & J. Gallardo, Emittance exchange using the helical
sweeping method, NuMu note 187, 2001.

\[rpsf\] R. Palmer, Emittance exchange with super FOFO lattice, 8/17/00;
http://pubweb.bnl.gov/people/palmer/nu/exchange/exch2.ps

\[rpvan\] R. Palmer, Emittance exchange & full system, Particle
Accelerator School, Vanderbilt, TN, 1999;
http://pubweb.bnl.gov/people/palmer/course/9exch.ps

\[sr\] status report

\[vbaip\] V. Balbekov & A. van Ginneken, Ring cooler for muon collider,
AIP Conf. Proc. 441:310-3, 1998.

\[vbhel\] V. Balbekov, Helical channel studies and simulations, NuMu
note 146, 2000.

\[vbmuc\] V. Balbekov, Ring cooler update, NuMu note 189, 2001.

\[vbpac\] V. Balbekov, Possibility of using a ring accelerator for
ionization cooling of muons, Proc. 1999 Part. Acc. Conf., p. 315-7.

\[yfstack\] Y. Fukui, Stacking mini-μ bunches for emittance exchange,
BNL Emittance Exchange Workshop, 2000;
http://www.cap.bnl.gov/mumu/exchange/Fukui3.pdf