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\begin{document}
\renewcommand{\arraystretch}{1.5}
%\begin{titlepage}
\begin{flushright}
%Princeton/$\mu\mu$/97-5\\
K.T.~McDonald \\
November 5, 1998 \\
%DRAFT\\
\end{flushright}
\smallskip
\begin{center}
{\LARGE\bf Comments on Ionization Cooling}
%{\Large\bf Abstract}
\end{center}
%\medskip
\section{Introduction}
This note is a commentary on Sec.~V of the draft Status Report, and perhaps
the beginning of a document that might fill a conspicuous gap.
Namely, {\bf there appears to be no accessible general discussion of the
principles of ionization cooling that also reflects our current understanding
of its application to a muon collider.}
This is situation is very unfortunate. It would not matter so much if
cooling were a solved problem, for which a sketch of a viable technical
solution would suffice for most readers. But, I believe, we need significant
improvement in our understanding of cooling. Such understanding might well
come from the contributions of people who read the Status Report to whet their
interest in muon colliders.
Therefore, I suggest that Sec.~V be revised to include more of the insights
into cooling that have developed by members of the Collaborations, as well
as better pointers into available literature. I suspect that we also need
a separate, new document specifically focused on cooling issues, in which
considerable detail, both theoretical and numerical, is presented.
\section{Comments}
\subsection{Accelerator Physics}
\begin{itemize}
\item
{\bf Where do equations SR(24) and SR(25) come from?}
[SR means Status Report. Equations without SR in front refer to the
present paper.]
Something close to these was given by Neuffer in 1983 \cite{ref2a}, but
Dave seems to draw different conclusions from them than does Bob Palmer.
A partial version of these equations was given earlier by Skrinsky
\cite{ref1a}, who also drew inferences from them closer to those of Neuffer
than to Palmer.
These early works assumed that fairly high energy muons were being cooled, and
so approximated $E_\mu = P_\mu c$.
The earliest appearance of SR(24) that I have found is in two useful papers by
Fernow and Gallardo \cite{Fernow94,ref24b}; however, the derivation there
includes the phrase ``it can be shown''. I believe that SR(24) is meant to
improve on earlier versions by being applicable for muons of any $\beta$.
Equation (25) seems to have first appeared in \cite{ref2b} by Neuffer, and
differs from an earlier version by a factor of 2 in the first term.
\item
{\bf Are equations SR(24) and SR(25) correct?}
I give a detailed derivation of SR(24) in sec. 3 below.
\item
{\bf Has a numerical study ever been performed with the purpose of verifying
equations SR(24) and SR(25)?}
I think not. This might make a good introductory project for someone who
wishes to combine the insights of analytic and numerical calculations.
\item
{\bf Why don't we try anymore to obtain longitudinal ionization cooling?}
Both \cite{ref2a,ref1a} discuss longitudinal cooling before transverse cooling.
They note that the $dE/dx$ curve has positive slope for muon momenta above
about 400~MeV/$c$, so longitudinal cooling might be possible here. Hence,
they emphasize muon cooling at momenta higher than this.
My answer to this question is contained in the next comment.
\item
{\bf Why are we now at momenta lower than 400 MeV/$c$ in the cooling section?}
One answer is: {\bf money}. It costs more in rf systems to operate at
higher energy, since the process of cooling takes the beam energy away and
replenishes it about 30 times. But, if we choose a parameter region in
which cooling proves to be impossible for physics reasons, the cost savings
is illusory. [Plus, I will comment later that cost optimization might well
take us far from the present parameter set.]
I have thought of {\bf another answer}, which I have never heard discussed,
although
it is a variant of a remark by Bob. We plan to replenish the lost beam energy
with rf cavities, which requires the beam to be bunched. The average
velocity of this bunch along the rf axis, $z$, must obey $\bar\beta_z < 1$.
But if the bunch has large emittance, some particles will have large angles.
Even if these particles had $\beta = 1$, their longitudinal velocity would be
only $\beta_z = \cos\theta_z$. To keep these particles in the bunch, we
must have $\bar\beta_z \leq \cos\theta_{z,\rm max}$.
A practical fact is needed here: what is a realistic $\theta_{z,\rm max}$?
If we choose $30^\circ$, then we immediately find $\cos\theta_{z,\rm max} =
0.866 = \bar\beta_z$, so the central $\gamma = 2$, and the
central momentum $\bar P_\mu = 182$~MeV/$c$.
In sum, {\bf cooling of a beam with large angular spread $\Leftrightarrow$
low central momentum $\Rightarrow$ must give up longitudinal ionization
cooling} (of the simplest sort).
Note that we have also demonstrated the important result that {\bf particles
with large angles must have higher momentum to stay within the bunch}.
Furthermore, there must by some kind of transverse confinement of the beam
particles, or they would wander off. In general, we expect that larger
angle particles will have trajectories with larger transverse amplitude.
Hence, {\bf particles with large amplitudes of transverse oscillation must
have higher momentum to stay within the bunch}.
Near eq.~SR(23) of the draft Status Report, it is said that we will ignore
correlations in phase space. I think this is misleading, since very basic
arguments show that we will have to maintain some particular correlations
for a cooling channel to work. Perhaps we should note that SR(23) is
an overestimate of the emittance if correlations are present, so a reduction
in the value of SR(23) surely corresponds to 6-d cooling.
\item
{\bf Why not cool at even lower momentum?} It might be easier to hold
a large-emittance bunch together if the highest momentum didn't have to
correspond to $\beta$ very close to 1.
We first look for an answer from eq.~SR(24), from which we learn
that there
is an equilibrium emittance below which one cannot cool (with a given
apparatus), obtained when SR(24) vanishes:
\begin{equation}
\epsilon_{N, \rm min} = {\beta_\perp (13.6\ \mbox{MeV}/c)^2 \over 2 \beta m_\mu
\langle -dE_\mu/ds \rangle L_R}.
\label{eq1}
\end{equation}
[This result is so useful it should go into the Status Report.]
{\sl To be consistent with the notation used in this Comment, I have
added the minus sign to $dE/ds$.}
%However,
%it should be pointed out that in (\ref{eq1}),
%$\langle dE_\mu/ds \rangle$ represents an
%average over a whole cooling ``cell'', not just inside the absorber.]
The idea of an equilibrium emittance appeared already in \cite{Ado}.
It is claimed in \cite{Fernow98}
that the betatron function for a particle of charge $e$ and
longitudinal momentum $P_z$ in a solenoid with magnetic field $B_z$ obeys
\begin{equation}
\beta_{\perp,\rm solenoid} = {c P_z \over e B_z}.
\label{eq2}
\end{equation}
See sec.~5 below for discussion of this.
[Equation (3) of \cite{Fernow98} mistypes
this result. {\sl It appears that the latter half of sec.~V of
the Status Report follows the latter half of \cite{Fernow98} fairly closely;
more of the first half of \cite{Fernow98} should go into the Status Report.}]
%www
We learn 3 key things about transverse cooling from this one result:
\begin{enumerate}
\item
The radiation length $L_R$ should be long. This is fairly obvious, and was
noted by Ado and Balbekov \cite{Ado}. Liquid hydrogen is best.
\item
The absorber should be placed at a low-$\beta_\perp$ point. This follows
also by noting that the ``heating'' term in SR(24), due to multiple scattering,
is less dangerous if the particles' angles are large, which occurs when
the $\beta_\perp$ is small.
This insight can be traced to Skrinsky \cite{Skrinsky71}.
Equation (\ref{eq2}) reminds us that low-$\beta_\perp$ requires strong
magnetic fields.
\item
$\beta \langle dE_\mu/ds \rangle$ should be as big as possible.
For $\beta$
slightly less than 1, $dE_\mu/ds \propto 1/\beta^{5/3}$. See Figure 23.1
of the particle data tables. Then, in the region of interest to us,
$\beta (dE_\mu/ds) \propto 1/\beta^{2/3}$, \ie, $\epsilon_{N,\rm min}
\propto \beta^{2/3}$. Hence, it appears from SR(24) that there might indeed
be some advantage in going to lower $\bar \beta$.
If the bunch is transported in a solenoid, (\ref{eq2}) indicates that
an additional factor of $\bar\beta$ appears in the expression for
$\epsilon_{N,\rm min}$, which reinforces the interest in lower $\bar\beta$.
\end{enumerate}
I don't know the answer to the question of the optimum $\bar\beta$.
Could it actually be a serious question?
\begin{itemize}
\item
Some guidance comes from eq.~SR(25). For low $\beta$ the first term becomes
positive and adds to the undesirable ``heating'' of straggling.
The first term changes sign at $\gamma\beta = 4$ (for hydrogen).
This suggests that going lower may aggravate our already difficult problem
with longitudinal emittance.
In sec.~3.2.1, we show that the ``heating'' of $(\Delta E)^2$ varies as
$1/\beta^4$ for low $\gamma$, which is probably the most quantitative
argument against use of very low $\beta$.
For $\beta$ too low, particles with lower than average energy would just
stop in the absorber. It would be interesting to have a quantitative
statement of where this problem sets in.
\item
The money argument favors lower $\beta$.
\item
Lower $\beta$ means lower $\gamma$, so the muons decay faster in the lab.
This appears to favor higher $\beta$.
But the length of the rf cavities needed to maintain the energy vary as
$\gamma$ (really $\gamma - 1$), so the decay loss is independent of
$\gamma$ to a first approximation.
% (and in the next approximation, the
%survival probability varies like $\exp(-(\gamma - 1)/\gamma)$, which favors
%smaller $\gamma$).
\end{itemize}
\item
{Should we include variants of SR(24) and SR(25) that show the functional
dependence on $\gamma$ and $\beta$ via approximate analytic expressions
for $dE/dz$ and $L_R$?}
Examples are given below in (\ref{d.11a}-\ref{d.11b}) and (\ref{e.11}).
Equations (\ref{d.11e}) and (\ref{e.11b}) give simple results for a specific
numerical example of possible relevance.
\item
{\bf What is $\theta_{x,\rm max}$?}
This question occurred in a previous comment, but deserves elaboration.
I follow \cite{Fernow98} here.
[I am changing notation here: $\theta_x$ is the projection onto the $x$-$z$
plane of what was called $\theta_z$ earlier.]
%qqq
Using the relation $\sigma_{\theta_x} = \sqrt{\epsilon/\beta_\perp}
= \sqrt{\epsilon_N/\gamma \beta \beta_\perp}$, our (\ref{eq1}) becomes
\begin{equation}
\sigma^2_{\theta_{x,\rm min}} =
{(13.6\ \mbox{MeV})^2 \over 2 \beta E
\langle -dE_\mu/ds \rangle L_R}.
\label{eq4}
\end{equation}
Suppose we are cooling a bunch with emittance $\epsilon_N =
n \epsilon_{N,\rm min}$. Then $\sigma^2_{\theta_{x}}$ is $n$ times that given
in (\ref{eq4}). To keep transport losses small, we will need to have
angular acceptance $m \sigma_{\theta_{x}}$, where $m \gsim 4$.
Together, we infer that the angular acceptance must be
\begin{equation}
\theta_{x,\rm max} = m \sqrt{n (13.6\ \mbox{MeV})^2 \over 2 \beta E
\langle -dE_\mu/ds \rangle L_R}.
\label{eq5}
\end{equation}
For example, with muons of $\gamma = 2$, $\beta = 0.866$, $E = 210$ MeV,
and liquid hydrogen absorbers for which
$\langle -dE_\mu/ds \rangle = 4.4$ MeV\, g$^{-1}$cm$^2$,
and $L_R = 61.3$ g/cm$^2$, we have
\begin{equation}
\theta_{x,\rm max} = m \sqrt{0.0019 n} = 0.35\ \mbox{radian,\ for}\ n = m = 4.
\label{eq6}
\end{equation}
This angle is unusually large for particle beam transport, and emphasizes
the technical challenge of ionization cooling. We also see that it will
be almost impossible to design a single apparatus that cooled an initial
emittance 10 times the minimum (\ref{eq1}). Rather, the cooling channel
should consist of a sequence of sections $i$, with
focusing strength
\begin{equation}
\beta_{\perp,i} = {\beta_{\perp,0} \over i},
\label{eq7}
\end{equation}
with lengths chosen so that section $i$ cools from
$4 \epsilon_{N,i,\rm min}(\beta_{\perp,i})$ to
$2 \epsilon_{N,i,\rm min}(\beta_{\perp,i}) =
4 \epsilon_{N,i+1,\rm min}(\beta_{\perp,i+1})$.
\item
{\bf Shouldn't we mention Robinson's ``law'' of damping decrements?}
A version of this appeared in \cite{ref1a},
and another appeared in \cite{Palmer94},
but it hasn't been discussed much since \cite{Neuffer94} in 1994. Why?
The idea is, I believe, that if one pushes too hard on transverse cooling,
one will
inevitably be in a situation where longitudinal heating becomes very severe.
Since I've never followed a derivation of this ``law'' before, I present one
in sec.~3.3.
\item
{\bf Shouldn't we make brief mention of frictional cooling?}
However, I don't have the reference for this.
\item
{\bf Shouldn't we refer to our separate studies of space-charge effects?}
In particular, has the work of Christine Celata appeared in referenceable form?
A related question: Is there any referenceable note on wakefield effects?
\item
{\bf Why do we use a solenoid channel to confine the muons during cooling
(+ capture and phase rotation)?}
This question seems to come up over and over as new people start thinking about
a muon collider. So, although the answer is relatively straightforward, it
would be good to include it. My version follows. See also \cite{Fernow98}.
Equation (\ref{eq1}) shows that the absorbers should be placed at a
low-$\beta_\perp$ point, \ie, where the confining forces are as strong as
possible. Between the absorbers we must have accelerating cavities, whose
transverse dimensions are relatively large. To keep costs down, the confining
forces should be weaker in the accelerating cavities. Hence, we need
a confining structure with a periodic variation in the strength of the
confining forces.
We also wish to transport with low losses a bunch with extremely large
transverse emittance, \ie, one with particles at large values of the
transverse coordinates. Use of a quadrupole channel to provide the
confining forces would result in large numbers of particles passing through the
quadrupole fringe fields at large radii, leading to significant interaction
with the higher-order terms in the transport equations, and poor transport
efficiency. In contrast, a solenoid channel provides confining forces that
are essentially continuous in $z$ and
which are independent of radius. This permits effective transport over the
full aperture.
Of course we must still address the following:
\item
{\bf Why do we use alternating solenoids?}
Bob's recent note \cite{whyalt} is an important step toward understanding
this question, and should be referenced in the Status Report (after some
technical oversights are fixed).
Indeed, on reading that note, I come to several conjectures beyond Bob's
conclusions. These follow from two basic observations:
\begin{itemize}
\item
If we desire particles to have zero mechanical angular momentum after exiting
a solenoid, they must have zero canonical angular momentum when inside the
solenoid. (These angular momenta are measured with respect to the symmetry
axis of the magnetic field.)
\item
For a particle to have zero canonical angular momentum when inside a solenoid,
the trajectory must pass through the symmetry axis of the magnet.
\end{itemize}
The first observation follow from conservation of canonical (rather than
mechanical) angular momentum; the second is discussed in sec.~4 along with
other related issues.
We draw several conclusions:
\begin{enumerate}
\item
Note that if the primary target were entirely along the magnetic
symmetry axis, then all secondary particles would be produced with very low
canonical angular momentum. That is, we could arrange for the beam to start
with the desired special condition. Our task would then be to keep from
generating canonical angular momentum later.
\item
Some canonical angular momentum is generated when the pion decays. (If the
pion decays at a point not on the magnetic symmetry axis, the
neutrino carries some angular momentum.)
\item
If the muon passes through an absorber when its trajectory is away from the
magnetic axis, is canonical angular momentum is changed. There is some
probability that the canonical angular momentum is ``heated'' even though
the mechanical transverse momentum is ``cooled''. Hence, it is possible
that more optimal cooling of angular momentum would involve absorbers whose
thickness decreases with radius.
\item
A speculation that might be worthy of simulation is that a cooling channel
with a single sign of the magnetic field
could be designed that emphasizes keeping the canonical angular momentum
small.
\item
If this is only partially successful, it might still be the case that a
single reversal of the magnetic field near the end of the channel would
suffice.
\end{enumerate}
\item
{\bf Shouldn't we give more details as to the sophistication of the
simulations?}
For example, mention the recent efforts to understand tails of the multiple
scattering distribution. The present text could leave the impression we
are still working in the Gaussian approximation.
\item
{\bf What is the relation between the rf frequency and the emittance?}
We have established that higher magnetic fields are needed to cool lower
transverse emittances. As I understand it, the rf frequency is related to
the bunch length, which is related to the longitudinal emittance.
Roughly, I expect that the rf frequency will increase as we move down the
cooling channel. Do we need to up the rf frequency every time we up the
magnetic field?
I note that the two examples, VD and VE, use the same rf frequency while the
field of E is twice that of D, but no comment is made here about the broader
issue.
In the first paragraph of sec.~VG, I read ``It should be pointed out
that in the earlier and later stages....''
This sentence is very interesting. It not only should be pointed out, it
should be explained! Why does the bunch length grow in the `later stages'?
Later than what? The 31-T example is presented as the last stage of
a Higgs factory. Perhaps the bunch lengthening in ``later'' stages is a
holdover from earlier designs in
which we skipped longitudinal emittance exchange near the end. I was never
clear as to how well motivated that choice was. Is it still active?
Discussion is warranted!
\item
{\bf Are wedges the best way to deal with the problem of longitudinal/transverse
emittance exchange?}
We now have separate simulations of 6-d cooling by a factor of 2 in 20 meters
of an alternating solenoid channel,
and of exchange of longitudinal and transverse phase space in a double bent
solenoid section. Each of these requires rather particular phase-space
correlations for efficient operation.
As I understand it, a procedure for ``matching'' from one set of correlations
to another does not yet exist, or requires a very long distance (hence,
significant cost in dollars and in muon decay loss).
Further, as near as I can tell, any system involving wedges and momentum
dispersion in dipoles or bent solenoids requires separate cooling channels
for positive and negative muons. This doubles the cost of the cooling
channel compared to that needed for one sign. Since my estimate of the
cost for a single cooling channel is a large fraction of \$1B, this is a
major effect.
Hence, {\bf finding a new solution to the emittance-exchange problem is the
major issue in muon collider design in the near future.}
But, the Status Report gives little indication that we are aware of this
issue.
My conclusion is that some of the late studies of the FOFO scheme deserve
further investigation. I elaborate.
It has long been realized \cite{ref2a,Oneill} that longitudinal phase space
could be damped by passing higher-momentum particle through longer absorbers:
\ie, use wedge absorbers plus momentum dispersion.
We also realize that transverse cooling only works well if there is a
reasonably strong correlation between momentum and amplitude. That is,
transverse cooling requires a kind of momentum dispersion.
Hence, there appears to be a golden opportunity to use a requirement of
transverse cooling to solve a major problem of longitudinal cooling.
Namely, use absorbers whose thickness varies with radius to combine
transverse cooling with continual longitudinal-transverse emittance
exchange.
This scheme has the major merit of working for both charges in a single
cooling channel. This is a billion dollar savings, if it can be realized.
The hoped-for solution to the matching problem is not to match between two
very different sets of correlations, but to find a single set of correlations
that permits effectively simultaneously
transverse and longitudinal cooling (the latter via emittance exchange rather
than direct cooling).
Some time ago, Bob made a heroic effort to implement this notion. It seemed
to me that success was close, when a problem of longitudinal instabilities
emerged. As I understand it (there is no surviving documentation from this
effort that I could find), the infamous synchrobetatron coupling was too
strong in the FOFO scheme -- and the whole idea was dropped.
It is far from obvious to me that the synchrobetatron problem had anything to
do with the use of variable thickness absorbers.
However, the use of variable thickness absorbers has its price.
The transverse cooling
per meter will be less. But, the law of damping decrements advises us that
any scheme that, in effect, cools simultaneously in all 3 subspaces will
cool more slowly in some of those subspaces than a scheme that cools in
some subspaces while heating in others.
It may well take twice as long to reduce the 6-d emittance by a factor of 2
with variable thickness absorbers, compared to the present scheme. But no
bent solenoid sections would be required, so the total length of the cooling
channel would be about the same. More rf acceleration will be required.
[Recall that rather general arguments based on angular acceptance and
cooling efficiency indicate that cooling apparatus of a given peak magnetic
field and corresponding rf frequency should be used for cooling only over a
factor of 2 in 2-d transverse emittance. Hence, a cooling ``section'' would
be longer in the new scheme.]
The use of variable thickness absorbers might, however, aggravate the
angular momentum problem. There is a hint (sec.~4) that the angular
momentum problem would be better addressed if the absorbers were limited to
small radii. This is a use of variable thickness absorbers, but not in the
manner needed for emittance exchange. For the latter, the absorber must
be thicker at larger radii, where the momenta are higher.
This may reinforce the need for the alternating solenoid scheme -- which
possibly is overly powerful for the simple case of uniform absorbers.
Recall that Bob's attempt to implement variable thickness absorbers took
advantage of the variation of the betatron function along the FOFO lattice,
and (I think) absorbers that were thickest on axis were placed that the
high-$\beta_\perp$ points, while ``complementary'' absorbers that were
thickest well off axis were placed at the low-$\beta_\perp$ points. The
latter required absorbers inside the rf cavities. In that scheme, the
absorbers were LiH, so they were relatively compact. If we use liquid
hydrogen absorbers, the rf cavities would probably have to be split into
two sections on either side of the low-$\beta_\perp$ points.
\bigskip
While my particular conjecture as to how cooling studies might evolve is
perhaps too speculative for the Status Report, we should give more indication
that we understand the need for improved cooling scenarios.
%aaa
\end{itemize}
\subsection{Editorial Issues}
\begin{enumerate}
\item
A major gap in sec.~V of the Status Report is the lack of references to
our own work on cooling. Many references are listed with little discussion
in sec.~I, but
sec.~V should include systematic pointers to the literature, including
brief phrases as to what topic is elaborated on where.
I include a candidate list of references in the bibliography of
this note (although the pointers to these are spread out over the note,
including my sec.~3).
\item
The abbreviations of units such a seconds should be a text font s, not a
math font $s$.
\item
Symbols that are used in a sentence should be in math font, not text font.
Thus, $(dE/dx)$, not (dE/dx). [Why do we use $dE/dx$ one place and $dE/ds$
another?]
\item
The speed of light is symbolized by math font $c$, not text font c.
\item
The use of $\beta$ to mean two different things in
eq.~SR(24) is distracting to the uninitiated. Plus, experimental physicists
are often less familiar with the betatron function than they should
be. (I'm no exception.) Therefore, two things might be useful in the Status
Report. First, refer to sec.~21 of the Particle Data Tables \cite{pdg},
which has a good brief introduction to the language of particle accelerators
for particle physicists. Second, add a statement to the effect that
$\beta_\perp = \sigma/\sigma'$, and that all of these are related to the rms
geometric emittance $\epsilon = \epsilon_N/\gamma\beta = \sigma\sigma'$, so
$\sigma = \sqrt{\epsilon \beta_\perp}$ and $\sigma' = \sqrt{\epsilon/
\beta_\perp}$
(if correlations are ignored).
\item
Equations SR(22) and SR(25) of the Status Report are typeset
using $<$ rather than $\backslash$langle $= \langle$, which latter is
preferable.
\item
Abbreviations should be written out once, with the abbreviation following
in parentheses, before they are used later in the text.
This need not apply to ``common'' abbreviations.
Is ``rms'' a self-evident abbreviation? Physical Review generally considers
it not to be so. (Likewise, ``rf'' is jargon, and should be written out
once at the beginning of the paper.) In any case, the abbreviation ``rms''
should always appear in text font, not math font.
\item
The first sentence of sec.~VA includes the symbol $\epsilon_{x,n}({\rm rms})$,
but this is defined only much later. This is bad practice.
\item
I still find the form of SR(23) distracting. The preceding text implies that
we are trying to be careful about the meaning of $\epsilon_{6,N}$; then
without justification we abandon that care and write $\gamma\beta \sigma_E$
where I would have expected the ``simplified'' version to read
$\sigma_P$.
I am not sure what is really intended here. I note that $zP_z \approx zP$
and that $zP$ has the same dimensions as $tE$. Hence we might write
$\epsilon_{\parallel,N}$ as $\sigma_t \sigma_E/mc$ with some justification.
But the present form makes little sense to me.
\item
I also find the superscript rms superfluous in SR(23);
$\sigma$ always stands for an
rms quantity in statistical discussions. (Plus, rms was set in math font,
when abbreviations for words should be set in text font. Plus, rms is
written as an argument of $\epsilon$ in SR(22) [in math font!], but as a
superscript in SR(23). I find both positionings awkward.)
\item
``6-d'' or ``6-D''? (Plus, the edict from Finley, Geer and Raja is to use
no hyphens.) I note that the text includes ``6-dimensional'', so I'd would
have expected the corresponding abbreviation to be ``6-d''.
\item
In my derivation below of SR(24) and SR(25)
(and in the notation of the Particle Data
Group \cite{pdg}), $dE/dz$ is negative. But in the notation of the Status
Report, $dE/ds$ is positive and is called the energy loss. The notation of
the Status Report, while nonstandard in the larger community, was no doubt
chosen so that the ``cooling'' term is easily identified by the minus sign
in front of it. Since we are trying to communicate with the larger
community, shouldn't we use the standard notation?
\item
``dE/dx'' $\to$ ``$dE/dx$''
\item
The last sentence of sec.~VC says `...but again, full simulations ...
have not been successfully demonstrated.'' The literal interpretation is that
we have no successful simulation at present. Is this what we mean to convey?
\item
In the first sentence of sec.~VD, ``diameter ).'' $\to$ `diameter).''
\item
``short 805 MHz high gradient linacs'' $\to$ ``short, 805-MHz, high-gradient
linacs''. [Five adjectives in a row, only some of which modify the noun.]
\item
``beta function'' or ``betatron function''? Isn't the beta function a
complicated version of the gamma function?
\item
What are ``analytic vacuum calculations''?
\item
I like the phrase ``naturally occurring correlations'', but I bet not a single
new reader will find the following 3 examples ``natural''. The new discussion
of the 2nd case is helpful.
\item
What is an ``alpha rf bucket''?
\item
``just equals'' $\to$ ``equals''.
\item
``The canonical angular momentum is defined such that it removes the axial
field dependence''. This makes no sense to me.
\item
``Without the absorbers the beam...'' $\to$ ``Without the absorbers,
the beam...''
\item
``the presence of absorber causes...'' $\to$ ``the presence of absorbers
causes...'' (I agree with the physics of this sentence.)
\item
``growth at the end of a long channel'' $\to$
``growth by the end of a long channel''
\item
``...as shown in the figure''. What figure?
\item
The caption for Fig.~23 is garbled. (c) is missing.
\item
The sentence ``Simulations have shown that 2 m is reasonable (half) period...''
is no doubt true, but not very decisive. My guess is that significantly
longer periods would work also. I believe that the value of 2~m arises more
from the need to reaccelerate after about 60 cm of hydrogen; We could
probably alternate the fields only every 2nd or 3rd 2-m cell, but this would
require additional magnetic configurations that are beyond our present
ambitions.
In any case, there should be a comma after ``field'' in this sentence.
\item
``....it forces the average betatron wavelength...'' This phrase is somewhat
circular, since ``it'' refers to choosing the betatron wavelength to be 2~m.
\item
How do we know that the synchrotron-oscillation wavelength is 14~m? I doubt
that this emerges from the simulation. Is the relation
$\lambda_\beta/\lambda_{\rm synchrotron} = 7$ invariant over all 20 cooling
sections, or just a particular value for the section being discussed?
\item
Caption of Fig.~24, ``B$_z$'' $\to$ ``$B_z$''
\item
``P$_z$'' $\to$ ``$ P_z$''. (use math mode)
\item
``x and y transverse'' $\to$ ``$x$ and $y$ transverse''
\item
``codes double precision GEANT and PARMELA'' $\to$ ``double-precision GEANT and
PARMELA codes'' The references seem out of order here; an issue for file
prst00.bbl.
\item
Several entries in table V and VI need review. ``6D'' or ``6d'' or ``6-D'' or
``6-d''? ``$\pi\ m \ rad$'' $\to$ ``$\pi$ m-rad''; ``98\%'' $\to$ ``0.98'';
``beam size'' and ``beam rad'' would both be better as ``beam radius'';
``$dp/p$'' $\to$ ``$dP/P$'', or $\sigma_P/P$; ``99.6\% $\to$ ``0.996''.
\item
Where does ``the required emittance for a Higgs factory'' come from? Refer
to Table I. However, when I look there, I seem to find differ values.
(0.8)(290) = 232.
\item
``20 \%'' $\to$ ``20\%''. The paper is very inconsistent about spacing
between numbers and the \% sign. The Physical Review style guide says to
omit the space.
\item
In sec.~VF, ``low Z wedge'' $\to$ ``low-$Z$ wedge''
\item
I debate the use of the word ``requires'' in the first sentence. More to the
point is that all we have thought of is the use of wedges in a dispersive
channel.
\item
``lattice'' is used in the 2nd sentence in a rather generalized way, since
the example does not involve a periodic structure.
\item
split infinitive: ``to exactly cancel this drift'' $\to$
``to cancel this drift exactly''
\item
``Particles above and below this reference momentum then drift up and down..''
This is awkward, since ``above'' and ``below'' refer to momentum space, but
``up'' and ``down'' refer to physical space. Also, in our double bent
solenoid we have both vertical and horizontal drifts. So the characterization
of the drift as ``up'' or ``down'' is disorienting (to use a word with ``east''
buried in it).
We might just leave this sentence out, since it's hard to follow and isn't
critical for the sentences that follow.
\item
In the text, and in the caption to Fig. 29(a), ``B$_z$'' $\to$ ``$B_z$'', \etc
\item
``Discontinuities....can excite Larmor oscillations...'' It's not clear
what is meant by ``Larmor oscillations''. The trajectory is a helix with
the (local) Larmor period whether or not there are ``discontinuities''
(something that is not physically possible, but can happen only in simulation).
For nonadiabatic changes in the bend radius of the solenoid, the guiding rays
are dispersed more than in the adiabatic case, which results in an emittance
growth if the simplified form $\epsilon = \sigma_x \sigma_{x'}$ is used.
(The true, but correlated, emittance is unchanged in this Hamiltonian
system.)
Whether this process can/should be described as an ``oscillation'' is not
evident.
\item
``adds unnecessary length''. This certainly is ``undesirable''; it remains
to be seen whether or not it is ``unnecessary''.
\item
``found to effect'' $\to$ ``found to affect''
\item
``and similar solutions have been successfully found'' Similar to what?
Since the following sentence elaborates on this, it would be better to drop
the phrase ``and similar...''
\item
In the caption of Fig.~30, ``$\delta p_z$'' $\to$ ``$\delta P_z$''
Also, ``length'' of what?
\item
In the text and in the caption to Fig.~31, ``y-p'' $\to$ ``$y$-$P$'', \etc\ \
The period is missing at the end of the
caption to Fig.~31.
\item
A \ is needed after the ``\vs'' in the caption of Fig.~31.
\item
Better not to start a sentence with the abbreviation ``RF''.
\item
``5 dimensional'' or ``5-d''?
\item
In Table VII. ``MeV/c'' $\to$ ``MeV/$c$''; ``mm mrad'' $\to$ ``mm-mrad'';
``Emit$^2_{trans}$'' $\to$ ``Emit$^2_{\rm trans}$'';
``$\Delta p_{long}$'' $\to$ ``$\Delta P_{\rm long}$''
\item
Beginning of sec.~VG,
better not to start a sentence with the abbreviation ``RF''.
\item
``The focusing fields'' would be better as ``The magnetic fields''.
There is possible ambiguity with the electromagnetic fields of the cavities
themselves. Plus, we aren't really using solenoids as focusing devices
(for which short solenoids and drift spaces combine to make an effective
lens), but as containing devices.
\item
In sec.~VG the use of Be foils as rf cavity windows is discussed. However,
at least once earlier, Be foils were was mentioned, and the new reader is
likely to be confused. Pointers to sec.~VG should be placed at the
earlier mentions of the foils.
\item
In Table VIII, ``Q/1000'' $\to$ ``$Q/1000$''; ``Zt$^2$'' $\to$ ``$Zt^2$''
\item
2nd sentence of sec.~VH, might be good to change ``focusing strength must
increase and $\beta_\perp$'s must decrease'' to ``focusing strength must
increase, \ie, $\beta_\perp$' must decrease'', since these two are
equivalent, while the present grammar suggests that they are unrelated.
\item
In the text and in the caption of Fig.~35, ``$x - p_x$'' $\to$ ``$x$-$P_x$''.
(Bring the - outside of math mode.)
\item
As before, ``MeV/c'' $\to$ ``MeV/$c$''
\item
``$\sigma_{p_x}$'' $\to$ $\sigma_{P_x}$'', \etc
\item
I don't like the large superscript ``$\epsilon_{x,N}^{\rm in}$'',
although someone worked hard to generate this.
I myself would have written this ``$\epsilon_{x,N,\rm in}$''.
Also, here but not elsewhere
in the paper, the emittance is qualified with a subscript $x$. The context
does not particularly call for this here, so it would be more consistent to
leave it out. (However, the subsection was probably written by someone
else, whose taste is different....)
\item
The sentence beginning ``A liquid Li lens consists of...'' should probably
be the start of a new paragraph. (Although apparently, Finley, Geer and
Raja like long paragraphs, whereas I prefer more shorter ones.)
\item
Put a comma after ``for example''
\item
``$p$'' $\to$ ``$P$'' for momentum; ``c'' $\to$ ``$c$'' for the speed of light
\item
At the end of sec.~VH I read of the ``Ionization Cooling Demonstration
Facility''. Then Sec.~VI is called ``Ionization Cooling Experiment''.
Aren't these two the same thing? The first phrase is perhaps better.
\item
The sentence beginning ``Following is a brief description...'' seems
redundant with the following sentence: `` The proposed R\&D program....''
\item
``Developing the appropriate rf'' $\to$ ``Developing an appropriate rf''
\item
``100 -- 300'' $\to$ ``100-300''. (This is an issue of taste among the several
TeX variants of dashes, hyphens and minus signs.
\item
$c$ for the speed of light
\end{enumerate}
%bbb
\section{Derivation of the Cooling Equations}
\subsection{Transverse Cooling}
I now attempt to derive SR(24).
The rms normalized 2-d transverse emittance in coordinates $x$ and $P_x$ of
a bunch of particles moving along the $z$ axis is
related by
\begin{equation}
m^2c^2 \epsilon_N^2 = \ave{x^2} \ave{P_x^2} - \ave{x P_x}^2.
\label{d.1}
\end{equation}
Consider the propagation of the bunch in the $z$ direction through a thin
absorber. In the impulse approximation, the particles' momenta change, but
their transverse positions do not. Hence, the rate of change of the
$\epsilon_N^2$ along $z$ may be written
\begin{equation}
2m^2c^2 \epsilon_N {d \epsilon_N \over dz} =
\ave{x^2} \ave{{d P_x^2 \over dz}} - 2 \ave{x P_x} \ave{x {d P_x \over dz}}.
\label{d.2}
\end{equation}
\subsubsection{$\ave{x P_x} = 0$}
If we ignore correlations such as $\ave{x P_x}$, we obtain the simpler form
\begin{equation}
2m^2c^2 \epsilon_N {d \epsilon_N \over dz} \approx
\ave{x^2} \ave{{d P_x^2 \over dz}}.
\label{d.2a}
\end{equation}
We wish to relate this to energy loss and multiple scattering caused by the
absorber. We introduce the particle's angle to the $z$ axis in the $x$-$z$
plane,
\begin{equation}
\theta_x \approx {P_x \over P_z} \approx {P_x \over P},
\label{d.3}
\end{equation}
where the approximations suppose that $P_x$ and $P_y$ are much less than $P_z$.
Then,
\begin{equation}
P_x \approx \theta_x P.
\label{d.4}
\end{equation}
We also note that
\begin{equation}
c P = \beta E, \qquad \mbox{and} \qquad
c^2 P^2 = E^2 - (m_\mu c^2)^2,
\label{d.5}
\end{equation}
leading to
\begin{equation}
dP^2 = {dE^2 \over c^2} \qquad \mbox{and} \qquad
{dP \over dE} = {E \over c^2 P} = {1 \over v}.
\label{d.5a}
\end{equation}
Thus,
\begin{eqnarray}
{dP_x^2 \over dz} & \approx &
\theta_x^2 {dP^2 \over dz} + P^2 {d \theta_x^2 \over dz}
= {\theta_x^2 \over c^2} {dE^2 \over dz}
+ P^2 {d \theta_x^2 \over dz}
= 2{\theta_x^2 E \over c^2} {dE \over dz}
+ P^2 {d \theta_x^2 \over dz}
\nonumber \\
& = & 2{P_x^2 \over \beta^2 E} {dE \over dz}
+ P^2 {d \theta_x^2 \over dz}.
\label{d.6}
\end{eqnarray}
We average (\ref{d.6}) over the bunch, but suppose that we may replace
$\beta$, $E$ $P$ by their bunch averages. In effect, we are neglecting
correlations between transverse momentum and total momentum. We have already
noted in sec.~2
that the existence of such correlations is, however,
critical to the success of cooling. But for correlations that are
``not too large'', we proceed:
\begin{equation}
\ave{{dP_x^2 \over dz}} \approx
2{\ave{P_x^2} \over \beta^2 E} {dE \over dz}
+ \ave{P}^2 \ave{{d \theta_x^2 \over dz}}
\approx 2{\ave{P_x^2} \over \beta^2 E} {dE \over dz}
+ {(13.6\ \mbox{MeV}/c)^2 \over \beta^2 L_R},
\label{d.7}
\end{equation}
using the standard form for the rms projected multiple scattering
(for example, (23.9) of \cite{pdg}),
where $L_R$ is the radiation length of the absorber material.
When we insert (\ref{d.7}) into (\ref{d.2a}), we find
\begin{equation}
2m^2c^2 \epsilon_N {d \epsilon_N \over dz} \approx
2{ \ave{x^2} \ave{P_x^2} \over \beta^2 E} {dE \over dz}
+ {\ave{x^2} (13.6\ \mbox{MeV}/c)^2 \over \beta^2 L_R}.
\label{d.8}
\end{equation}
In the first term on the right of (\ref{d.8}), the averages contain
$\epsilon_N^2$ according to (\ref{d.1}) with the neglect of correlations.
In the second term, we follow Neuffer and write
\begin{equation}
\ave{x^2} = \epsilon \beta_\perp = {\epsilon_N \beta_\perp \over \gamma \beta}
\label{d.9}
\end{equation}
where $\epsilon$ is the rms 2-d geometric transverse emittance, and
$\beta_\perp$ is the value of the betatron function of the transversely
confining beam optics at the position of the absorber. Hence,
\begin{equation}
2m^2c^2 \epsilon_N {d \epsilon_N \over dz} \approx
2{(mc)^2 \epsilon_N^2 \over \beta^2 E} {dE \over dz}
+ {\epsilon_N \beta_\perp (13.6\ \mbox{MeV}/c)^2 \over \beta^3 \gamma L_R},
\label{d.10}
\end{equation}
and so
\begin{equation}
{d \epsilon_N \over dz} \approx
{\epsilon_N \over \beta^2 E} {dE \over dz}
+ {\beta_\perp (13.6\ \mbox{MeV}/c)^2 \over 2 \beta^3 \gamma (m_\mu c)^2 L_R}
= {\epsilon_N \over \beta^2 E} {dE \over dz}
+ {\beta_\perp (13.6\ \mbox{MeV}/c)^2 \over 2 \beta^3 E m_\mu L_R}.
\label{d.11}
\end{equation}
{\bf This confirms SR(24) of the Status Report}, with sufficient neglect of
correlations.
A picky point: in my derivation (and in the notation of the Particle Data
Group \cite{pdg}), $dE/dz$ is negative. But in the notation of the Status
Report, $dE/ds$ is positive and is called the energy loss. The notation of
the Status Report, while nonstandard in the larger community, was no doubt
chosen so that the ``cooling'' term is easily identified by the minus sign
in front of it.
It is instructive to use approximate analytic expressions for $dE/dz$ and
$L_R$ so that the two terms of (\ref{d.11}) are more readily compared.
The Bethe-Bloch formula for $dE/dz$ is given as eq.~(23.1) of \cite{pdg}:
\begin{equation}
{dE \over dz} \approx - 4 \pi r_e^2 m_e c^2 N_0 \rho {Z \over A}
\left( {1 \over \beta^2} \ln{2 \gamma^2 \beta^2 m_e c^2 \over I} - 1 \right).
\label{e.9}
\end{equation}
Here, $N_0$ is Avagadro's
number (per mole), $\rho$ is the density of the absorber in, say, g/cm$^3$,
the atomic ``weight'' $A$ is in g/mole, and $I$ is the ionization potential
of the absorber material. We have set the maximum
kinetic energy imparted to an electron in a collision with a muon to
$2\gamma^2 \beta^2 m_e c^2$, which is valid for $\gamma m_e/m_\mu \ll 1$,
as holds in the muon cooling channel. We have also neglected the
density-effect term, which is significant only for $\gamma \gsim 3$.
A useful fit for the radiation length $L_R$ has been given as (23.19) of
\cite{pdg}:
\begin{equation}
{1 \over L_R} = 4\alpha r_e^2 N_0 \rho {Z(Z + 1) \over A}
\ln {287 \over \sqrt{Z}},
\label{e.7a}
\end{equation}
where $\alpha$ is the fine-structure constant.
With (\ref{e.9}-\ref{e.7a}), eq.~(\ref{d.11}) can also be written as
\begin{equation}
{d \epsilon_N \over dz} \approx
{1 \over \beta^2 E L_R}
\left[ - {\pi m_e c^2 \epsilon_N \over \alpha (Z + 1) \ln{287 \over \sqrt{Z}}}
\left( {1 \over \beta^2} \ln{2 \gamma^2 \beta^2 m_e c^2 \over I} - 1 \right)
+ {\beta_\perp (13.6\ \mbox{MeV}/c)^2 \over 2 \beta m_\mu } \right]
\label{d.11a}
\end{equation}
For example, with a hydrogen absorber we take $I = 15$ eV, and find
\begin{equation}
{d \epsilon_N \over dz} \approx
{1 \over \beta^2 E L_R}
\left[ - 19.2 \epsilon_N \left( {12 \over \beta^2} - 1 \right)
+ {0.88 \beta_\perp \over \beta } \right],
\label{d.11b}
\end{equation}
where the value 12 holds for $\gamma\beta \approx 2$. The minimum value of
$\epsilon_N$ that can be achieved with a hydrogen absorber at a location
where the betatron function is $\beta_\perp$ is then,
\begin{equation}
\epsilon_{N,\rm min} = {72 \beta \beta_\perp \over 1 - \beta^2/12}.
\label{d.11c}
\end{equation}
Equation (\ref{d.11b}) can be rewritten in terms of $\epsilon_{N,\rm min}$ as
\begin{equation}
{1 \over \epsilon_N} {d \epsilon_N \over dz} \approx
- {230\ \mbox{MeV} (1 - \beta^2/12) \over \beta^4 E L_R}
\left( 1 - {\epsilon_{N,\rm min} \over \epsilon_N } \right).
\label{d.11d}
\end{equation}
For example, with $\gamma = 2$, $\beta = 0.866$, $E = 210$ MeV,
$P = 182$ MeV/$c$, then
\begin{equation}
{1 \over \epsilon_N} {d \epsilon_N \over dz} \approx
- {1.8 \over L_R}
\left( 1 - {\epsilon_{N,\rm min} \over \epsilon_N } \right).
\label{d.11e}
\end{equation}
To cool $\epsilon_N$ from, say, $4 \epsilon_{N,\rm min}$ to
$2 \epsilon_{N,\rm min}$ would require about $0.6 L_R \approx 480$ cm of
liquid hydrogen, using (\ref{d.11e}) with $\ave{1 - \epsilon_{N,\rm min} /
\epsilon_N} \approx 2/3$. I believe ICOOL indicates that about 600 cm
would be required.
%ccc
\subsubsection{$\ave{x P_x} \neq 0$}
In this case, we also need the average $\ave{ x dP_x/dz}$,
still in the impulse approximation, (\ref{d.2}). From (\ref{d.7}), we can write
\begin{equation}
{dP_x \over dz} \approx
{P_x \over \beta^2 E} {dE \over dz}
+ { (13.6\ \mbox{MeV}/c)^2 \over 2 P_x \beta^2 L_R}.
\label{d.12}
\end{equation}
The second term, however, is ill behaved as $P_x \to 0$. Ignoring this,
we would then find
\begin{equation}
\ave{x {dP_x \over dz}} \approx
{\ave{x P_x} \over \beta^2 E} {dE \over dz}
+ \ave{{x \over P_x}} { (13.6\ \mbox{MeV}/c)^2 \over 2 \beta^2 L_R}.
\label{d.13}
\end{equation}
For this to make physical sense, we must declare that $\ave{x/P_x} = 0$.
This also can be justified as follows. A nonzero correlation $\ave{x P_x}$
in the initial particle distribution means that $\ave{P_x}$ varies with $x$.
After passing through the thin absorber, the $P_x$ are smeared by multiple
scattering, but at a given $x$, the $\ave{P_x}$ remains unchanged, and
so the correlation $\ave{x P_X}$ is unchanged by multiple scattering.
Hence, (\ref{d.8}) is now
\begin{eqnarray}
2m^2c^2 \epsilon_N {d \epsilon_N \over dz} & \approx &
2{( \ave{x^2} \ave{P_x^2} - \ave{x P_x}^2) \over \beta^2 E} {dE \over dz}
+ {\ave{x^2} (13.6\ \mbox{MeV}/c)^2 \over \beta^2 L_R}
\nonumber \\
& = & 2{ (mc)^2 \epsilon_N^2 \over \beta^2 E} {dE \over dz}
+ {\ave{x^2} (13.6\ \mbox{MeV}/c)^2 \over \beta^2 L_R},
\label{d.14}
\end{eqnarray}
using (\ref{d.1}-\ref{d.2}). In the presence of a correlation
$\ave{x P_x}$, it would not be proper to use (\ref{d.9}). So we would just
write
\begin{equation}
{d \epsilon_N \over dz} \approx {\epsilon_N \over \beta^2 E} {dE \over dz}
+ {\ave{x^2} (13.6\ \mbox{MeV}/c)^2 \over 2 \epsilon_N \beta^3 E m L_R}.
\label{d.15}
\end{equation}
However, without the simple relation (\ref{d.9}), we do not obtain as much
insight from this equation as we can from (\ref{eq1}), which holds when
$\ave{x P_x} = 0$.
\subsubsection{Thick Absorbers}
Thus far we have assumed the absorber is thin, and made the impulse
approximation that a particle's $x$ (and $y$) are unchanged during passage
through the absorber. The case of thick absorbers has been considered
by Juan and Rick in \cite{ref24b}, with the general conclusion that if
the confining fields are ``strong enough'', there is little qualitative
change in the form of the transverse cooling equation.
\subsection{Longitudinal Cooling}
The argument here is little different from that in \cite{ref2a}.
In the thin absorber limit there is no change in $z$ of a particle as it
cross an absorber. So if suffices to consider changes in $P_z$, or nearly
equivalently, in $E$. More precisely, since the central energy $E_0$ is
nonzero, we consider changes $\Delta E = E - E_0$ and desire an expression
for $\ave{d(\Delta E)^2/dz}$.
There are two effects to consider: the variation in the mean energy loss with
particle energy, and fluctuations about the mean. We calculate these
separately, and add them in quadrature. First, a particle of energy
$E$ that traverses an absorber of thickness $\delta z$ has mean energy loss
$\delta E_{\rm mean}$
given by
\begin{equation}
\delta E_{\rm mean} = {dE \over dz} \delta z \approx \left( {dE_0 \over dz} +
\Delta E {d^2 E_0 \over dEdz} \right) \delta z.
\label{e.1}
\end{equation}
Then change in $\Delta E$ is then
\begin{equation}
\delta(\Delta E)_{\rm mean} \approx \Delta E {d^2 E_0 \over dEdz} \delta z.
\label{e.2}
\end{equation}
Hence,
\begin{equation}
{d(\Delta E)^2_{\rm mean} \over dz} \approx 2(\Delta E)^2 {d^2 E_0 \over dEdz},
\label{e.3}
\end{equation}
Second, we consider fluctuations in the energy loss in the absorber.
To the first approximation,
it suffices to consider this only for the central energy $E_0$. Using the
nomenclature ``straggling'' for this effect, we have an additional term
\begin{equation}
{d(\Delta E)^2_{\rm straggling} \over dz}.
\label{e.4}
\end{equation}
Combining this with (\ref{e.3}), we have
\begin{equation}
{d(\Delta E)^2 \over dz} \approx 2(\Delta E)^2 {d \over dE}{d E_0 \over dz}
+ {d(\Delta E)^2_{\rm straggling} \over dz}.
\label{e.5}
\end{equation}
This is eq.~SR(25) of the Status Report, again noting that my $dE/dz$ has the
opposite sign to that used there.
\subsubsection{$d(\Delta E)^2_{\rm straggling}/dz$}
It is hard to find a crisp reference for the form of the energy straggling
fluctuations. The basic calculation is due to Bohr \cite{Bohr}. Our past
reference has been to Fano \cite{Fano}, but this paper is quite hard to
read. It might be better to point to sec.~13.3 of \cite{Jackson}, or at least
to add this reference.
Neuffer \cite{Neuffer94} quotes the desired result as
\begin{equation}
{d(\Delta E)^2_{\rm straggling} \over dz} =
4 \pi (r_e m_e c^2)^2 N_0 {Z \over A} \rho \gamma^2 (1 - \beta^2/2)
= 2 \pi (r_e m_e c^2)^2 N_0 {Z \over A} \rho (\gamma^2 + 1),
\label{e.6}
\end{equation}
to display the dependence on muon energy. This result applies only
for ``thick'' absorbers, which is reasonable for the muon collider where
we take the energy away 30 times by ionization loss, although each absorber
is only about 5\% of a radiation length.
When the muons are later accelerated, $\Delta E$ remains constant. Thus,
we seek to minimize $\Delta E$ and not $\Delta E/E$. Hence, the
undesirable ``heating''
due to straggling is minimized by operating at the lowest possible $\gamma$,
according to (\ref{e.6}).
For the record, I note that (\ref{e.6}) can be recast in a way that
emphasizes the radiation length $L_R$ of the absorber by using the fit
(23.19) of \cite{pdg} (given above as (\ref{e.7a})):
\begin{equation}
{d(\Delta E)^2_{\rm straggling} \over dz}
= {\pi (m_e c^2)^2 A (\gamma^2 + 1) \over 2 \alpha (Z + 1) L_R
\ln (287/\sqrt{Z})}.
\label{e.7}
\end{equation}
Equation SR(25) of the Status Report could thus be written
\begin{equation}
{d(\Delta E)^2 \over dz} \approx 2(\Delta E)^2 {d \over dE}{d E_0 \over dz}
+ {\pi (m_e c^2)^2 (\gamma^2 + 1) \over 2 \alpha (Z + 1) L_R
\ln (287/\sqrt{Z})}.
\label{e.8}
\end{equation}
A sense of the relative importance of the two terms in eq.~SR(25) can be
gotten from the Bethe-Bloch formula (\ref{e.9}).
With $E = \gamma m_\mu c^2$,
the leading term in the derivative of (\ref{e.9}) with respect to $E$ is
\begin{equation}
{d \over dE} {dE \over dz} \approx 8 \pi r_e^2 {m_e c^2 \over m c^2} N_0
{Z \over A} {\rho \over \gamma^3 \beta^4}
\left[ \ln{2 \gamma^2 \beta^2 m_e c^2 \over I} - \gamma^2 \right]
\approx 8 \pi r_e^2 {m_e c^2 \over m_\mu c^2} N_0
{Z \over A} {\rho \over \gamma^3 \beta^4} (12 - \gamma^2),
\label{e.10}
\end{equation}
where the final approximation assumes $I \approx 15$ eV for the ionization
potential of hydrogen. (This puts the $dE/dz$ minimum at $\gamma = \sqrt{12}$,
which is a bit low.)
Then, recalling (\ref{e.6}), eq.~(\ref{e.5}) becomes
\begin{eqnarray}
{d(\Delta E)^2 \over dz} & \approx &
2 \pi (r_e m_e c^2)^2 N_0 {Z \over A} \rho \left[
{(\Delta E)^2 \over m_e c^2 m_\mu c^2} {4 (12 - \gamma^2) \over \gamma^3
\beta^4} + (\gamma^2 + 1) \right]
\nonumber \\
& \approx &
{\pi (m_e c^2)^2 \over 2 \alpha (Z + 1)
\ln (287/\sqrt{Z}) L_R} \left[
{48(\Delta E)^2 \over m_e c^2 } {(1 - \gamma^2/12) \over \gamma^2
\beta^4 E} + (\gamma^2 + 1) \right].
\label{e.11}
\end{eqnarray}
This form reveals that the heating due to the variation in $dE/dz$ with
energy is proportional to $1/\beta^4$, which is perhaps the strongest
argument against cooling at very low $\beta$.
One or the other versions of (\ref{e.11}) would be a useful addition to
the Status Report.
For a hydrogen absorber, we can write (\ref{e.11}) as
\begin{equation}
{1 \over (\Delta E)^2 } {d(\Delta E)^2 \over dz} \approx
{466\ \mbox{MeV} (1 - \gamma^2/12) \over \gamma^2 \beta^4 E L_R} \left(
1 + {1.1\ \mbox{(MeV)}^2 \gamma^3 \beta^4 (\gamma^2 + 1) \over
(1 - \gamma^2/12) (\Delta E)^2 } \right).
\label{e.11a}
\end{equation}
For example, with $\gamma = 2$ (which is about the largest we can
consider for transverse cooling) and $\Delta E \approx 10$ MeV,
the first term in (\ref{e.11a}) is about twice the second, and
\begin{equation}
{1 \over (\Delta E)^2 } {d(\Delta E)^2 \over dz} \approx {1 \over L_R}.
\label{e.11b}
\end{equation}
Recalling the example at the end of sec.~3.1.1, the transverse emittance
$\epsilon_N$ was estimated to cool by a factor of 2 in $0.6 L_R$. Equation
(\ref{e.11b}) estimates that $(\Delta E)^2$ would grow by a factor of 1.8
over the same distance.
It looks to me like there is no value of $\gamma$ for which the approximation
(\ref{e.11}) predicts longitudinal ionization cooling. However, our
approximation underestimates the slope of $dE/dz$ for $\gamma >3$, due to our
neglect of the density effect.
It is noteworthy that cooling (heating) scales as the radiation length with
coefficients near unity (for example, (\ref{d.11e}) and (\ref{e.11b})).
Perhaps we could say loosely that ionization cooling is a manifestation of
the very low energy tail of bremsstrahlung, and is in some sense a form
of radiative cooling. This suggests we can find other aspects of
ionization cooling in common with radiative cooling, as in the following
section.
\subsection{The Law of Damping Decrements}
I read in sec.~8.2.3, p. 287 of \cite{Wiedemann} that Robinson
\cite{Robinson} showed that for a process that damps the 6-d emittance
of a bunch, the sum of the damping decrements of all 3
2-d subemittances is a constant. {\sl Robinson's paper does not give a
general ``proof'', but an argument specific to radiative damping. Has
someone else given a more general argument?}
In sec.~3.3.1, I look at momentum damping times (but don't complete the
argument), and in sec.~3.3.2, I consider emittance damping times.
\subsubsection{Momentum Damping Times}
We follow \cite{ref1a} here.
If we ignore
the ``heating'' effects of multiple scattering and straggling, an
interesting relation between transverse and longitudinal ``cooling'' can
be demonstrated.
In the first approximation, the effect of passage of a charged particle
through an absorber is to reduce the magnitude of a particle's momentum {\bf P}
without changing its direction. That is,
\begin{equation}
{d{\bf P} \over dt} = {dP \over dt} \hat{\bf P}.
\label{f.1}
\end{equation}
We note that
\begin{equation}
{dP \over dt} = {dP \over dE} {dE \over dz} {dz \over dt}
= {v_z \over v} {dE \over dz} \approx {dE \over dz},
\label{f.2}
\end{equation}
recalling (\ref{d.5a}).
The transverse part of (\ref{f.1}) can now be written
\begin{equation}
{dP_\perp \over dt} = {dE \over dz}{P_\perp \over P},
\label{f.3}
\end{equation}
so the transverse momentum is damped in time according to $\exp(-t/\tau_\perp)$,
where
\begin{equation}
{1 \over \tau_\perp} = -{1 \over P} {dE \over dz}.
\label{f.4}
\end{equation}
Recall that in our notation, $dE/dz < 0$. Remember also that muon cooling
does not proceed by damping the total momentum $P$ to zero. Rather,
the energy lost to ionization is continually replenished by accelerating
cavities between the absorbers, such that $P$ remains essentially constant
at some value $P_0$.
Turning to the longitudinal momentum, we are not concerned with damping
$P_z$ to zero, but damping the difference, $\Delta P_z = P_z - P_{z,0}$,
while $P_{z,0} \approx P_0$.
Much as in sec.~3.2, we then write
\begin{equation}
{d\Delta P_z \over dt} = {d \over dP_z}{dE \over dz} \Delta P_z
\approx {d \over dP}{dE \over dz} \Delta P_z
= {dE \over dP} {d \over dE}{dE \over dz} \Delta P_z
= v {d \over dE}{dE \over dz} \Delta P_z.
\label{f.5}
\end{equation}
Thus, the longitudinal damping time $\tau_\parallel$ is given by
\begin{equation}
{1 \over \tau_\parallel} = - {d \over dP}{dE \over dz}
= - v {d \over dE}{dE \over dz}.
\label{f.6}
\end{equation}
For $\gamma < 3$-4, $\tau_\parallel < 0$, the the longitudinal momentum
spread is not damped, but grows with time.
{\sl The above is true, but are we making any money from it?}
\subsection{Emittance Damping Times}
This section follows \cite{Palmer94}.
We could also consider the damping of the emittances, again with the
neglect of multiple scattering and straggling.
Thus, eq.~(\ref{d.11}) tells us that the transverse emittance $\epsilon_N$
has a damping distance $z_\perp$ given by
\begin{equation}
{1 \over z_\perp} = - {1 \over \beta^2 E} {dE \over dz}.
\label{g.1}
\end{equation}
When a particle traverse one damping distance in the absorber, it loses
energy $E_{\perp}$ related by
\begin{equation}
{1 \over E_\perp} = - {1 \over z_\perp {dE \over dz}} = { 1 \over \beta^2 E}.
\label{g.2}
\end{equation}
Similarly, eq.~(\ref{e.3}) for $(\Delta E)^2$ leads to a damping distance
$z_{\Delta E}$ given by
\begin{equation}
{1 \over z_{\Delta E}} = - 2 {d \over dE} {dE \over dz},
\label{g.3}
\end{equation}
and the energy loss $E_{\Delta E}$ over this distance is
\begin{equation}
{1 \over E_{\Delta E}} = {2 {d \over dE} {dE \over dz} \over {dE \over dz}}
\approx - {2 \over \gamma^2 \beta^2 E} \left( 1 - {\gamma^2 \over 12} \right),
\label{g.4}
\end{equation}
where the approximation follows from (\ref{e.9}-\ref{e.10}).
We now consider the sum of the energy damping decrements of the 2-d emittances
in $x$, $y$ and $\Delta E$. Noting that $E_x = E_y = E_\perp$, we have
\begin{equation}
\sum {1 \over E_i} = {2 \over E_\perp} + {1 \over E_{\Delta E}}
= {2 \over \beta^2 E} + {2 {d \over dE} {dE \over dz} \over {dE \over dz}}
\approx {2 \over E}
\left( 1 + {1 \over 12 \beta^2 } \right)
\approx {2 \over E},
\label{g.5}
\end{equation}
where the first approximation is based on (\ref{g.4}), and the second
approximation is reasonable for $\beta$ near 1.
It is implied in \cite{Palmer94} that the final result of (\ref{g.5}) is
exact, but my derivation is not powerful enough to reveal this.
In our case, the claim of Robinson could be satisfied by
any function of $E$, not just $2/E$. However, it is helpful to know that
the result (\ref{g.5}) is not an accident.
The impact of (\ref{g.5}) for muon cooling is that strong transverse
cooling implies strong longitudinal heating, even when neglecting
multiple scattering and straggling!
\section{Canonical Angular Momentum}
\subsection{Kinematic Facts}
The canonical angular momentum
of a charge $e$, with mechanical momentum ${\bf P} = (P_r,P_\phi,P_z)$ in a
solenoid field ${\bf B} = B_z \hat{\bf z}$ is
\begin{equation}
L_z = r \left( P_\phi + {e A_\phi \over c} \right) =
r P_\phi + {e r^2 B_z \over 2c},
\label{l.1}
\end{equation}
in Gaussian units, where $r$ is the distance from the magnetic symmetry axis
in cylindrical coordinates $(r,\phi,z)$, and the vector potential of the
solenoid field is ${\bf A} = (0,rB_z/2,0)$.
If the particle has transverse momentum $P_\perp$, the
radius $R_B$ of its helical trajectory in the magnetic field is
\begin{equation}
R_B = {cP_\perp \over eB_z},
\label{l.2}
\end{equation}
and the sense of rotation of the trajectory is $-\hat{\bf z}$ (Lenz's law).
We label $R_G$ as the distance from the magnetic axis to the ``guiding ray''
of the helical trajectory (axis of the helix).
\begin{figure}[htp] % h = here, t = top, b = bottom, p = new page
\begin{center}
\includegraphics[width=6.5in, angle=0, clip]{angmom2.eps}
\parbox{5.5in} % change 5.5in to \hsize for full-width caption
{\caption[ Short caption for table of contents ]
{\label{angmom2} The projection onto a plane perpendicular to the magnetic
axis of the helical trajectory a charge particle of transverse momentum $P$.
The magnetic field $B_z$ is out of the paper, so the rotation of the helix
is clockwise for a positively charged particle.
a) The trajectory does not contain the magnetic axis, and $L_z > 0$.
b) The trajectory contains the magnetic axis, and $L_z < 0$.
}}
\end{center}
\end{figure}
Since the canonical angular momentum is a constant of the motion, we can
evaluate it at any convenient point on the particle's trajectory. In
particular,
we consider the point at which the trajectory is closest to the magnetic
axis. As shown in Fig.~\ref{angmom2}, this point obeys $r = R_G - R_B$, and
so (\ref{l.1}) tells us that
\begin{equation}
L_z = \left( R_G - R_B \right) P_\perp +
{e B_z \over 2 c} \left( R_G - R_B \right)^2
= \left( R_G^2 - R_B^2 \right) {e B_z \over 2 c},
\label{l.3}
\end{equation}
using (\ref{l.2}).
Note that $R_G^2 - R_B^2$ is the product of the closest and farthest distances
between the trajectory and the magnetic axis.
Hence, the canonical angular momentum vanishes for motion in a solenoid
field if and only if $R_G = R_B$,
\ie, if and only if the particle's trajectory passes through the magnetic axis.
We also see that if the trajectory does not contain the magnetic axis, the
canonical angular momentum is positive; while if the trajectory contains
the magnetic axis, the canonical angular momentum is negative.
\subsection{The Effect of Energy Loss in an Absorber}
We ``cool'' the transverse momentum of the muon with absorbers placed in their
path. Then the projected trajectory of the muon after the absorber is a
circle of smaller radius, according to (\ref{l.2}).
In the first approximation, we ignore multiple scattering. Then the
circle after the absorber is tangent to the circle before the absorber,
and the point of tangency is at the absorber, as shown in Fig.~\ref{angmom1}.
\begin{figure}[htp] % h = here, t = top, b = bottom, p = new page
\begin{center}
\includegraphics[width=6.5in, angle=0, clip]{angmom1.eps}
\parbox{5.5in} % change 5.5in to \hsize for full-width caption
{\caption[ Short caption for table of contents ]
{\label{angmom1} The effect of energy loss in an absorber on particle
trajectories in a solenoid magnetic field. The outer (dash-dot)
circles represent the
coil of the magnet, seen end-on. The solid and dashed circles are the
projections of particle's trajectory before and after passing through the
absorber, respectively. a) The initial trajectory has zero
canonical angular momentum, and therefore passes through
the magnetic axis. The point of absorption is on the axis also. The
final transverse momentum is lower, but the canonical angular momentum is
still zero. b) The canonical angular momentum is initially zero, but becomes
nonzero after the absorber. c) The canonical
angular momentum is initially nonzero, but the magnetic axis is within the
trajectory. The absorber reduces the magnitude of the
canonical angular momentum.
d) The canonical angular momentum is
initially nonzero, and the magnetic axis is not within the trajectory.
The absorber decreases or increases the canonical angular momentum,
depending on the location of the absorber.
}}
\end{center}
\end{figure}
There are several cases:
\begin{enumerate}
\item
The initial canonical angular momentum is zero, and the absorber is on the
magnetic axis (Fig.~\ref{angmom2}a). The canonical angular momentum remains
zero.
\item
The initial canonical angular momentum is zero, but the absorber is not on the
magnetic axis (Fig.~\ref{angmom2}b). The final trajectory does not
contain the magnetic axis, and the final canonical angular momentum
is greater than zero.
\item
The initial canonical angular momentum is less than zero, so the
trajectory contains the magnetic axis (Fig.~\ref{angmom2}d).
The final trajectory lies within the initial trajectory. Unless the energy
loss is large, the final trajectory will still contain the magnetic axis, and
the final canonical angular momentum remain negative.
The magnitude of the closest and farthest distances of the trajectory from
the axis are both reduced, so the magnitude of the canonical angular
momentum is reduced.
zero.
\item
The initial canonical angular momentum is greater than zero, so the
trajectory does not contain the magnetic axis (Fig.~\ref{angmom2}d).
The final trajectory lies within the initial trajectory, so the canonical
angular momentum remains positive and its value can either increase or
decrease. If the absorber is close to the magnetic axis, the closest
distance of the trajectory to the magnetic axis is little changed, but
the farthest distance is decreased; hence, the canonical angular momentum would
decrease. However, if the absorber is far from to the magnetic axis, the
farthest distance of the trajectory to the magnetic axis is little changed,
while the closest distance is increased hence, the canonical angular momentum
would increase.
\end{enumerate}
In general, the effect of a series of absorbers not on the magnetic axis
would be to ``cool'' the transverse momentum $P_\perp$ and the
trajectory radius $R_B$ towards zero, but to leave the particle with a
nonzero guide radius $R_G$, and hence with a
a nonzero canonical angular momentum.
One option is to restrict the absorbers to a maximum radius $R_A$. In the
limit that $R_B$ is cooled to zero, we would still have $R_G \approx R_A$,
Hence the canonical angular momentum would converge on a value of order
$e R_A^2 B_z/2c$. The particle would emerge from the solenoid with this
value for its mechanical angular momentum. Hence, (\ref{l.3}) tells us
\begin{equation}
P_{\phi,\rm outside}[\mbox{MeV/}c] \approx {e R_A B_z \over 2c}
= 150 R_A[\mbox{m}] B_z[\mbox{T}].
\label{l.4}
\end{equation}
For example, if $R_A = 1$ cm and $B_z = 15$ T, then $P_{\phi,\rm outside}
\approx 22.5$~MeV/$c$. This is still rather high.
Another option is to use alternating solenoids.
\subsection{Alternating Solenoids}
This section follows \cite{whyalt}. {\sl Bob's definition of canonical
angular momentum has a sign error for the term involving the magnetic field,
if charge $e$ is taken as positive.}
The possible advantage of alternating the direction of the magnetic field
in the solenoids is sketched in Fig.~\ref{angmom3}.
\begin{figure}[htp] % h = here, t = top, b = bottom, p = new page
\begin{center}
\includegraphics[width=6.5in, angle=0, clip]{angmom3.eps}
\parbox{5.5in} % change 5.5in to \hsize for full-width caption
{\caption[ Short caption for table of contents ]
{\label{angmom3} The effect of alternating the sign of the magnetic field in
the solenoid.
}}
\end{center}
\end{figure}
Suppose the transverse momentum of a particle has been cooled to zero, but
the radius of the guiding ray is nonzero, say $R_G = R_0$. The radius $R_B$ of
the helical trajectory is zero. The
canonical angular momentum of that particle in a field $B_z = +B$ is
$L_z = +eR_0^2 B/2c$. The
projection of the motion on a plane perpendicular to {\bf B} is shown in
Fig.~\ref{angmom3}a.
If the particle then exited the solenoid, the situation would be as sketched in
Fig.~\ref{angmom3}b. In the impulse approximation, the radius of the particle
does not change, but the fringe field of the solenoid gives it an azimuthal
kick. This is easily calculated via conservation of canonical angular
momentum, and we find that $P_{\phi,\rm outside} = eR_0B/2c$. This is very
undesirable.
Suppose instead, the particle exited the solenoid with $B_z = +B$ and
immediately entered another solenoid with $B_z = -B$, as sketched in
Fig.~\ref{angmom3}c. Again the canonical angular momentum of the particle
is conserved, and in the impulse approximation, the position of
the particle does not change. Hence the transverse momentum is kicked by
the fringe fields to double the amount in case b), namely $P_\phi =
eR_0^2 B/c$. Since the particle is in a magnetic field, its projected
trajectory is a circle, but now $R_B = R_0$, and the circle is centered on
the magnetic axis, so that $R_G = 0.$
If we can cool the newly created transverse momentum to half its value
while in the field $B_z = -B$, then the helix radius will shrink to $R_B =
R_0/2$. If in this process, the radius of the guiding ray rises from zero
to $R_0/2$, then the final canonical angular momentum would be zero and
the particle could exit the magnet without experiencing an azimuthal kick.
This is shown in Fig.~\ref{angmom3}d.
If we have enough control over the growth of $R_G$ during the cooling to
guarantee the desired final condition, $R_G = R_0/2$, a single reversal of
the solenoid field would suffice. This scenario has not been explored yet
in simulation.
Rather, the present thinking is that frequent reversal of the solenoid field
will best accomplish the desired goal of ending with very small canonical
angular momentum. An ICOOL (?) simulation, Fig.~\ref{angmomfig4}, shows
hows a ionization cooling in a sequence of alternating solenoids can
keep the canonical angular momentum always near zero, while the mechanical
angular momentum drops by a factor of two in 20~m.
% Sample LaTex Figure. Refer to this figure in the text at fig.~\ref{ NAME }
\begin{figure}[htp] % h = here, t = top, b = bottom, p = new page
\begin{center}
\includegraphics*[bb = 50 200 550 375, width=6.5in, clip]
{palmer:angmomfig4.eps}
\parbox{5.5in} % change 5.5in to \hsize for full-width caption
{\caption[ Short caption for table of contents ]
{\label{angmomfig4} Cooling of angular momentum in a channel of solenoids
whose fields are reversed every 2~m.
}}
\end{center}
\end{figure}
\section{Betatron Function of a Solenoid}
Here I try to reconcile the language of betatron functions with a separate
understanding of helical trajectories in a solenoid.
I read (in, for example, sec.~21 of \cite{pdg})
that the amplitude $x(z)$ for a transverse coordinate of a particle
in a beam transport system along the $z$ axis is represented as
\begin{equation}
x(z) = A \sqrt{\beta_\perp(z)} \cos(\phi(z) + \delta),
\label{b.1}
\end{equation} where $A$ and $\delta$ are constants, $\beta_\perp$ is the
betatron function, and $\phi$ is the phase-advance function which obeys
\begin{equation}
{d \phi \over dz} = {1 \over \beta_\perp}.
\label{b.2}
\end{equation}
The projected slope $x'(z)$ of the trajectory then obeys
\begin{eqnarray}
x'(z) & = & - {A \over \sqrt{\beta_\perp(z)}} \sin(\phi(z) + \delta)
+ A {\beta'_\perp(z) \over 2 \sqrt{\beta_\perp(z)}} \cos(\phi(z) + \delta),
\nonumber \\
& \approx & - {A \over \sqrt{\beta_\perp(z)}} \sin(\phi(z) + \delta),
\label{b.2a}
\end{eqnarray}
where the approximation holds for ``slowly varying'' betatron functions.
For the record, if we consider a bunch of particles with various $A_i$,
then the rms bunch parameters are
\begin{equation}
\sigma_x = \sigma_A \sqrt{\beta_\perp}, \qquad
\sigma_{x'} = {\sigma_A \over \sqrt{\beta_\perp}},
\qquad \mbox{and} \qquad
\epsilon = \sigma_x \sigma_{x'} = \sigma_A^2,
\label{b.2b}
\end{equation}
ignoring correlations between $x$ and $x'$. The usual relations follow:
\begin{equation}
\epsilon = {\sigma_x \over \sigma_{x'}}, \qquad
\sigma_x = \sqrt{\epsilon \beta_\perp},
\qquad \mbox{and} \qquad
\sigma_{x'} = \sqrt{\epsilon \over \beta_\perp}.
\label{b.2c}
\end{equation}
\subsection{Uniform Solenoid}
We first consider a solenoid with a uniform field $B_z$. The a particle
with charge $e$ and transverse momentum $P_\perp$ moves in a helix of
radius
\begin{equation}
R_B = {c P_\perp \over e B_z},
\label{b.3}
\end{equation}
and with angular velocity (the Larmor frequency)
\begin{equation}
\omega_B = {e \beta_\perp B_z \over P_\perp} = {e c B_z \over E} =
{e \beta_z B_z \over P_z}.
\label{b.4}
\end{equation}
{\sl I remember things like this from a `relativistic' form of $F = ma$ for
circular motion due to the Lorentz force: $\gamma m v_\perp^2/r = e v_\perp
B_z/c$.}
Of course, the particle moves in $z$ with velocity $\beta_z c$. Hence,
the $x$ projection of a helix centered on the $z$ is then
\begin{equation}
x = R_B \cos \left( \omega_B t + \delta \right)
= R_B \cos \left( {\omega_B z \over \beta_z c} + \delta \right)
= R_B \cos \left( {e B_z z \over c P_z} + \delta \right),
\label{b.5}
\end{equation}
Comparing with (\ref{b.1}-\ref{b.2}), it is natural to identify the
betatron ``function'' as
\begin{equation}
\beta_\perp = {c P_z \over e B_z},
\label{b.6}
\end{equation}
based on the form of the phase (not of the amplitude).
This result was quoted earlier as (\ref{eq2}).
We can, of course, write (\ref{b.3}) as
\begin{equation}
R_B = {c P_z \over e B_z} {P_\perp \over P_z}
= \sqrt{c P_z \over e B_z} {P_\perp \over P_z} \sqrt{\beta_\perp},
\label{b.7}
\end{equation}
so the constant factor $A$ (whose dimensions are [length]$^{1/2}$) in
(\ref{b.1}) is
\begin{equation}
A = \sqrt{c P_z \over e B_z} {P_\perp \over P_z}
=\sqrt{c P_\perp^2 \over e B_z P_z} .
\label{b.8}
\end{equation}
\subsection{Slowly Varying Field $B_z(z)$}
For a solenoid whose field varies ``slowly'' in $z$, it is reasonable to
define the local radius $R_B(z)$ of the (nonuniform) helix and the
local Larmor frequency $\omega_B(z)$. Then (\ref{b.5}) continues to have
meaning, and again we identify the betatron function as (\ref{b.6}).
But does the radius of the helix obey the form $R_B(z) = A
\sqrt{\beta_\perp(z)}$, as required for description (\ref{b.1}) to apply?
Our use of the betatron function in (\ref{eq2}) emphasized this aspect.
For motion in a ``slowly varying'' field, the magnetic flux through the
orbit is an adiabatic invariant:
\begin{equation}
R_B^2 B_z = {c^2 P_\perp^2 \over e^2 B_z} = K =\ \mbox{constant},
\label{b.9}
\end{equation}
using (\ref{b.3}).
Inserting (\ref{b.9}) in (\ref{b.8}), we have
\begin{equation}
A = \sqrt{e K \over c P_z}
\approx \sqrt{e K \over c P} = \ \mbox{constant},
\label{b.10}
\end{equation}
since the total mechanical momentum $P$ is constant in any static magnetic
field.
We conclude that the description (\ref{b.1}) of both amplitude and phase
in terms of the betatron function (\ref{b.6}) is valid for motion in a
slowly varying solenoidal field so long as $P_z \approx P$, \ie, so long as the
angles $\theta_x$ and $\theta_y$ are not too large.
{\sl I am intrigued by a feature of the above discussion. We have shown that
the form (\ref{b.1}) is a reasonable description for the projected trajectory
of a charged particle in a ``slowly varying'' solenoid field. Nowhere in
the argument was there a requirement that the solenoid field be periodic
in $z$. Yet the classic derivation of (\ref{b.1}) makes heavy use of this
assumption. Section 5.5 of \cite{WiedemannI} gives a good discussion
of (\ref{b.1}) as an insightful guess to the solution of the differential
equation $x'' + k(z) x = 0$, without requiring $k(z)$ to be periodic.}
%xxx
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R.~C.~Fernow and J.~C.~Gallardo,
{\em Validity of the differential equations for ionization cooling},
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R.~C.~Fernow and J.~C.~Gallardo,
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%rrr
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\end{document}
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\subsection{Adiabatic Invariance}
This section is prompted by the remark just above, and a reading of Chap.~VII\
of \cite{Landau}.
Consider a particle of mass $m$ moving in the $x$-$z$ plane subject to a
confining force in the $x$ direction whose strength is a function of $z$.
That is, the force is derivable from a potential
\begin{equation}
U(x,z) = {k(z) x^2 \over 2}.
\label{b.11}
\end{equation}
If the motion is paraxial, we may approximate the velocity $\dot z$ as a
constant. The position $z$ of the particle is then a known function of time
$t$. Hence, the transverse equation of motion of the particle of mass $m$ is
\begin{equation}
m\ddot x + k(t) x = 0.
\label{b.12}
\end{equation}
In this approximation, the time-dependent Hamiltonian of the system is
\begin{equation}
H(x,P,k) = {P^2 \over 2m} + {k(t) x^2 \over 2},
\label{b.13}
\end{equation}
where $k(t)$ is presumed to be slowly varying on the scale of the
(quasi)period of the motion.