Report on

Plans of the Beams Division

for

TeV33

April 16, 1997

J. Marriner, editor

Contributors:

C. Ankenbrandt

P. Bagley

F. Bieniosek

M. Church

J. Johnstone

I. Kourbanis

D. McGinnis

J. Marriner

S. Ngaitsev

S. O'Day

R. Pasquinelli

S. Shukla

S. Werkema

**1. Colliding beam parameters & Overview**

1.1 Collider Luminosity

1.2 Importance of Antiproton Production

1.3 Importance of Beam-Beam Effect

1.4 TeV33 overview and parameters

2\. Assumed conditions

2.1 Main Injector

2.2 Recycler

2.3 Tevatron Energy

2.4 Proton Removal

3\. Antiproton Production

3.1 Proton Intensity at the Antiproton Target

3.1.1 Linac & Booster performance

3.1.2 Main Injector performance

3.1.3 Slip stacking

3.2 Antiproton Target

3.2.1 Beam sweeping

3.2.2 Lithium lens mechanical issues

3.2.3 Radiation damage issues

3.2.4 Antiproton target radiation shielding and environmental issues

3.2.5 Antiproton Target Conclusions

3.3 Antiproton Acceptance

3.3.1 Lithium Lens Gradient

3.3.2 Transverse aperture increase to 32π x 32π

3.3.3 Matching the Beam Phase Space to the Debuncher Admittance

3.3.4 Debuncher Aperture

3.3.5 Momentum aperture increase

3.4 Antiproton Bunch Rotation

3.5 Antiproton Cooling Overview

3.6 Transverse Cooling in the Debuncher

3.7 Debuncher Momentum Cooling

3.8 Longitudinal Stacking in the Accumulator

3.9 Stochastic Cooling Technology

3.9.1 4-8 Ghz Pickups

3.9.2 4-8 GHz Signal Transmission

3.10 Antiproton Electron Cooling

3.10.1 Electron Cooling Rate

3.10.2 Electron Beam Optics

3.10.3 Technical Considerations

3.10.4 Alternative approaches

3.11 Transfers Between the Accumulator and Recycler

4\. Colliding beams issues

4.1 Tevatron Injection

4.2 Proton Intensity

4.3 Antiproton Intensity

4.4 Tevatron rf voltage considerations

4.5 Single beam stability

4.6 Beam abort

4.7 Interaction Regions

4.7.1 Lattice

4.7.2 Magnitude of the crossing angle

4.7.3 Effect of the crossing angle on the luminosity

4.7.4 Effect of the crossing angle on the z distribution

4.8 Beam-beam effects

4.8.1 Bunch-by-bunch tune shifts

4.8.2 Choice of equal horizontal and vertical crossing angles.

4.8.3 Choosing our favorite combinations

4.8.4 subsection \-- An example

4.9 Luminosity Leveling

5\. Summary of upgrades required

6\. Risks & Alternative approaches

6.1 Risks

6.2 Alternatives

7\. Conclusion

#  Colliding beam parameters & Overview

The Tevatron antiproton collider achieved a peak luminosity of
2.5x10^31^ cm^−2^sec^−1^ in Run I, which ended in February 1996. The
Main Injector, the Recycler Ring, and the associated colliding beams
upgrades are expected to enable luminosities in the range of
8×10^31^--2×10^32^ cm^‑2^sec^‑1^ in Run II, which is scheduled to begin
late in 1999. These upgrades are sometimes collectively referred to as
Fermilab III. The addition of the Recycler Ring to the Main Injector
project is expected to improve the antiproton utilization efficiency and
accounts for a factor of 2 to 3 of the luminosity increase..

The goal of the TeV33 project is to increase the peak luminosity to the
vicinity of 10^33^ cm^‑2^sec^−1^. A more specific goal of obtaining 30
fb^−1^ by the year 2006 was suggested in the TeV2000 committee
report.[^1] The TeV33 period of collider running is also referred to as
Run III.

Table I is the working parameter table for Run II. It illustrates the
changes required to achieve the Run II luminosity goals and also the
benefits of antiproton recycling. Run II requires a modest improvement
in proton intensity and about 8 times more antiprotons (spread over 6
times more bunches). The antiproton stacking rate is required to
increase substantially (about a factor of 3) to produce the necessary
antiprotons. The Run II luminosity also benefits from the smaller (2
eV-sec) longitudinal emittances expected from the Main Injector, the
higher energy (1000 GeV instead of 900 GeV) Tevatron, and the higher
antiproton transmission efficiency of the Main Injector.

##  Collider Luminosity

The luminosity of the Tevatron collider may be written as

$L = \frac{3g_{r}f_{0}}{\beta^{}}\left( \text{BN}_{\overline{p}} \right)\left( \frac{N_{p}}{e_{p}} \right)\frac{F\left( \beta^{},q_{x},q_{y},e_{p},e_{\overline{p}},s_{z} \right)}{(1 + e_{\overline{p}}/e_{p})}$
\[1\]

where *γ~r~=E/mc^2^* is the relativistic energy factor, *f~0~* is the
revolution frequency, and $\beta^{}$ is the beta function at *s*=0
(where it is assumed to attain the same minimum in each plane). The
proton (antiproton) beam transverse emittance
$e_{p}(e_{\overline{p}})$is defined to be
$e = 6\text{pg}_{r}s^{2}/\beta$ for a bunch with a gaussian distribution
and assumed to be the same in both transverse planes, *B* is the number
of bunches, $N_{p}(N_{\overline{p}})$ is the number of protons
(antiprotons) per bunch,* θ~y~* and *θ~y~* are the crossing half-angles,
*σ~z~* is obtained from the rms proton and antiproton bunch
lengths$s_{{}_{z}}^{2} = \left( s_{\text{zp}}^{2} + s_{z\overline{p}}^{2} \right)/2$
and *F*≤1 is a form-factor that accounts for the depth of focus
(hourglass) and crossing angle effects on the luminosity caused by
non-zero bunch lengths. The bunch lengths depend on the longitudinal
emittance and the rf voltage, but the luminosity depends only on the
bunch lengths. In Run II, the form-factor is dominated by the hourglass
effect (the design crossing-angle is 0). For gaussian beams the
hourglass effect may be written as:

$F = \frac{\sqrt{p}\beta}{s_{z}}e^{\frac{\beta^{2}}{s_{z}^{2}}}\text{erfc}\left\lbrack \frac{\beta}{s_{z}} \right\rbrack$
\[2\]

where the complementary error function is related to the error function
by $\text{erfc}(z) = 1 - \text{erf}(z)$. For TeV33 the crossing angle
effect is large and the luminosity comes mainly from the *z*=0 region
where the hourglass effect is small. In this case the form-factor *F*
may be written as

$F = \frac{1}{\sqrt{1 + s_{z}^{2}\left( q_{x}^{2}/s_{x}^{2} + q_{y}^{2}/s_{y}^{2} \right)}}$
\[3\]

where
$s_{x}^{2} = \left( s_{\text{xp}}^{2} + s_{x\overline{p}}^{2} \right)/2$
and similarly for *y*.

The luminosity formula ( \[1\]) is written to emphasize the major issues
in achieving high luminosity. The first quantity in parentheses is the
total number of antiprotons. Under current and probably future operating
conditions, the most important factor contributing to the achievable
luminosity is the total number of antiprotons in the ring,
$\text{BN}_{\overline{p}}$. The second most important factor is the
proton phase space density, *N~p~/ε~p~*, which is constrained by the
need to limit the beam-beam tune shift. The form-factor (*F*) and the
emittance ratio factor $e_{p}/\left( e_{p} + e_{\overline{p}} \right)$
are important, but they cannot exceed unity and the amount of luminosity
that can be gained using these factors is limited.

##  Importance of Antiproton Production

Of the many technical issues involved with high luminosity
proton-antiproton colliders, there is probably no more fundamental
limitation than the requirement that antiprotons must be produced at
least as rapidly as they are consumed in beam-beam collisions. The
minimum production rate is

$F_{\overline{p}}^{(\text{min})} = n_{c}s_{a}L$ \[4\]

where *n~c~* is the number of collision points and L is the luminosity.
The cross-section is the cross-section for scattering outside the
acceptance of the Tevatron. This cross-section is only slightly less
than the total cross-section. We assume that *σ~a~* is 70 mb at 1000
GeV. With 2 collision points a luminosity of 10^33^ cm^−2^-sec^−1^ is
sustained with a minimum antiproton production rate of 50×10^10^ hr^−1^.

A more realistic estimate of the antiproton flux must take into account
the fact that antiprotons beam-beam collisions are not the only
mechanism for antiproton loss. We define the antiproton utilization
efficiency as the number of antiprotons lost through beam-beam
collisions divided by the total number of antiprotons produced. During
the latter part of Run Ib the antiproton utilization efficiency was
about 7%.[^2]

The second consideration in determining the antiproton flux required is
that neither the luminosity nor the stacking rate is constant. For
example, during the Run Ib period referred to above the Tevatron was
producing beam-beam collisions for the experiments 51% of the time. The
average initial luminosity of these stores was 1.25×10^31^
cm^−2^-sec^−1^, but the average rate of accumulating luminosity during a
store (29 nb^−1^/hr) corresponds to a luminosity 64% lower. Thus, the
Run Ib experience is consistent with the "Snowmass Criterion": that the
integrated luminosity obtained is equal to the peak luminosity times the
length of the run divided by 3. The peak stacking rate during Run Ib was
7.2x10^10^ hr-1. During the same Run Ib running period the antiproton
source was stacking 62% of the time at an average rate of 4.3x10^10^
hr-1 (60% of the peak value). Thus, the total number of antiprotons
accumulated was 37% (also roughly 1/3) of the peak rate times the length
of the run. It should be noted that the percentages of time given for
both the Tevatron and the Antiproton Source have no corrections for
effects such as scheduled and unscheduled maintenance; they represent
actual operating experience during an extended run.

The antiproton utilization efficiency must increase dramatically for Run
II when the luminosity is expected to increase to 2×1032 cm−2sec−1 from
the Run Ib value of 2×1031 cm−2sec−1 while the stacking rate increases
from 7×10^10^ hr^−1^ to 20×10^10^ hr^−1^. If the duty factors and
efficiencies experienced in Run Ib were to remain the same, then the
antiproton utilization efficiency would have to increase to 42%.
However, the use of the Recycler as post-Accumulator should side-step
the problem of reduced stacking rate when the Accumulator stack size
increases above about 50 mA. Accounting for the increase in average
stacking rate and assuming a negligible inefficiency in the process of
transferring beam to the Recycler, an antiproton inefficiency of perhaps
32% would be sufficient to achieve the Run II goals. The increase in
antiproton utilization efficiency is expected to arise from improved
transmission through the Main Injector, from avoiding the inefficiency
of coalescing the antiproton bunches, and from the recovery of unspent
antiprotons by the Recycler at the conclusion of a Tevatron store.

For the purposes of the TeV33 design, we assume that the Run II goals
will be met but that there will be no further increases in the
antiproton utilization efficiency. Under these assumptions, the increase
in luminosity is directly proportional to the increase in stacking rate,
and we conclude that peak stacking rates of about 10^12^ antiprotons per
hour are required to support two interaction regions at 10^33^
cm^−2^-sec^−1^. This rate is a 10-fold increase in stacking rate over
the Tevatron I design, a 14-fold increase over the best stacking rate
achieved, and a 5-fold increase over the projected Run II stacking rate.
Clearly, dramatic increases in the antiproton production rate are an
essential element of any plan to achieve a luminosity of 10^33^
cm^−2^-sec^−1^ in the Tevatron proton-antiproton collider.

##  Importance of Beam-Beam Effect

The formula for the (linear) antiproton beam-beam tune shift with no
crossing angle is:

$\begin{array}{r}
D\nu = 6\frac{r_{p}}{4p}n_{c}\frac{N_{p}}{e_{p}} \\
0\text{.}\text{0073}(p\text{ mm-mrad/10}^{\text{10}})n_{c}\frac{N_{p}}{e_{p}}
\end{array}$ \[5\]

where *r~p~* is the classical proton radius (1.535x10^-18^ m) and *n~c~*
is the number of interaction points. Operating experience in the
Tevatron suggests that the maximum tolerable beam-beam tune shift lies
in the range 0.02 to 0.025.

When the beam-beam tune shift is caused primarily by head-on
interactions at zero crossing-angle, the beam-beam tune shift determines
the maximum value of the factor$N_{p}/e_{p}$, which appears in Eqn.
\[1\]. For TeV33, the formula (Eqn. \[5\]) does not apply. In TeV33, the
beams cross at an angle to avoid unwanted beam-beam interactions near
the interaction region . The crossing angle at the interaction region
dramatically reduces the beam-beam tune shift (some higher order effects
increase), and the sum of the long range interactions cause tune shifts
comparable to those at the interaction points. These crossing angle and
long range effects depend on both *N~p~* and *ε~p~* separately, and may
partially cancel depending on the detailed geometry of the beams and
their orbits. These issues are discussed in considerably more detail in
section 4.8. Despite the complicated nature of the beam-beam
interaction, the proton beam parameters for TeV33 are consistent with a
naive application of Eqn. \[5\].

## TeV33 overview and parameters

The general strategy to increase the luminosity in the Tevatron
proton-antiproton collider is to produce antiprotons at a much higher
rate and to maintain the Run II antiproton utilization efficiency. This
goal can be accomplished if the antiproton production rate can be
increased and the emittances held at their Run II values. Some possible
TeV33 parameters are given in Table II. The antiprotons are distributed
into more bunches (90 for TeV33 versus 36 for Run II). The number of
bunches is determined primarily by the desire of the experiments to keep
the number of interactions per beam crossing as low as possible.
However, at least for the parameters shown in Table II, the number of
bunches would have to increase to avoid excessive proton beam tune
shifts (caused by high intensity antiproton bunches). The TeV33
parameters are, at this point, still speculative. It is quite possible
that we will choose to obtain higher luminosity by making the bunches
smaller in some dimension. One version (with improved rf focusing) is
shown in Table II. For the parameters given, the bunch length is reduced
from 35 cm to 15 cm and the form-factor is increased from 0.52 to 0.79.
Other factors appearing in the luminosity formula can also be
manipulated by various techniques. However, the *integrated* luminosity
can be increased by at most a factor of 2 over the assumed antiproton
utilization efficiency of 50%, because the integrated luminosity is
limited by the production rate of antiprotons.

# Assumed conditions

##  Main Injector

The Main Injector is a project that will result in the replacement of
the Main Ring by a smaller, larger aperture ring known as the Main
Injector. The Main Injector project is nearing completion and is
described in detail in the Main Injector Design Handbook.[^3] Some of
the major goals of the Main Injector project, relevant to TeV33 are
listed in Table III. This document assumes that all of these goals will
be met.

##  Recycler

The Recycler is described elsewhere.[^4] The most important design goal
is to recover, on average, 50% of the antiprotons that could potentially
be recovered. This goal could be met, for example, if 75% of the
antiprotons are recovered from 75% of the stores that end normally[^5].
We assume that we will continue to achieve the 50% antiproton recovery
efficiency for TeV33 despite the increased number of bunches, the higher
intensities, and (possibly) somewhat larger emittances.

The performance of the Recycler will be enhanced beyond the Run II goals
that are described in Ref. Error: Reference source not found. The
electron cooling system will greatly increase the antiproton
accumulation rate and the addition of a dedicated beam line between the
Accumulator and the Recycler will enable frequent and reliable transfers
to the Recycler.

## Tevatron Energy

We have specified that the Tevatron will run at 1000 GeV in Run II and
future runs. We have finished an upgrade to the cryogenic cooling system
and have accelerated protons to 980 GeV. The plan to achieve 1000 GeV
operation involves running some satellite refrigerators at lower
pressure (and therefore a lower temperature) and some shuffling of
magnets. It is clear that the ultimate energy limit of the Tevatron lies
near 1000 GeV, but it is not clear whether we will be able to run
reliably at 1000 GeV. The precise operating energy will depend on our
success at optimizing the operation of the cryogenic systems (including
possible improvements) and at mitigating the problem of low
quench-current magnets.

## Proton Removal

In order to recycle the antiprotons, they must be separated from the
protons. We plan to eliminate the protons at the end of a Tevatron
store, before deceleration. This plan has the advantage of making the
deceleration process much easier because of the absence of beam-beam
interaction effects. In addition, this scheme allows the deceleration of
the antiprotons on the central orbit which has better field quality and
more aperture than the helical orbit used for acceleration. However, the
plan does require removal of the protons from the Tevatron at high
field, when the Tevatron magnets have the least margin against quenches
induced by beam loss. While we have substantial experience with removing
the protons with scrapers for special experiments (the proton and
antiproton beams are spatially separated), it typically takes half an
hour to complete the process. Improvements both in technique and speed
would be highly desirable. At the moment, it is uncertain how this goal
will be accomplished in Run II.

The Tev33 parameters require the removal of about 3 times the number of
proton bunches. The techniques established for Run II may require
modification. We assume that an adequate solution will be found based on
Run II experience.

#  Antiproton Production

The strategies for increasing the antiproton production fall into three
categories:

1)  Increase the number of protons targeted per pulse

2)  Increase the proton beam pulse rate

3)  Increase the antiproton collection efficiency

Relative to the Run II parameters, we plan to increase the number of
protons targeted (by a factor of 1.9), to decrease the pulse rate by
25%, and to increase the antiproton collection efficiency (by a factor
of 3.4), for a net increase in the antiproton flux of 4.8. The decrease
in repetition rate is largely to allow more time per cycle for
stochastic cooling although some decrease is required to accommodate the
increased number of Booster cycles for slip stacking (see 3.1.3)

Table IV shows the numerology of antiproton production during Collider
Run 1b and the production that would be expected in Collider Run II in
the absence of upgrades to the target station, the injection line
(AP-2), and the Debuncher apertures (assuming also that only a single
Booster batch is targeted). The gain in production is due to the
increased proton intensity on target and the increased repetition rate.
Without the target sweeping system (described below), the yield
(antiprotons produced per proton) would decrease because the proton beam
spot size would need to be increased in order to keep the deposited
energy density in the nickel target below 800 J/g. Assuming perfect
Debuncher to Accumulator transfer efficiency, the maximum stacking rate
in Run II will be 21.4x1010/hour. In Run 1b, the Debuncher to
Accumulator transfer efficiency was 80%, and the Accumulator stacking
efficiency (rf stacking plus stack tail momentum cooling) was 90%.
Upgrades to the Debuncher and Accumulator stochastic cooling systems are
expected to increase these two efficiencies for Run II.

There are several upgrades which greatly increase the antiproton flux
into the Debuncher in Run II and beyond:

1)  Beam sweeping: By sweeping the proton beam in a small circle on the
    target during the 1.6 μsec spill, the energy deposition can be
    spread out over a larger volume of the target, and therefore the
    beam spot size can be made smaller, which increases the yield. The
    antiproton beam downstream of the target is then swept to remove the
    circular motion. In the absence of beam sweeping, each factor of 2
    increase in proton intensity on target requires that the beam spot
    size be enlarged causing the antiproton yield to drop by 15-20%.

2)  AP-2/Debuncher transverse aperture increase: The measured transverse
    emittance of beam entering the Debuncher is 15π x 16π, and the
    measured acceptance of the Debuncher is 29π x 26π.  Increasing the
    AP-2/Debuncher transverse aperture to 32π x 32π will increase the
    yield into the Debuncher by a factor of 2.6.

3)  Lithium lens improvements: Currently the lens operates at 750 T/m.
    Modest improvements to the original design may permit reliable
    operation at 900 T/m, which will increase the yield by 17% under
    current conditions.

4)  Development of new lens design involving liquid lithium could allow
    reliable operation at 1200 T/m, with a further increase in yield. We
    are still investigating the feasibility of the liquid lithium lens.

5)  Momentum aperture increase: The full momentum acceptance of the
    Debuncher is currently measured to be 4%. By ramping the slip factor
    η to .003 at injection, the momentum aperture can be increased by
    18%, which increases the yield by 18%.

In addition to the above improvements, \"slip stacking\" in the Main
Injector (longitudinally stacking more than one Booster batch of protons
into the MI in each 120 GeV beam cycle) may increase the integrated
number of protons on target by a factor of 1.9 (see section 3.1.3).
Table V summarizes the possible gains that can be made in antiproton
production and collection with the above upgrades. The table lists the
cumulative gains in yield and production. The order is arbitrary; the
factors that increase the antiproton flux are largely independent.

While there are a number of techniques available to increase the
antiproton flux, there is no realistic alternative to improved
antiproton cooling. The plan for TeV33 requires higher frequency
stochastic cooling in the 4-8 Ghz band and the introduction of electron
cooling in the Recycler. The cooling is discussed in section 3.5.

## Proton Intensity at the Antiproton Target

Increasing the intensity of the proton beam at the antiproton target is
one technique to increase the antiproton flux. The higher intensity
proton beam is difficult to accommodate in the target station because of
the high peak energy deposition in the target, possible radiation damage
to the target station components, and the need to shield personnel and
the environment from excessive radiation. The TeV33 plan calls for an
increase in proton intensity by a factor of 1.9 through the slip
stacking technique described in section 3.1.3.

### Linac & Booster performance

Run II specifications call for the Booster to produce 5x10^12^ protons
per pulse with a maximum transverse emittance of 20π mm-mrad. The
maximum Booster intensity achieved to date is 4.4×10^12^, but the
Booster normally operates at lower intensity in order to achieve the
smaller emittance required by the Main Ring, where the effective
normalized acceptance is about 13π mm-mrad. The Booster is expected to
reach its goal of 5×10^12^ after a period of operation with the Main
Injector.

Improvements in Booster intensity beyond those already expected would be
useful in producing lower emittance proton beams in collisions and for
producing antiprotons at a higher rate. The Linac intensity is probably
not of primary importance because beam can be injected for multiple
turns using H^--^ ions. Most proposals for increased Booster performance
involve fairly expensive Linac energy upgrades or construction of a
larger aperture Booster to overcome space-charge effects. The more
modest plans to improve the Booster aperture for Run II may be more
effective than now anticipated. For the moment, it seems prudent not to
rely on anything but incremental improvements in Booster intensity.

We now plan that the Booster longitudinal emittance will be 0.1-0.2
eV-sec for Main Injector operation, but the Booster appears to be
capable of producing beams[^6] of 0.07 eV‑sec. Figure 1 shows the
measured longitudinal emittance versus intensity in the Booster. The
solid curve represents measurements taken with the old, 200-MeV Linac;
the dashed curve was taken after the upgrade to 400 MeV. The improvement
in longitudinal emittance is not the direct result of the 400 MeV
upgrade, but the result of suppressing a longitudinal coupled bunch
instability.

![](media/image1.emf){width="5.25in" height="3.8854166666666665in"}

### Main Injector performance

The Main Injector was designed, in part, to accept the large emittance
beams that one might expect on the basis of the extrapolation of the
solid curve in Figure 1. We are left with the rather pleasant situation
that the momentum spread of the Booster beam is considerably smaller
than the momentum acceptance of the Main Injector. We expect to take
advantage of the unexpected improvement in beam emittance by stacking
multiple pulses into the Main Injector.

The ultimate Main Injector intensity limitation is not known either
experimentally or theoretically. However, compared to the Main Ring, the
Main Injector has a much larger aperture and more attention has been
paid to achieving a low beam impedance. These features are expected to
allow substantially higher beam currents. One known limitation is the
amount of rf power available: the Main Injector can support 10^11^
particles per bunch without modification (the nominal design intensity
is 6×10^10^).

The Laslett tune shift for the Main Injector (space charge plus image
charge) has been calculated (see reference Error: Reference source not
found) to be -0.08 for 6×10^10^ protons per bunch, but the maximum
tolerable tune shift is unknown. The tune shift limit could possibly be
increased with an improved resonance compensation scheme in the Main
Injector, or the transverse or longitudinal emittances could be
increased to decrease the tune shift. The increased emittances would
have an effect on antiproton production (the effect of the longitudinal
emittance on bunch rotation is discussed in section 3.4), but the
emittance effects are weaker than the intensity effect.

Other collective effects, such as coherent beam instabilities, could
make it difficult for the Main Injector to operate at higher intensities
per bunch. We have extensive experience damping instabilities in the
Main Ring and we should be able to damp dipole instabilities. Other
instabilities, such as microwave type instabilities, could be more
difficult to handle.

### Slip stacking

The most promising strategy to increase the beam intensity seems to be
to stack Booster pulses in the Main Injector. The most promising
technique appears to be to coalesce bunches in the Main Injector with a
technique known as \"slip stacking\". Slip stacking is particularly
attractive because it requires no major hardware and could be
implemented at the *beginning* of Run II. A cartoon of the stacking
process is shown in Figure 2. A similar technique has been used at the
CERN Proton Synchrotron (PS).[^7]

According to the current plan for Run II, the 120 GeV protons used in
antiproton production will be obtained by transferring one Booster batch
into the Main Injector at 8 GeV and accelerating it to 120 GeV. The
proton intensity can be nearly doubled using \'slip stacking\' in the
Main Injector. This involves stacking two Booster batches end to end,
but with slightly differing momenta, into the Main Injector. The two
batches have different periods of revolution and \'slip\' relative to
each other azimuthally and finally overlap. When they overlap they are
captured using a single rf which is the average of the initial
frequencies associated with the two batches. Since the Booster and Main
Injector acceleration cycles are 66 ms and 1.5s respectively, the need
for an extra Booster cycle increases the total antiproton production
cycle by less than 0.1 sec and is a small effect on the total cycle
time. We have planned for the TeV33 production cycle to be 2.0 sec. This
cycle time is long enough to load the Main Injector with 6 Booster
batches, slip stack one batch (to be used for antiproton production) and
to extract the other 5 batches for fixed target experiments. Slip
stacking on all 6 batches (12 Booster cycles) is, in principle,
possible, but requires a longer Main Injector cycle.

![](media/image1.emf){width="4.802777777777778in"
height="3.140277777777778in"}

The following is a list of factors that determine the optimum momentum
separation between the two batches, initially and before they are
coalesced, and the rf voltages involved.

1)  A larger momentum separation reduces the time before the batches can
    be coalesced.

2)  A larger momentum separation requires a larger horizontal aperture.

3)  A smaller momentum separation just before the batches are coalesced
    leads to a smaller longitudinal emittance for the final beam
    (neglecting the effect of the second rf system).

4)  The rf buckets for the two batches get more distorted as the
    separatrices move closer together. The losses become fairly high if
    the separatrices overlap. So the beams should spend as little time
    with their separatrices close together as possible before they are
    coalesced.

The procedure used to find a way to obtain a final coalesced beam of
small emittance containing a reasonably large fraction of the initial
beams, consists of two steps. The first step, described in more detail
in section 3.1.3.1, consists of finding the approximate heights of the
initial buckets and that of the final bucket after coalescing, which
would result in a final beam of small longitudinal emittance with small
losses during coalescing. We ignore the distortion of the rf buckets due
to multiple frequencies in this step. In the next step, which is
described in section 3.1.3.2, we use these approximate heights of the
buckets as starting points and use rf simulations that include the
distortion of the buckets, to find an acceptable strategy of rf
manipulations to achieve our goals and to estimate the final emittance
of the beam and the losses during coalescing when we use the strategy.

In Section 3.1.3.3 we estimate the effects of beam loading and consider
ways to overcome its adverse effects. The effects are serious due to the
high intensity of the proton beam. Compensating for the effects is
complicated by the simultaneous presence of two beams and rf systems.

#### Optimum rf bucket heights

The optimum bucket heights before and after coalescing were found
assuming gaussian particle distributions for the initial beams. The
harmonic number is 588 and rf frequency is approximately 53 MHz
respectively. We assume a 0.15 eV-s longitudinal emittance for each of
the initial beams. This value is the measured emittance in the Main Ring
at injection. The emittance in the Main Injector is expected to be lower
due to improved Booster performance. The height of a bucket with area
0.15 eV-s is 6.15 MeV.

For given heights of the initial and final buckets after coalescing, the
area of the beam contour in the final bucket containing 95% of the
initial beam was found by integrating the part of the initial gaussian
distribution within the contour. The process was repeated for various
heights of the final bucket. Figure 3 shows the area corresponding to
various final bucket heights, for an initial bucket height of 6.2 MeV.
The height that gave the minimum area was chosen as optimum final bucket
height for the given initial bucket. The process was repeated for
various values of the initial bucket height. Figure 4 shows the optimum
final bucket heights and the heights of the corresponding beam contours
containing 95% of the beam for various initial bucket heights. Figure 5
shows the minimum area containing 95% of the beam for various initial
bucket heights.

![](media/image1.emf){width="4.972222222222222in" height="2.5in"}

Even if the injected beams are gaussian, the beam distributions before
coalescing are not expected to be gaussian if the two rf systems have
frequencies that are close together. The distortion of the distributions
due to the presence of the second rf was determined using a tracking
simulation and is described in section 3.1.3.2.

![](media/image1.emf){width="4.94375in" height="2.5965277777777778in"}

![](media/image1.emf){width="4.722222222222222in"
height="2.513888888888889in"}

#### Acceleration and Coalescing

The fractional difference in periods of revolution for the two batches
is given by

$\frac{DT}{T} = h\frac{Dp}{p}$, \[6\]

where  $Dp/p$ is the fractional momentum difference and *η* is the slip
factor. The slip factor is given by

$h = \frac{1}{g_{t}^{2}} - \frac{1}{g^{2}}$ \[ 7\]

For the MI, γt = 21.8, γ = 9.55 at injection , and η = 8.86×10^−3^. The
length of a Booster batch is 1.57 μs, and the period of revolution in
MI, *T* = 11.14 μs at injection. If the two batches are injected 46 MeV
apart and allowed to slip, they would overlap completely after half a
Booster cycle, i.e., 33 ms. Simulations show that for a bucket height of
10 MeV, the distortion of the particle distributions due to the presence
of the second rf is negligible. However, to obtain a small longitudinal
emittance of the final beam, the two beams have to be accelerated
towards each other before they are coalesced. The bucket height has to
be reduced so that the two beams can be brought close together. Were it
not for the effect of the second rf one could accelerate the beams and
reduce the bucket height very slowly to minimize particle loss. The
presence of the second rf encourages faster rf manipulations once the
beams are close to each other.

The rf curve program was determined by trial and error. After a few
trials, the variation of the rf voltage, frequency, and synchronous
phase angle depicted in Figure 6 was accepted as satisfactory. The two
beams are captured with a single rf while they are still accelerating.
The efficiency of acceleration and coalescing for a final longitudinal
emittance of 0.34 eV-s is 95%. Figure 7 shows the beam distributions
just before coalescing along with the initial and final buckets. The
dashed curve inside the final bucket is a contour containing 0.34 eV-s
of area.

![](media/image1.emf){width="4.790277777777778in"
height="2.876388888888889in"}

![](media/image1.emf){width="4.790277777777778in"
height="2.876388888888889in"}

![](media/image1.emf){width="4.790277777777778in"
height="2.876388888888889in"}

Figure 6. The frequency curve (a), the voltage curve (b), and the
resulting synchronous phase angle (c) that were used in the tracking
simulation.

![](media/image1.emf){width="5.25in" height="4.197916666666667in"}

#### Beam Loading

Because of the high beam current, the beam loading voltage in the rf
cavities is a serious concern. If the quality factor, *Q*, is high and
the bunch length is short, the cavity voltage *V*(*t*) following the
passage of a bunch of charge *q* is given by:

$V(t) = \frac{{qw}_{r}R}{Q}e^{- (\alpha + i)w_{r}t}$ \[8\]

where *R* is the cavity shunt impedance, *ω~r~* is the cavity resonant
frequency, and *α*=1/2*Q*. In the case that the bunches are spaced by
*τ*=2*π*/*ωr*, the voltage after the passage of n bunches is easily
found to be

$V(nt) = \frac{{qw}_{r}R}{Q}\frac{1 - e^{- n\text{pα}}}{1 - e^{- \text{pα}}}$
\[ 9\]

We can apply Eqn. \[ 9\] to estimate the beam loading voltage. As an
example, we consider the case where there are two batches of 84 bunches
each in the Main Injector and that the last 42 bunches of the first
batch and the first 42 bunches of the last batch overlap and are exactly
in phase. We ignore the difference in revolution frequencies of the two
batches and the difference between the resonance frequency of the cavity
and the revolution frequencies. Under these circumstances, one can use a
generalization of Eqn. \[ 9\] to estimate the beam loading voltage as
shown in Figure 8. The beam loading varies dramatically as the bunches
slip past each other. When the bunches from the two batches are in phase
(0° curve in Figure 8), the beam loading voltages add. But when the
bunches are out of phase (180° curve in Figure 8), the beam loading
voltages nearly cancel. The calculation is for a total of 18 cavities
with *R*/*Q*=100 Ω and *Q*=5000. The voltage increases when the beam
passes through the cavities. During the time that the two beams overlap
the voltage increases at twice the rate. When the beam is absent the
voltage decays at a rate determined by the time constant *α*.
Approximately 0.4 ms later the bunches are out of phase and the beam
voltage becomes very small.

![](media/image1.emf){width="5.25in" height="4.104166666666667in"}

This estimate of the beam loading voltage indicates that, if
uncompensated, the beam loading voltage (3 MV) would dwarf the rf
voltage (100 kV). We propose to control the beam loading voltage by:

1)  Tuning all cavities to the nominal 8 GeV frequency.

2)  Using a small number of cavities (2 or perhaps 4) to produce the
    required rf voltage and de-Qing the remaining cavities. One simple
    technique that appears to be moderately effective is to turn off the
    screen voltage to reduce the tube plate resistance. This technique
    is estimated to de-*Q* the cavities by a factor of 3.

3)  Feed-forward can be used on all the cavities. A resistive gap
    measures the wall current. This current, after being properly
    scaled, can be applied to the cavity drivers. Based on current Main
    Ring experience it is expected to achieve a factor of 10 reduction
    in the beam loading.

4)  Feedback can be used on all the cavities. A signal proportional to
    the gap voltage is amplified, inverted, and applied to the driver
    amplifier. This technique is expected to achieve a factor of 100
    reduction in beam loading voltage (based on previous experience in
    the Main Ring and results achieved elsewhere).

If all these efforts were successful, beam loading should be reduced to
a negligible value. Experiments in the Main Ring to measure the
suppression of the beam loading voltage and calculations of the
tolerance of the slip stacking process to large beam loading voltages
will determine the feasibility of slip stacking in the presence of
relatively high impedance rf cavities.

Other collective effects could be important. These include:

1)  Microwave instabilities at low voltages

2)  Feedback loop coupling

3)  Cavity tuning instabilities

We plan further theoretical work and are beginning experimental work
using the Main Ring beam.

##  Antiproton Target

The higher intensity proton beam expected at the antiproton target in
Run II and TeV33 results in an increasingly hostile environment at the
target station. Run II doubles the amount of beam targeted compared to
Run I; TeV33 results in a further doubling of the intensity. The
antiproton source target area is a high radiation area that contains a
number of high voltage devices.

### Beam sweeping

One issue that is fairly well understood is the allowable peak energy
deposition in the target. Figure 9 shows the relative yield of
antiprotons as a function of rms beam spot size. Also shown is the
energy density in a copper target as a function of rms beam spot size as
calculated by MARS10[^8] for 5×10^12^ protons per pulse[^9]. Estimates
of the peak density of energy deposition per pulse achieved to date are
in the range of 800 J/g. This is above the melting point of copper
(about 600 J/g), and close to the melting point of nickel (about 1000
J/g). In order to maintain peak energy deposition below present levels
the spot size would have to be increased to 0.3 mm for Run II (5×10^12^
protons in a 1.6 μs pulse) and the yield would be reduced 15-20%
compared to the nominal beam size of 0.10-0.15 mm. We plan instead to
sweep the beam on the target while synchronously sweeping the downstream
acceptance so that we can obtain an effective defocusing of the beam
with no reduction in yield.[^10] With slip stacking, the sweeping radius
will need to be increased to about 0.5 mm. The current design of the
beam sweeping system is compatible with a maximum sweeping radius of 0.5
mm, but uncertainties such as the breakdown voltage of the magnets in
the beam environment may limit the maximum sweeping radius to the design
value of 0.33 mm.

![](media/image1.emf){width="4.756944444444445in"
height="3.0458333333333334in"}

The beam sweeping scheme utilizes two upstream sweep magnets driven in
quadrature by a 625 kHz sinusoidal current wave-form to trace a circular
pattern on the target with the 120-GeV proton beam, followed by two
downstream magnets to redirect the 8 GeV antiprotons exiting the
collection lens parallel to the AP-2 transport line (see Figure 10). The
sweeping radius is much smaller than the 2 cm diameter of the lithium
collection lens and the aperture of AP-2. Two current-carrying plates,
roughly 3 cm wide, with an air gap of 3 cm, will provide the deflecting
magnetic field. An air gap is used since the beam is already transported
through air from upstream of the target to downstream of the pulsed
magnet. This system is currently an R&D project with expected completion
of a prototype kicker magnet and power supply in late 1997.

![](media/image1.emf){width="4.76875in" height="2.0902777777777777in"}

### Lithium lens mechanical issues

The collection lens[^11] focuses the antiprotons produced at the target.
The effect of the lens gradient on the yield is discussed in section
3.3.1. However, because the lithium lens is subject to the hostile
environment encountered in the target station, reliability issues are
discussed here. The current-carrying lithium portion has a diameter of
2 cm and a length of 15 cm. The lithium is encased in a cylindrical
water-cooled Ti-6Al-4V jacket. Fermilab lithium lenses of recent design
have survived over 7 million pulses at a lens gradient of about 750 T/m.
Increasing the field gradient of the lens increases the yield. But even
a small increase in repetitive stress in the Ti-6Al-4V cylinder (which
acts as both a cooling jacket and a pressure vessel) leads to a much
shorter fatigue life of the metal. Thus, operation at even 5% greater
field gradient has not yet proven possible beyond 1-2 million pulses.
Several improvements in the design of the lens are expected to further
improve reliability and field strength. Our ultimate goal is to improve
the current design to be reliable at 900 T/m. A number of improvements
to the mechanical design have been developed already. These improvements
include thicker end caps for the cooling jacket, a stronger beryllium
window, and improved handling and placement of welds during construction
of the cooling jacket. These improvements have been incorporated into
the latest lenses, which have been tested extensively in the target
vault at 750 T/m. These lenses will be tested at higher lens field
gradients in the future.

One particular critical parameter is the lens preload. During the
filling process, lithium is pumped under pressure into the evacuated
titanium vessel. The preload serves to insure that the lithium cylinder
maintains its shape at mid-pulse, when significant magnetic pinch forces
are present. Lower preload leads to lower operating stresses; a small
stress decrease on the titanium cooling jacket should result in a great
increase in the life of the lens for a given field gradient. A recent
analysis using ANSYS[^12] shows that it is possible to lower the preload
pressure in the lithium by at least 15%. The original lens design
preload ensured that the lithium would maintain its shape on the first
pulse when the lithium is at 20 ˚C. In its steady state, the lens is at
65 ˚C and the additional thermal stress on the titanium is approximately
the same as the stress due to the preload. Thus, at steady state almost
no preload is needed. If one could reduce the lithium preload from 2300
psi to 500 psi, then the lens would be able to operate at 1000 T/m
without deformation of the lithium cylinder. A lens with a preload of
1950 psi is currently in use in the target vault.

Lithium melts at the relatively low temperature of 180 ˚C. If liquid
lithium were used in the lens and circulated through a heat exchanger
with a pump, then water cooling of the Ti jacket would not be necessary.
This would allow for a much more robust Ti jacket that could withstand
much larger shock waves, allowing for larger current pulses and magnetic
field gradients. Some R&D has been done in this area in the past.[^13]
Gradients of 1200 T/m might be possible with this technique.

### Radiation damage issues

Information on the maximum allowable radiation dose of insulating
materials is sketchy, but we already typically exceed the high end of
the ranges specified. Because of the large uncertainty in using
published data, the final determination on the acceptability of
materials for use in our target station will necessarily be based on our
own experience. Recently, a failure of the torlon[^14] insulating
material in the \"pulsed magnet\" was experienced. This material had
survived a 4-year beam exposure at current intensities. Higher beam
intensities will result in accelerated failure rates or will further
restrict the range of materials that can be used in the target vault.

We have seen a large number of lithium lens failures over the years.
Few, if any, of these failures were the result of radiation damage.
However, we have realized that the lithium lens is susceptible to a type
of radiation damage that is unique to the lithium lens. Gaseous products
from the reaction ![](media/image1.emf){width="1.0555555555555556in"
height="0.16666666666666666in"} are expected to build up over time,
possibly affecting the operation of the lens.[^15] Swelling of the lens
is expected, due to the pressure of the contained gas, potentially
limiting the lifetime of the lens under an intense beam environment. To
avoid this problem, we have identified a source of 99% isotopically pure
^7^Li and are currently building all lenses with this material.

While there are no problems that have been identified as being
insurmountable, the susceptibility of target station components to
radiation damage remains a concern based on our past experience. In our
plans for TeV33, we have chosen to achieve some of the required increase
in antiproton flux by increasing the antiproton acceptance. This
strategy helps reduce the potential for radiation damage (compared to
the scenario where all the increase is achieved by targeting more
protons), but it does require higher performance from the stochastic
cooling systems.

### Antiproton target radiation shielding and environmental issues

The radiation shielding and environmental issues are fairly well
understood. Higher beam intensities will probably require modifications
to the shielding or to the accessibility of the target hall. It is
likely that improved air handling procedures (to reduce airborne
contamination) will also be required. Some of the major concerns are
listed below.

1)  The target vault is classified as a high radiation area (\<100
    mrem/hr). Some additional concrete may be required to maintain this
    classification (a standard rule of thumb is that 2.3 ft of concrete
    reduces the dose rate by a factor of 10).[^16]

2)  The AP0 building is classified as a radiation area (\<10 mrem/hr).
    Some additional concrete shielding may be required to maintain this
    classification. As an alternative, some critical areas could be
    enclosed by ropes or fences.

3)  The lithium lens is cooled by a closed loop water system. The levels
    of tritium may require additional measures to comply with
    established safety standards.

4)  Stairwells and penetrations have not been considered in detail, but
    may need some additional shielding measures.

5)  Some additional soil may be required over the beam lines.

6)  The ground water contamination potential has been considered with
    the "Concentration Model."[^17] The result of the calculation based
    on this model is that we can target 10^22^ protons per year in the
    existing antiproton target station. This is well within the TeV33
    parameters of 10^7^ pulses of 10^12^ protons per pulse.

7)  The limit on airborne emissions is 100 Ci per year. The
    measured[^18] target station emissions indicate that 10^20^ protons
    per year will saturate the 100 Ci limit.

### Antiproton Target Conclusions

In summary, our current understanding is that targeting twice the Main
Injector intensity (*i.e.*, 1013 per batch) is technically feasible
provided that the beam sweeping system operates reliably.

## Antiproton Acceptance

We plan to increase the antiproton yield by increasing the gradient at
which the lithium lens can reliably operate and by increasing both the
transverse and longitudinal acceptance in the Debuncher. Increasing the
acceptance has the advantage that it makes no new requirements on the
proton beam and target station components, but it has the disadvantage
that a larger phase space area must be cooled.

### Lithium Lens Gradient

Measurements of yield vs. gradient (Figure 11) show that the yield
increases approximately linearly over the range measured. These yield
data were obtained with the current acceptance, which has been measured
to be about 16π×16π mm-mrad with the AP-2 line matched to beta functions
*β~x~*=2.5 m and *β~y~*=1.5 m at the downstream focal point of the
lithium lens. We have established a goal of 900 T/m for the improved
lithium lens design (as described in section 3.2.2), which would result
in a 17% increase in yield if everything else remained the same.
However, if the acceptance is increased to 32π×32π mm-mrad and the AP-2
match is reoptimized to *β~x~*=* β~y~*=4.5 m, we calculate a 28%
increase in yield for the same increase in lens current using the Monte
Carlo computer program MCLENS[^19]. We plan to achieve the latter
increase in yield with improvements to the AP-2 line and the Debuncher
Ring.

![](media/image1.emf){width="5.0in" height="4.0in"}

### Transverse aperture increase to 32π x 32π

The transverse phase space area of antiprotons accepted into the
Debuncher during stacking has been measured to be about 16π x 16π (see
Figure 12).  This is substantially smaller than the measured admittance
of the Debuncher of 29π x 26π, and we have suspect that the AP-2 beam
transport line may be limiting the acceptance. The apertures in the AP-2
line are large enough for a 32π x 32π beam but the acceptance could be
limited by misalignment of the apertures, beam steering, or focusing
errors. These types of errors are coupled. We suspect steering errors
are partially compensated by changing quadrupole gradients (which
results in focusing errors). Recently, the quadrupoles in the upstream
end of AP-2 were surveyed and realigned in an effort to minimize
quadrupole steering, but no increase in yield was observed. Work is
presently underway to develop an accurate beam line model and improve
the beam line tune, steering, and matching into the Debuncher.

![](media/image1.emf){width="6.010416666666667in"
height="3.5861111111111112in"}![](media/image1.emf){width="6.010416666666667in"
height="3.5861111111111112in"}

### Matching the Beam Phase Space to the Debuncher Admittance

Figure 13 shows a Monte Carlo simulation (see Ref. Error: Reference
source not found) of the antiproton phase space density in the
horizontal dimension at the downstream focus of the lithium lens. A fit
to the measured yield vs. target-to-lens distance indicates that the
aspect ratio of the phase space ellipse at the lens which is accepted
into the Debuncher is now β=2.5 m horizontally and β=1.5 m
vertically.[^20] However, for apertures up to about 32π × 32π and for a
lens gradient of 750 T/m the optimum phase space ellipse (maximum yield)
is β=4.5 m in both dimensions. If the 32π × 32π mm-mrad acceptance is
achieved and if AP-2 is optimally matched to the Debuncher, the yield is
predicted to increase by a factor of 2.59 at the current lens gradient
of 750 T/m. A major uncertainty in this extrapolation is the assumption
that the current acceptance is an ellipse centered in phase space. (It
might not be centered.) Figure 14 shows the beam envelope for 32π x 32π
beam, initial βx,y=4.5m, and ∆p/p=4% in an AP-2 model. This model
(assuming perfect steering) shows that the beam easily fits in the AP-2
beam pipe, except possibly at the injection septum. The narrowest
apertures are horizontally at the upstream end of the septum (4.2 × beam
sigma), vertically at the downstream end of the septum (6.5 × beam
sigma), and horizontally at the upstream end of the injection kicker
(6.7 × beam sigma).

![](media/image1.emf){width="5.525in" height="3.825in"}

### Debuncher Aperture![](media/image1.emf){width="5.525in" height="3.6125in"}

The Debuncher transverse aperture has been measured to be as large as
29π x 26π mm-mrad at the central momentum[^21] although it generally
deteriorates with time to something more like 25π x 23π mm-mrad.
Aperture scans indicate aperture limitations at the stochastic cooling
pickup and kicker arrays. The Debuncher stochastic cooling upgrade for
Collider Run II will increase the aperture of these devices to a nominal
40π mm-mrad. Previous measurements[^22] indicated that even with the
cooling tanks removed, the apertures were still limited to about
29π mm-mrad. The Debuncher model indicates (ignoring steering ,
focusing, and alignment errors) that without the cooling tanks, the
aperture is greater than 40π everywhere except at the extraction kicker
(Ay=36π) and the injection septum (Ay=29π). A stronger injection kicker
would allow for a greater separation between the circulating beam and
injected beam at the downstream end of the septum. Then, bumping the
closed orbit away from the septum would allow for a vertical aperture
larger than 40π at that point.  The issue of whether or not new kickers
are required needs more careful study. A further possible cause of the
limited aperture is quadrupole misalignment, which causes beam steering,
which cannot be corrected because there are very few correction elements
in the bend sections of the Debuncher. Figure 15 summarizes a recent
Debuncher quadrupole survey. These alignment errors are large compared
to the original survey tolerances, but their effect on the beam has not
been studied in any detail. These elements will be realigned during the
long shutdown for Main Injector installation. If this does not open up
the aperture, then a program of installing motorized quadrupole mounts,
dipole shunts, and possibly trim elements in conjunction with beam-based
alignment techniques may be appropriate to obtain an acceptance of 32π ×
32π mm-mrad. Beam studies with the goal of defining the program to
increase the acceptance of the AP-2 line and Debuncher are currently
underway.

### Momentum aperture increase![](media/image1.emf){width="5.525in" height="3.4in"}

By decreasing the slip factor η in the Debuncher from its design value
of .006 to .003 at injection time, the momentum aperture of the
Debuncher can be increased by 17% at injection. This is accomplished by
reducing the dispersion in the bend sections of the Debuncher. Figure 16
shows the dispersion function in the Debuncher for 2 different values of
η. The η change is implemented in the lattice with a series of
overlapping local dispersion bumps in the high dispersion regions which
reduce the dispersion there. After bunch rotation and adiabatic
debunching, which occurs in the first 60 msec of the beam cycle, η will
be ramped to .009 to improve stochastic cooling. A ramp from .006 to
.009 has already been successfully tested in the Debuncher,[^23]
although some work is still required on the quadrupole power supply
regulation circuits and quadrupole shunt circuits to improve tune
control and prevent crossing resonance lines during a fast ramp. This
ramp does not change the beta functions anywhere in the ring by more
than 5%.

The current momentum aperture of the AP-2 left bend is at least 4%. The
limitation is the dispersion in the left bend, where the beam is scraped
from nominally ∆p/p=10% to ∆p/p=4%. The question of whether the AP-2
beam line will need to be modified to accommodate a 17% larger momentum
spread needs to be investigated experimentally.

![](media/image1.emf){width="4.95in" height="3.15in"}

## Antiproton Bunch Rotation

Bunch rotation is a crucial emittance preservation technique that is
used in the production of antiprotons.[^24] The effectiveness of the
bunch rotation depends on the longitudinal proton emittance in the Main
Injector. The process has been simulated using the existing 5 MV rf
system and an initial antiproton momentum spread of ±2.4%. The final
momentum spread of the antiprotons versus the initial proton
longitudinal emittance is shown in Figure 17. For larger longitudinal
emittances the Debuncher momentum spread depends linearly on the initial
time spread in the proton beam. For very small emittance,
non-linearities in the bunch rotation process are responsible for the
final momentum spread. In this case, the momentum spread can be reduced
by using a higher harmonic rf system in the Debuncher. Figure 17 also
shows the momentum spread of the Debuncher beam measured in Run Ib and
the bunch lengths expected in the Main Injector for longitudinal
emittances of 0.15 eV-sec (nominal Main Injector emittance) and 0.5
eV-sec (slip stacking goal).

![](media/image1.emf){width="5.597916666666666in"
height="4.353472222222222in"}

## Figure 17. The Debuncher momentum spread after bunch rotation versus the initial bunch length. The width is determined from the full width of the Schottky signal measured at 10 dB down from the peak (FW@10 dB). The Run Ib operating point (measured) is shown as well as a calculated curve.Antiproton Cooling Overview

For Run II, an upgrade of the existing 2-4 GHz cooling system in the
Debuncher and an upgrade of the 1-2 GHz stack tail cooling system to a
2-4 GHz cooling system will support stacking rates of 2x1011/hour for
Run II. . We plan to use stochastic cooling in the Recycler initially as
described in the Recycler Design Handbook (see ref. Error: Reference
source not found).

An additional factor of 4 increase in antiproton flux can be
accommodated by additional upgrades in the Debuncher and the Accumulator
to 4-8 GHz. The Debuncher systems would achieve the factor of 4 increase
in cooling rate by doubling the bandwidth (a factor of 2) and by
reducing the mixing factor (the second factor of 2). The stack tail
system in the Accumulator would benefit from an increased bandwidth (a
factor of 2), but would have half its cooling load assumed by the
Recycler Ring. An important ingredient of this effort is an R&D effort
to produce high sensitivity 4-8 GHz pickups and kickers.

In the Recycler an electron cooling system will be used to
longitudinally stack the antiprotons for TeV33. Electron cooling
promises much higher cooling rates than would be possible with the
multi-GHz stochastic cooling systems used in the Debuncher and
Accumulator rings. Stochastic cooling rates are inversely proportional
to the number of particles being cooled while electron cooling is
independent of the number of particles to be cooled. The high beam
intensities in the Recycler favor electron cooling by a wide margin. The
rf manipulations and beam transfers are not described in this report but
are essentially identical to those described in the Recycler Design
Report (see ref. Error: Reference source not found) and the Tevatron I
Design Report (see Ref. Error: Reference source not found).

The cooling calculations assume that 6×108 antiprotons are delivered at
2 sec intervals to the Debuncher in 1 nsec bunches with a full (100%)
momentum ∆p/p=4.8% and a transverse emittance (100%) in each plane of
ε=32 π mm-mrad. The beam is debunched to a momentum spread of 0.3% and
cooled to 0.2%. The beam is cooled by about a factor of 4 transversely
to 6 π mm-mrad (95%). Table VI summarizes the beam parameters in the
Debuncher.

The beam is transferred to the accumulator ring where it is stacked
longitudinally and cooled transversely by a factor of 7. Every 60 sec
the most dense 8 eV-sec of longitudinal phase space is transferred to
the Recycler. The frequency of transfers and the size of the
longitudinal emittance depend on the performance of the cooling systems
in the Recycler and the Accumulator, and will probably be adjusted in
the future. The Accumulator beam parameters are shown in Table VII.

The beam is stacked longitudinally in the Recycler with electron
cooling. The nominal stacking cycle is 60 sec, but this time can be
varied according to the strength of the electron cooling and the ability
of the Accumulator to store beam. The projected Recycler beam emittance
shown in Table VII are upper limits, but are anyway smaller than the
parameters required for the Tevatron beam parameters (Table II). The
smaller emittance, if preserved to low beta, will produce a higher
luminosity than those shown in the table. On the other hand, the
emittance could be intentionally diluted to avoid instabilities if
necessary.

## Transverse Cooling in the Debuncher

The beam is cooled transversely in the Debuncher by a conventional
stochastic cooling system. The process has been simulated by a computer
program that is similar to the one used to design the present Debuncher
cooling system. This computer program comprehensively describes the
stochastic cooling process by solving a Fokker-Planck equation for the
beam distribution as a function of betatron amplitude. The solution
includes the effects of good and bad mixing, thermal noise, and signal
suppression. The initial and final beam distribution functions are shown
in Figure 18. Figure 18 also illustrates the distribution that would be
obtained with half the cooling power.

![](media/image1.emf){width="5.945833333333334in"
height="5.945833333333334in"}

Figure 18. Debuncher transverse cooling simulation.

Since broad band microwave power is expensive, it is important that the
power level be kept as low as possible. One of the largest uncertainties
in the determination of the required power level is the coupling of the
kicker power to the beam. For the purposes of the calculation, a model
of an array of strip-line pickups and kickers has been used. The
structure actually used will be determined by the 4-8 GHz pickup and
kicker R&D program that is described in 3.9.1. It has been assumed that
the pickups and kickers are of the \"plunging\" type and that the
aperture is continuously adjusted to maximize the cooling rate while
minimizing the loss of particles from scraping on the pickups and
kickers. If the impedance is sufficiently high, it may not be necessary
to utilize plunging pickups. The parameters of the beam and the cooling
system are given in Table IX.

The cooling system described above provides about 12× the cooling rate
as the system described in the Tevatron I Project Design Report. The
major reasons for the improved capability are given in Table X.

## Debuncher Momentum Cooling

The Debuncher momentum cooling will be similarly improved by using the
4-8 GHz band. The momentum cooling uses the same pickups and kickers
that will be installed for the transverse cooling system. It only
requires additional electronics: low noise amplifiers, medium level
electronics including a notch filter, and additional traveling-wave tube
(TWT) power amplifiers. None of these components present additional
challenges beyond those described in the section on Debuncher Transverse
Cooling. The performance requirements for momentum cooling are lower
(cooling by less than a factor of 2) and the signal to noise ratio is
better because the longitudinal signal is stronger than the transverse
signal and because the momentum system uses pickups from both the
horizontal and vertical systems.

The performance of the upgraded simulation is illustrated in Figure 19.
The cooling system parameters used in the simulation are given in Table
XI.

![](media/image1.emf){width="5.495833333333334in"
height="5.495833333333334in"}

## Longitudinal Stacking in the Accumulator

The TeV33 longitudinal stacking system is similar in concept to the
current stack tail system. A major design consideration is the amount of
cooling done. A model, due to van der Meer,[^25] relates the rate of
accumulation of flux (Φ) to other system parameters

$F\text{=-}\frac{\text{TW}^{2}|h|E_{d}}{pL}$ \[ 10\]

where *W* is the cooling system bandwidth, *T* is the revolution period
of the storage ring, $h = 1/g_{t}^{2} - 1/g^{2}$, *p* is the nominal
beam momentum, *Λ* is a numerical factor *Λ*=ln2, and *E~d~* is the
logarithmic slope of the density distribution *Ψ(E)*:

$\frac{1}{E_{d}} = \frac{1}{Y}\left| \frac{dY}{\text{dE}} \right|$.
\[11\]

With the other parameters fixed, an arbitrarily large antiproton flux
can be accommodated in the accumulator by tailoring the logarithmic
slope of the gain to arbitrarily small values. The consequence of large
values of *E~d~* is that less cooling is done in the Accumulator. The
cooling deficit (compared to the Run I and Run II designs) will be
absorbed by the electron cooling system that will be available in the
Recycler as discussed in section 3.10.

The system contains pickups in the high dispersion regions of the
Accumulator, correlator notch filters, and kickers in zero dispersion
region. The stacking process has been simulated by a computer program
that is similar to the one used to design the Debuncher momentum cooling
system and the Stack Tail momentum cooling system in the Accumulator. A
tutorial description of this process was given in the Tevatron Project I
Design Report (see Ref. Error: Reference source not found). The computer
program comprehensively describes the stochastic stacking process by
solving a Fokker-Planck equation for the distribution function. A block
diagram of the system is shown in Figure 20.

![](media/image1.emf){width="5.538888888888889in"
height="2.2944444444444443in"}

Figure 20. Block diagram of the TeV33 stack tail cooling system.

It is planned that beam would be stacked in the Accumulator for 60 sec.
After this interval, the beam would be extracted and transferred to the
Recycler Ring. Figure 21 shows the antiproton energy density function at
0 sec (immediately following a transfer to the Recycler) and after 20
and 40 seconds of stacking. The parameters of the beam and the cooling
system are given in Table XII.

![](media/image1.emf){width="5.945833333333334in"
height="5.945833333333334in"}

## Stochastic Cooling Technology

There are two technologies that are crucial to the success of these
upgrades. The first technology is the 4-8 GHz pickup and kicker loops.
The second technology is signal transmission which we intend to base on
a laser modulation/demodulation scheme. Other technologies including
microwave components and amplifiers are commercially available for the
4-8 GHz band.

### 4-8 Ghz Pickups

Present stochastic cooling electrode technology is based on octave
bandwidth planar loops. The gain of these falls off quickly at high
frequencies. For example, a 100Ω characteristic strip-line loop has a
measured sensitivity of 3Ω at 2-4 Ghz. A similar 100Ω loop has a
measured sensitivity of 0.75Ω at 4-8 Ghz.

Many of these low sensitivity broad band loops are combined in binary
arrays to form a broad band array. There are a number of problems with
binary combiner boards. Even at 2-4 Ghz, they may have large insertion
losses and microwave modes. Also, microstrip combiner boards require
relatively large areas to avoid undesired coupling between circuit
elements. These large circuit boards are difficult to cool efficiently
and require large vacuum tanks. Although stripline is electrically
superior to microstrip it is much more difficult to deal with
mechanically

However, 4-8 GHz slots have been measured to be more sensitive than 3-D
loops or planar loops. A single waveguide slot has been calculated to
have an effective impedance of 37 Ω over a bandwidth of 0.7 GHz at 6 Ghz
as shown in Figure 22. The slot couples electromagnetic energy from the
wakefield of the beam into the output waveguides as shown in Figure 23.
The energy of a number of slots can be added together constructively if
the phase velocity of the wave in the outer waveguides matches the
velocity of the beam. A waveguide without any slots has a phase velocity
faster than the velocity of light. The slots act to "slow down" the
phase wave in the outer waveguides. The number of slots per length will
determine the phase velocity of the wave in the outer waveguides.
Research has been started to calculate what line density of slots is
needed to match phase velocities. This type of structure is commonly
known as a slow wave Floquet structure. This structure is also similar
in concept to the 8-10 GHz CERN design of the slow wave ridged pickup
array used for bunched beam cooling.

![](media/image1.emf){width="5.99375in" height="3.1368055555555556in"}

Figure 22. Coupling impedance versus frequency for a single slot of
varying width.

![](media/image1.emf){width="5.9944444444444445in"
height="3.3784722222222223in"}

Figure 23. A schematic drawing of a 4-8 Ghz slot array.Each slot array
will be narrow band (\<1.0 GHz) but tuned to a separate center
frequency. The arrays will be summed together to form a broad band
response as shown in Figure 24. The use of narrow band channels
simplifies the design of the equalizer, which is necessary to obtain the
optimum gain at all frequencies in the cooling band. To design the
equalizer, each channel would have its own gain and phase knob.

Just as in the 2-4 Ghz design, the pickup arrays will be cooled to 20
°K. The 4-8 Ghz slot arrays could be designed to plunge and will
probably have a much smaller cross-sectional area than a 2-4 Ghz array
by about a factor of three. This would allow the vacuum tanks to be
smaller and the cryogenics to be more efficient. Also, there would be
only one termination per array instead of one termination per loop as in
the 2-4 Ghz design. This single termination could be placed outside the
tank. For the kickers, placing the termination outside the vacuum tank
would eliminate the need for water cooling inside the kicker tank.

### 4-8 GHz Signal Transmission

It is impossible to transmit high frequency microwave signals over long
distances with single-mode coaxial cable. As the frequency increases,
the cable diameter must decrease to avoid dispersion from higher order
modes. At 4 GHz the signal loss in crossing the Accumulator ring is
about 40 dB. At 8 GHz it is 80 dB. Light can be transmitted long
distances with low attenuation, so a laser modulated at microwave
frequencies is an attractive choice for transmitting microwave signals.
We already have considerable experience in transmitting signal in
optical fibers (for use in microwave notch filters). However, the
propagation velocity in optical fibers is too slow to transmit signals
across a ring; this transmission must be made in vacuum or air. A
disadvantage of using the optical technique is the conversion loss in
converting the microwave signal to an optical modulation and the
consequent reduction in dynamic range.

Development of an optical microwave link that can propagate a signal in
the air has progressed to the bench prototype. Procurement of optical
telescopes, beam expanders, and microposition equipment has been
completed. The test setup shown in Figure 25 was mounted on a 3 by 6
foot light table. The network analyzer is used in averaging mode. The
system is microphonic with ±1 dB of vibration jitter. Some of the
important parameters are: the DFB (Distributed FeedBack) laser emits
3.24 mW at 1310 nm, 2.92 mW is transmitted through the second telescope
and focused on the second lens/beam collimator. With careful tuning we
were able to couple 2.1 mW of this optical power into the single mode
fiber. A careful optical/microwave calibration was performed on the
system. The microwave calibration was taken at 5 GHz with 16 averages
and 3% bandwidth smoothing. The dials on the micropositioner are in
microns with one full turn of the coarse adjustments equal to 500
microns. The nanotrak system has piezo motion control of 30 microns in
*x* and *y* planes on the micropositioner.

![](media/image1.emf){width="5.997916666666667in"
height="5.381944444444445in"}

The transmission (amplitude and phase) obtained with this setup across
the 0.5-10.5 GHz band is shown in Figure 26. The stability of the
optical link is critical to successful performance of the stochastic
cooling. The variations in the optical power and microwave signal with
respect to perturbations caused by drifts in alignment, temperature, and
other time dependent phenomena are being studied in bench tests. A field
test of the optical link will occur this year.

![](media/image1.emf){width="5.15in" height="3.275in"}

Figure 26. Microwave transmission in the 0.5-10.5 GHz band. The top
trace is obtained with a fiber to fiber connect ion. The bottom traces
are free space connections obtained over a 24 hour period with position
tracking. The vertical scale is 2 dB per division.

## Antiproton Electron Cooling

Electron cooling of the 8 GeV antiprotons in the Recycler provides an
attractive solution to the problem of cooling large stacks of
antiprotons. The proposed electron cooling system is similar to the one
described in the IUCF proposal to cool 12 GeV protons in the SSC Medium
Energy Booster.[^26] Although electron cooling has now become a routine
tool in many laboratories, its use has been restricted to lower energy
accelerators (\<500 MeV/nucleon). An R&D program is currently underway
at Fermilab to extend electron cooling technology to the GeV range.

Since the electron energy has to be approximately 4 MeV, the use of
traditional electron cooling technology with a Cockcroft-Walton (C-W)
power supply and a magnetically-confined electron beam is impractical.
In fact, compact commercial C-W voltage generators are limited to about
1 MV, about a factor of 3 times higher than the IUCF, CELSIUS, and GSI
electron cooling systems. In the medium energy regime the Pelletron[^27]
(Van de Graaff generator type) electrostatic accelerators, having an
operating range of about 2-20 MV, would replace the C-W generators. In
this regime the continuous longitudinal magnetic field is no longer
strictly necessary for focusing although longitudinal focussing still
has advantages. The beam focusing requirements are discussed in more
detail below.

### Electron Cooling Rate

Electron cooling is accomplished by merging an antiproton beam with a
co-moving electron beam in a short region of a storage ring. Ions moving
in the accompanying electron beam rest frame lose energy by Coulomb
interactions. The transverse
($1/t_{\hat{}} = 1/e_{n}{de}_{n}/\text{dt}$) and longitudinal
($1/t_{||} = 1/sds/\text{dt}$) electron cooling times of an ion beam
with transverse normalized emittance ε~n~ and relative momentum spread
σ=∆p/p are well approximated as

$t_{\hat{}»\frac{A}{Z^{2}}\frac{g^{2}}{4\text{ph}L}\frac{M}{m}\left( \frac{\text{kT}_{e}}{\text{mc}^{2}} \right)^{\frac{3}{2}}\frac{1}{\text{nr}_{e}^{2}c}}$
\[12\]

$t_{||}»\frac{A}{Z^{2}}\frac{g^{2}}{4\text{ph}L}\frac{M}{m}\left( \frac{\text{kT}_{e}}{\text{mc}^{2}} \right)^{\frac{3}{2}}\frac{u_{{}_{||}}}{\text{nr}_{e}^{2}c}\left\lbrack 1 + u_{{}_{||}}^{2} \right\rbrack\text{.}$
\[13\]

Eqn. \[12\] and Eqn. \[13\] include the effects of the "flattened"
electron velocity distribution. *A* and *Z* are the ion atomic number
and charge state; *γ* is the usual relativistic parameter; *Λ* is the
Coulomb logarithm (≈10); *k* is the Boltzmann constant; *m* and *M* are
the electron and proton masses; *c* is the speed of light; *n* is the
laboratory frame electron beam density; and *r~e~* the classical
electron radius; all other symbols are defined in Table XIII. The
quantities *u~⊥~* and *u~~* are the dimensionless transverse and
longitudinal velocities normalized to the electron beam rms transverse
velocity:

\[Ed note: Explain flattened\]

$u_{\hat{} = \sqrt{\frac{\text{βge}_{n}\text{mc}^{2}}{\beta_{I}\text{kT}_{e}}}\text{;   }u_{||} = \beta\frac{Dp}{p}\sqrt{\frac{\text{mc}^{2}}{\text{kT}_{e}}}}$
\[14\]

where *β* is the usual relativistic parameter and *β~Ι~* is the ion
beta-function in the cooling region. Electron cooling is most effective
when *u* ≤1; in this regime the first addends in the square brackets of
Eqn. \[12\] and Eqn. \[13\] dominate. Note that the longitudinal cooling
time (2) is roughly proportional to the ion beam momentum spread. This
approximation is only valid for a "flattened" electron velocity
distribution and when the longitudinal ion velocity in the moving frame
is greater than the electron beam longitudinal rms velocity spread.

### Electron Beam Optics

In evaluating Eqn. \[12\] and Eqn. \[13\] we have taken the electron
beam temperature to be 0.2 eV. This number is somewhat arbitrary but is
defined by the following considerations. Let's assume that a standard
tungsten dispenser cathode of 7 mm diameter is used, which gives the
intrinsic electron beam temperature at the cathode of 0.11 eV, or 1050
˚C. If we now assume that the electron beam is accelerated and expanded
to a diameter of 2 cm, the electron beam temperature becomes 13 meV, or
12 μrad transverse one-dimensional rms angular spread. However, if there
exists a misalignment between the electron and antiproton beam
trajectories of this order or greater, or if the electron beam optics is
not adjusted so as to make the beam parallel to this order or greater,
there will be effective temperatures in excess of the cathode
temperature. We feel confident that the two beams can be aligned to
better than approximately 40 μrad, and that the angular spread due to
the electron beam space charge can be kept below this value, thus giving
us a conservative estimate for the electron beam temperature of 0.2 eV.
This is not a trivial goal when one considers that the earth's magnetic
field alone would cause an electron beam deflection of hundreds of mrad
over the 66 m cooling length.

The electron beam is generated in the 4.3 MV terminal of a Pelletron
accelerator. Two solenoids at the beginning of the cooling straight
produce the required beam size and convergence. Following the cooling
straight section, the beam is then transported back to the 4.3 MV
terminal and collected. Figure 27 illustrates the layout of the 66 m
long cooling section. The simplest electron focusing channel in the
cooling region is a series of very weak solenoids with focal length
*f~sol~* spaced by the distance *L~sol~*. Figure 28 shows a typical 2-m
long module incorporating the electron beam optics, alignment, vacuum,
and diagnostics system in the cooling region. Each solenoid in this
module provides just enough focusing to locally correct the electron
beam expansion due to its space charge. Consequently, this section is
optically equivalent to a drift: to first order, a particle entering off
axis, but with no transverse momentum, will leave the section with the
same transverse position and with no transverse momentum. Assuming that
the electron beam current density is uniform one can obtain a simple
relation, connecting the solenoid focal length, *f~sol~* , with other
parameters:

$f_{\text{sol}} = \frac{r_{b}^{2}}{2\text{KL}_{\text{sol}}};K = \frac{I_{e}}{\beta^{3}g^{3}I_{o}},$
\[15\]

where I~0~≈17 kA. Thus, by choosing the 2 m long module, this expression
gives the solenoid focal length *f~sol~* ≈184 m and the maximum beam
divergence in this case is *r~b~*/ (2 *f~sol~*) ≈ 27 μrad.

![](media/image1.emf){width="5.99375in" height="2.4159722222222224in"}

Such a seemingly simple focusing system is not without drawbacks,
however. For example, if the space charge of the electron beam is
compensated by a factor of *γ^−2^* (≈1% ) due to residual gas ions, the
electron beam will converge in an uncontrollable way (the solenoids can
only focus the beam). In addition, this focusing scheme is susceptible
to various electron beam instabilities, which might turn out to be
detrimental for the high current electron beam transport through the
cooling section. One might also argue that in the case of weak periodic
focusing the Coulomb logarithm *Λ* could be a factor of 2 lower than the
value used for evaluation of the cooling times in the previous section.
This effect is especially obvious for the "slow" (compared to the
time-of-flight through the cooling section) collisions between the
electron and antiproton. In this regime the electron is "smeared" by the
periodic focusing, thus increasing the minimum impact parameter and,
consequently, reducing the Coulomb logarithm. This effect is similar to
the reduction of *Λ* for magnetized cooling. All these phenomena as well
as alternate focusing schemes (*e.g.*, a solenoidal field) are currently
under investigation.

### Technical Considerations

In addition to the electron beam optics design in the cooling region
there are two more principal technical problems which need to be
addressed in order to build a 4 MeV electron beam cooling system: (1)
the electron beam energy regulation and (2) the efficient and stable
recirculation of an electron beam. Below, each of these problems is
briefly discussed.

####  Electron beam energy regulation

Pelletrons are commercially available with the short-term stability
(ripple) of ±20 ppm (parts per million) using a generating voltmeter
(GVM) for feedback[^28]. This small amount of ripple would have a
negligible effect upon the cooling time. Long-term regulation of a
Pelletron, however, is substantially worse.[^29] Using the standard
corona probe feedback system with an error signal from the GVM, a
non-temperature-stabilized Pelletron will drift by as much as 800
ppm/hr, and a thermally-insulated Pelletron by about 75-100 ppm/hr.

Thus, the standard short-term Pelletron regulation should be more than
adequate for use in the Recycler. The long-term stability can be greatly
enhanced by using an external energy measurement system, for example, a
BPM system in a region of high dispersion in the electron beam line.

#### High Collection Efficiencies and Stable Recirculation

Achieving stable recirculation of a nonmagnetically confined electron
beam is the most significant technical obstacle. Since a Pelletron
accelerator will only source on the order of a few hundred μA, the
system must have collection efficiencies of at least 99.99% for
operation of a several ampere electron beam. With magnetically-confined
beams, such collection efficiencies are easily achieved. In fact, at
IUCF, collection efficiencies of 99.9999% have been demonstrated.[^30]
Our goal is to demonstrate reliable, high efficiency, dc recirculation
of a 2 A electron beam using the 2 MV Pelletron accelerator at NEC. Our
R&D plan to achieve stable recirculation has been described by
Nagaitsev.[^31]

### Alternative approaches

Much of the R&D has concentrated on the Pelletron approach to electron
cooling. However, a number of alternatives are also being explored. Many
potential alternatives offer substantial cost advantages of the
Pelletron. However, it is premature to speculate which if any of these
alternatives will be feasible.

## Transfers Between the Accumulator and Recycler

The current plan for Run II calls for the beam to be transferred from
the Accumulator to the Recycler through the existing AP3-AP1 line, a
remnant of the Main Ring (from F14 to F0), through the Main Injector to
the Recycler. In order to accomplish this operation, the AP1 line, which
also delivers Main Injector protons to the antiproton target, must be
changed from its nominal 120 GeV energy to 8 GeV. It has proven
difficult and time consuming in past collider runs to change between 8
and 120 GeV operation in AP1. Even if the tuning problem could be
overcome, it would still be necessary to interrupt stacking for a
sufficient period of time to change the magnetic fields and establish
the timing signals to accomplish the transfer. Å simpler and probably
more effective solution is the construction of a dedicated transfer line
between the Accumulator and the Recycler. This new line, known as the
AP-5 line, would also have operational benefits for Run II. It is
proposed to construct such a line as soon as practical in Run II.

#  Colliding beams issues

## Tevatron Injection

The bunches in TeV33 will probably be spaced at 7 rf bucket (39.5 m)
intervals. The trigger electronics for the two major detectors (CDF and
D0) are being upgraded for a 7 rf bucket spacing. Smaller spacing
between bunches or even unbunched beams have been discussed but are not
considered here. A gap in the beam to accommodate injection and
extraction (the beam abort) is required.

At the termination of a normal store, only the protons are removed, and
they will probably be removed without the use of the abort kickers.
However, we plan to use the abort kickers in abnormal situations to
remove the beams quickly without quenching the superconducting magnets.
The existing beam abort kickers require a minimum gap of 2.6 μsec (139
rf buckets) to rise to a field which is adequate to steer the beam onto
the abort dump. The major experimental areas are located at B0 and D0:
1/3 of the circumference of the ring apart. Loading the ring in a
three-fold symmetric fashion provides an identical pattern of
interactions at the two interaction regions. Non-symmetric patterns are
possible and can have some advantages; a non-symmetric pattern would
almost certainly be used if there were only one detector.

The antiproton injection kicker has been built for use in Run II and has
a rise time of 400 nsec (21 rf buckets). We plan to modify the proton
injection kicker to have a similar rise time. The rise times of the
injection kickers requires additional gaps in the bunch trains. The
bunch loading scheme used in the present analysis, which satisfies the
above constraints, calls for three groups of bunches, each 371 rf
buckets long to fill the 1113 Tevatron rf buckets. A group consists of 3
batches of 10 bunches spaced at 7 rf bucket intervals. The batches are
separated by 20 rf buckets for the injection kicker and the last batch
is followed by the abort gap of 139 empty buckets. This scheme is
illustrated in Figure 29

The gaps in the beam cause an undesirable spread in the beam-beam tune
shift parameters. With the scheme illustrated in Figure 29, only 90 out
of 159 potential rf buckets are filled. If some of the gaps were
eliminated or reduced, and the total number of antiprotons were held
constant, the number of interactions per beam crossing could be reduced.
We expect that the actual configuration of bunches will change as we
study the beam-beam interaction and the beam loading constraints in more
detail.![](media/image1.emf){width="5.988888888888889in"
height="1.4875in"}

## Proton Intensity

The TeV33 proton intensities are specified to be the same as those used
in Run II and, in fact, already achieved in the Main Ring. Simulations
predict that 5 bunches, each with 6x10^10^ particles in a longitudinal
emittance of 0.15 eV-sec can be coalesced into a single 1.38 eV-sec
bunch with 30x10^10^ particles.[^32]

A simulation for seven bucket spacing has also been performed. In order
to achieve good coalescing efficiency with 5 bunches, the rf wave-form
must be linearized using higher harmonics of the rf. The simulation used
first, second, and third harmonics of the basic 7.58 MHz frequency (7
bucket spacing). The results of the simulation are shown in Figure 30.
Each of these frequencies would be created by a new, albeit modest rf
system. The new rf systems are summarized in Table XIV.

Based on beam experiments with multi-batch coalescing, we know that beam
loading effects are important. The simulation has not taken these
effects into account. Some of the measures we plan to use to reduce
beam-loading effects are discussed in section 3.1.3.3. The number of
batches that can be simultaneously coalesced will affect the loading
scheme. At the moment, we plan to simultaneously coalesce 10 batches.

![](media/image1.emf){width="3.15in"
height="2.589583333333333in"}![](media/image1.emf){width="3.15in"
height="2.589583333333333in"}

Figure 30. A simulation of the proton coalescing process. The beam
distribution and rf wave form is shown (a) after rotation with the
(linearized) *h*=84 rf, (b) immediately after recapture with the *h*=588
(53 MHz) rf, and (c) as the beam approaches its equilibrium distribution
in the *h*=588 bucket.

## Antiproton Intensity

The proton coalescing hardware could be used to form the antiproton
bunches, but it is likely to be unnecessary. The Main Injector is
expected to be capable of accelerating 0.5 eV-sec bunches through
transition with the 53 MHz rf system. The total Recycler longitudinal
beam emittance will be less than 50 eV-sec in stacking mode (see Table
XIII). The Recycler beam could thus be packaged into batches of 12
bunches with 7 bucket spacing, each bunch having an emittance of
50/100=0.5 eV-sec or less. These bunch trains could be accelerated
directly by the Main Injector provided that collective effects do not
cause any serious problem. An alternative approach, that we plan to use
in Run II, is to use a lower harmonic rf system in the Main Injector.
The Run II plan, for example, calls for the acceleration of 1.5 eV-sec
bunches in the Main Injector with the 2.5 MHz (coalescing) rf system and
a reduced ramp rate.

In any event, the Recycler, with electron cooling, has the potential to
deliver much smaller emittance beams than the 2 eV-sec listed in Table
II. Such low emittance bunches could produce large tune shifts in the
proton beam, be subject to rapid growth from intrabeam scattering, and
be subject to instabilities (such as the fast head-tail mode). We may
choose to dilute the beam to the 2 eV-sec emittance listed in Table II
in order to avoid some of these effects, or we may choose to take
advantage of the increased luminosity available from lower emittance
beams. If we should find it necessary to blow up the emittance before
transition in the Main Injector, we would use the coalescing technique
described in section 4.2 for the protons also.

## Tevatron rf voltage considerations

One of the more important factors in the luminosity form-factor (*F* in
Eqn. \[1\]) is the bunch length. The bunch length is determined by the
rf voltage and the bunch emittance. For gaussian beams in a stationary
bucket having emittances small compared to the bucket area:

$s_{l} = \sqrt{\frac{\text{he}_{L}^{2}}{\text{18}p\left( \text{ehV} \right)E_{0}w_{0}^{2}}}$
\[16\]

As shown in Table II, the existing 1 MV of rf at 53 MHz results in an
rms bunch length of about 36 cm. This can be reduced to 13 cm with a 16
MV system operating at 212 MHz. Figure 31 shows a comparison of the
average integrated luminosity that could be obtained with and without
the rf upgrade[^33].

The evolution of the beam emittances during a store is expected to be
dominated by intrabeam scattering and should be rather different in the
cases with and without the rf upgrade. The evolutions of longitudinal
and transverse emittances are shown in Figure 32 and Figure 33.

![](media/image1.emf){width="6.0in" height="4.694444444444445in"}

![](media/image1.emf){width="5.4in" height="4.225in"}

![](media/image1.emf){width="6.0in" height="4.694444444444445in"}

Stronger rf focusing could be obtained with either conventional or
superconducting technology. Since no power is delivered to the beam when
the beams are stored, superconducting technology is a natural choice.
With an inductively tuned superconducting cavity, the beam current will
excite the cavity, producing the necessary voltage. While this approach
appears promising, it will be necessary to establish the techniques
required to capture the beam into the desired buckets and to examine the
beam stability. We have postponed a more detailed look at these beam
dynamics issues because the rf upgrade, if feasible, will be an
expensive proposition and will occur, if at all, in the later stages of
the TeV33 upgrade.

##  Single beam stability

Beam instabilities can be divided into two types: single-bunch and
multi-bunch instabilities. The single-bunch may be caused when a
bunch\'s wake fields act on itself. Multi-bunch instabilities may be
caused when the wakefield acts on subsequent bunches. The proposed
single bunch intensities for TeV33 are essentially the same as those
already achieved. Therefore, the instabilities expected are those known
from current operation. A head-tail transverse instability is
observed[^34] when the chromaticity is negative. In normal operation,
the chromaticity is maintained at about ξ=+10 to provide a margin
against this instability. When the beams are colliding, where the
instability is presumably stabilized by the tune spread from the
beam-beam tune shift, the chromaticity is reduced towards zero. The long
range beam-beam tune shift in TeV33 results in a spread in
bunch-by-bunch chromaticities, so we plan to suppress the instability
with the new Tevatron bunch-by-bunch damper. The damper may cause
excessive emittance growth if used for more than a few minutes, but we
expect beams should be stable when collisions are achieved.

We have also observed what appears to be a longitudinal head-tail
instability.[^35] This instability is suppressed by a longitudinal
bunch-by-bunch damper designed for 6 bunch operation.[^36] This damper
system will need to be upgraded to handle 36 bunches(21-rf bucket
spacing) in Run II and ultimately the 7-rf bucket for TeV33.

Multi-bunch instability thresholds depend primarily on the total
current. TeV33 operation will be significantly more susceptible to
multi-bunch instabilities than was the case in Run I (6 bunch
operation). However, based on fixed target operation (with 2.5x10^13^
particles), we expect transverse instabilities from the resistive wall
effect and longitudinal coupled bunch instabilities from the cavity
modes. The resistive wall instability is not seen in the Tevatron when
the chromaticity is tuned to be positive. The most severe longitudinal
coupled bunch instability appears to be a mode-1 instability. Since
neither of these instabilities is particularly vicious in fixed target
operation, we expect to be able to control them in Collider operation.

## Beam abort

The role of the Tevatron abort will change dramatically when antiproton
recycling becomes a reality. The existing A0 abort removes protons and
antiprotons from the accelerator simultaneously in a single turn and
consequently cannot be used if the antiprotons are to be retained at the
end of the store. The role of the A0 abort in Run II and beyond would be
to:

1)  Provide a beam dump when proton-only stores are used (including the
    normal tune-up of beam transfers.

2)  Remove colliding beams urgently--to avoid a quench, for example.

The effectiveness of the abort for colliding beams is limited by the
following factors:

1)  It is difficult to develop a signal that reliably indicates the need
    for an urgent abort. For example, the loss monitor system that is
    used in fixed target operation is not allowed to trigger the abort
    in colliding beams operation.

2)  Some faults (such as a refrigeration failure) typically result in
    quenches whether or not the beam is removed.

3)  There are times during the injection process and possibly during
    acceleration when the beams are not appropriately cogged to make
    possible an abort without excessive (quench inducing) losses.

Despite the limited use of the A0 abort, we tentatively plan to retain
it in its current configuration and to use it whenever possible to
minimize the number and the severity of quenches and to provide a more
controlled removal of the beam energy and better confinement of the
resulting radioactivity.

The intensity limitations[^37] of the A0 abort are summarized in Table
XV. The yearly dose rates assume a utilization of 7200 proton beam
aborts at 150 GeV, 6 proton beam aborts at 1 TeV, and 2 antiproton beam
aborts per day for 183 days per year. The most serious limitations are
the activation of the absorber and soil contamination. The soil
contamination problem can be mitigated by the addition of steel on the
outside of the existing absorber. The 0.1 rad/hr limit on the residual
activity is somewhat arbitrary but could probably be reached by adding
some additional shielding to the existing absorber.

## Interaction Regions

It has not yet been decided how many interaction regions will be
occupied by experiments during the TeV33 era. One or both of the major
interaction regions at B0 and D0 might be occupied. A proposed, new
interaction region at C0 might also be occupied. We have not considered
in any detail at all, any option other than the current operating
condition of having high luminosity experiments at B0 and D0.

The low beta insertion at C0, if any, has not been designed (although
there has been a significant effort towards this end). The low beta
insertions at B0 and D0 are likely to undergo some modification probably
at the time of Run II. These modifications, which are described below,
have been used in the calculation of the beam-beam effects.

### Lattice

A low beta insertion which is matched to the Tevatron and which yields
zero for both η and η\' in the IR has been described.[^38] This
insertion uses the current physical configuration of the IR and the
magnets gradients can be achieved with the already installed
quadrupoles. Implementation of this design would only require the
interchange of two power supplies and a reversing switch. For β\*=
0.35m, the IR lattice functions and the corresponding quad gradients for
this dispersion-free solution are shown in Figure 34 and Table XVI.
Compared to the Run I optics, this solution results a slightly smaller
beam in the IR and also appears to have a beneficial side effect on the
beam separation. Just after the separators, in the plane with the most
separation, the new insertion has stronger focusing, resulting in a
faster betatron phase advance, and a more rapid separation of the beams.
But, the lattice has not yet been optimized for the beam-beam effects.
One particular problem we hope to consider in the future in the
modification of the horizontal phase advance between the horizontal
separators at B11 and B17.

![](media/image1.emf){width="5.722222222222222in" height="3.94375in"}

### Magnitude of the crossing angle

The first parasitic crossing for TeV33 occurs just 19.8 m from the
interaction point (IP), placing it within the interaction region (IR)
triplet quadrupoles and before the first set of separators. It does not
appear possible to avoid collisions at these points without introducing
a crossing angle at the IP. A straightforward calculation \[5\] shows
that additional separators inboard of the IR triplet quadrupoles would
have little effect on beam separation at the first crossing. Essentially
all the betatron phase advance occurs in the immediate vicinity of the
IP, and is practically zero across the IR triplet quadrupoles. For
realistic gradients and lengths, electrostatic separators are incapable
of producing the 3-5σ beam separation desired, with θ=0 at the IP.
Without a crossing angle the additional crossing points would more than
triple the beam-beam tune shift arising from the interaction point.

Ultimately, the size of the crossing angle will be determined by
operational experience. A priori, this choice is not obvious. The
crossing angle represents a compromise between avoiding excessive
beam-beam effects from the parasitic crossings nearest the interaction
point and the reduction in luminosity resulting from a crossing angle.
The *half*-crossing angle *θ* that produces *nσ* beam separation at the
first parasitic crossing is approximately:

$q»\frac{n}{2} \cdot \sqrt{\frac{e}{6\text{pβ}^{}\beta_{r}g_{r}}}$
\[17\]

where $\beta_{r} = v/c»1$. For 3*σ* and 5*σ* separations *θ* becomes
approximately 142 and 236 μrad, respectively. We have chosen to divide
the crossing angle equally between the horizontal and vertical planes at
each IP (*θ~x~*=*θ~y~*=100 and 170 μrad, respectively).

### Effect of the crossing angle on the luminosity

Because our bunches are very long (either about 37 cm with the present
Tevatron rf system or about 14 cm if we implement the rf upgrade
described in section 4.4) with very small transverse beam sizes at the
interaction point (about 30. microns), the crossing angles we are
contemplating are not small. They have significant effects on the
overlap of the beams and so on the luminosity and on the beam-beam tune
shifts. Figure 35 shows the form-factor in the luminosity equation as a
function of crossing angle for the two bunch lengths assuming the
remaining parameters are those given in Table II. The third bunch length
(54 cm) can be obtained with a 4 eV-sec beam in a bucket formed by 1 MV
at 53 MHz.

### Effect of the crossing angle on the z distribution![](media/image1.emf){width="4.2in" height="3.286111111111111in"}

The *z* distribution at the colliding beam can be calculated easily once
the bunch length and the crossing angle are known. Figure 36 shows the
distribution of vertices for 170 μrad crossing angle assuming various
bunch lengths.

![](media/image1.emf){width="6.0in" height="4.333333333333333in"}

## Beam-beam effects

*Peter Bagley is working on section 4.8. The current text is just notes
for the final version.*

Introduction: Importance of the beam-beam effect.

### Bunch-by-bunch tune shifts

Give equations. Mention the fundamental variables.

(Assume x0n, y0n, s0n = 0, then xi/xi0 depends on thnx, thny, and
sigs/beta\*.)

I think we should also explicitly say that the snowmass result for xi vs
crossing angle is wrong due to its using a bunch length a factor of 2
too large.

Figures xxx to zzz show the luminosity and the tune shifts and
transverse coupling for zero amplitude particles.

These graphs have curves for bunch lengths of 37 cm and 14 cm.

Give the other parameters used for the graph.

At zero angle, the luminosity is a little lower than the simple
expression due to the hourglass effect. For a bunch length of 37 cm,
half horizontal and vertical crossing angles of 100, 136, and 170 μrad
decrease the luminosity by factors of xxx, yyy, and zzz, respectively.

Big hit in luminosity

For a bunch length of 14 cm, half horizontal and vertical crossing
angles of 100, 136, and 170 μrad decrease the luminosity by factors of
xxx, yyy, and zzz, respectively.

With the shorter bunch length there is a greater overlap of the bunches
and less luminosity is lost.

May want a picture of the bunches crossing at an angle. (MacDraft ??)

Comment that the

Because of the large crossing angles, the bunches only overlap near the
beta minimum and the hourglass effect is greatly reduced.

This is a good place to put in the bit on the longitudinal distribution
of the events.

Include a version of the graph of lum, xi vs crossing angle.

Since we have room I would tend to put in something closer to my
version, with curves for35 cm and 14 cm bunch lengths, and showing lum,
xi, and xisq.

If we show more than one bunch length, the reference lum should be the
head on, short bunch limit, so the difference in the hour glass effect
for the 2 bunch lengths can be seen.

### Choice of equal horizontal and vertical crossing angles.

Show and explain figures for sum of +-2, +-1, 0 points for

(theta = 140 and 240 urad) and for (bunch lengths of 37 cm and 14 cm).

Adjusting the signs of the crossing angles can change the signs of the
coupling

components, but does not change the sign of the tune shift components.

By making the crossing angle all horz at one IP and all vert at the
other IP, get similar total horz and vert tune shifts. But get smaller
total tune shifts if use about equal horz and vert crossing angles at
both IP\'s.

Give numbers for these cases.

Tune shift components never get very negative, can\'t get very much
cancellation.

With two IP\'s we can chose signs that make the xisqcos largely cancel.

### Choosing our favorite combinations 

Introduce idea that there isn\'t much room in the lattice for more
separators and very little flexiblity as to where they could be put. 11
and 49 locations are already filled with separators. 48 and some 17
locations available as well as some space near some 0 locations.

Put in sketch of dipoles and orbit ?

To keep machine closed, the bend center can not move. If you move a
\"space\", have to split it. End effects are expensive (need bypass
boxes for helium and bus work)

Move by one dipole, dr is 5.0 cm

Move by one cell (e.g. A22 to A23) dr is 24 cm

Move by one house (e.g. A22 to A32) dr is 1.9 m.

Can\'t move \"spaces\" very far.

Separators must be at efficient locations (High beta in kick plane.)

Main problems are from first few crossings near the IP\'s

Separations and behavior of these first few crossings are entirely
determined by the optics, the settings of 49 and 11 separators, and the
signs and sizes of the crossing angles at the IP.

Have many possible combinations of signs of crossing angles and size and
signs

of the pieces of the helix.

However not every combination is allowed at the 2 IP\'s. The helices in
the arcs must match together.

Discuss procedure for choosing favorite combinations.

2 series \-- For jj15c and a given size of the crossing angles, chose a
fairly large helix in the arcs, then ran through all 32 combinations of
the signs of the pieces of the helix and the signs of the crossing
angles. Main considerations were the min. diag sig sep, the no. of
points with dss \<4, the min ((diag sep)/(sigma max)).

3 and 4 series \-- For jj15c and a given size of the crossing angles,
looked at the combinations of sizes of the 11, 49 location separator
kicks and the signs on the crossing angles and the sep kicks that would
give the best results. Again main considerations were the min. diag sig
sep, the no. of points with dss \<4, the min ((diag sep)/(sigma max)).

(Don\'t show the gen tables, but try to give an idea of what they
represent.)

Example configurations shown in this figure are an attempt at producing
good behavior at these first few near misses. Don\'t worry too much
about separator settings being reasonable.

We are beginning to believe that the 100 urad case (corresponding to 3
sigma center to center separations at the first 2 crossings) is too
small of a crossing angle. So we have not made 3 and 4 series versions
for the 100 urad configuration. Despite this, we have not entirely ruled
out the 100 urad case.

Present basic parameter set.

Describe and give equations for zero amp calculations.

pppp2.100npnn

pnpp3css.136nppp

pnpp4ass.170nppp

### subsection \-- An example 

Present and discuss all the \"standard\" graphs for 3css.

This particular configuration was set up as an exercise to try to get
very good separation at the crossings near the IP\'s, within about 56
half buckets of either B0 or D0.

This is not an optimized solution.

Mention sep settings

Rather than use the horizontal separators at B17, this assumes we have 2
modules

at B48. In this lattice with 35 cm beta\*, the horz phase advance
between the B11 and

the B17 horizontal separators is about 2\*pi\*(.90). They are nearly
degenerate and

often end up at high voltage and fighting each other. The B48 location
is at a much better

phase location, about 2\*pi\*(3.32) in horz phase from the separators at
B11.

Rather than use the horizontal separators at F17, this assumes we have 2
modules at

D48. This is because with both the RF system and the injection point at
F0, space

around F0 will be extremely tight, whereas space near E0 will be freed
up.

Give table of sep. settings ?

HB48 is at -188 kV. If the C0 abort is ripped out, there may be room for
more

separators modules near here.

VA17 is at 217.6 kV. There is room at VA17 for more separator modules if
the

Schottky pick ups are moved.

VA49 is at -112 kV. This is close to what it can do.

Separator settings are not presently realistic, but we could probably
find ways to achieve

this basic configuration. Again though, keep in mind that this is not
intended to be a real

solution. Its only an exercise.

diag sigma sep

short vert helix +- 8.5 sigma

short horz helix +-10 sigma

long vert helix +-12 sigma

long horz helix +-6.5 sigma

Short vert helix and Long horz helix are somewhat smaller than we would
like. Again

these values are primarily intended to give good separation near the
IP\'s. One of the

questions is whether this is sufficient separation in the arcs.

xix, xiy, xisq cos, xisq sin vs arc

In the long helix, the vert sigma separation is larger than the horz
separation.

So the points with the smallest sigma separation are where the
separation is mainly horz

and the vert separation is near zero. This corresponds to positive
vertical tune shifts.

The largest peaks have the vertical sigma separations closest to zero at
locations of

peak vert. beta

Similarly in the short helix, positive horz tune shift peaks occur at
analogous places.

Negative horz tune shift peaks occur where the diag sigma sep is the
smallest.

Note the opposite signs of the bbsq cos components near the 2 IP\'s.

These contributions will largely cancel.

Through the arcs, the horizontal and vertical betatron phases advance
together and

very nearly 2pi\*(0.25) apart. Consequently the bb coupling component
mainly appears

as the sin like component. Only near the 0 locations are the horz and
vert phases nearly

in phase.

Usual comments that we are mainly concerned with the global coupling
components.

We really have not looked at the (phix + phiy) components.

One point apiece with large xix and xiy near D0. B0 looks much better.

We can essentially switch these and make the two points with large xix
and xiy appear

near B0, with D0 looking much better.

But can\'t make both B0 and D0 look as good as B0 because of links
between signs of kicks

needed, e.g. at B11 and C49. Also because of overall sign requirement on
crossing angles

at B0, D0 to make coupling cancel.

Show a separate graph for bunch by bunch xiy vs xix for zero amp
particles for say sigs of

37.1 cm.

Show the version with the points labeled with the bunch no.s The \"upper
tier\" of points are the points within 2 of the ends of the trains or
subtrains.

Comment that a different bunch length displaces this graph (give values)
but does not change the pattern or contribute to the differences between
the tune shifts for the different bunches (at least not the way that we
are handling it.)

Show for all crossings and show another version with arc contribution
only ?

Point here is that arcs have a significant contribution to the spread in
the tune shifts between the different bunches. As far as the tune shifts
for zero amp. particles are concerned, we need more separation in the
arcs. Probably particularly for the short vert and the long horz pieces
of the helices, which, as mentioned previously, are a little smaller
than we would like.

xisq sin vs xisq cos for zero amp part

usual caveats \-- we\'ve chosen to compute coupling and tune shifts
separately,

actual tunes will be determined by both\...

Although 5 points (1 above, 4 below) are separated from the main
cluster, the coupling

does not show the separation of the bunches within 2 of the ends of the
trains or subtrains

seen in the tune shifts.

range in coupling is probably acceptable.

If exactly decouple bunches in the middle, range of global tune splits
will be

sqrt((.0110/2)\^2 + (.0055/2)\^2) = .006

bunch by bunch positions and angles ?

Ranges look pretty reasonable.

Describe and give equations for tune shifts vs. amplitudes

No energy offsets.

Would be nice to at least take a quick look at the effects of an arrival
time offset on the tune shifts from the main IP\'s.

Define amplitude.

specifically mention the bugs in the calculations in the snowmass paper

bunch length factor of 2 too large

amplitudes factor of sqrt(2) too large. \-- could refer to this as not
clearly defining what was meant by amplitude. But because of the
population of particles for the histogram, snowmass is wrong.

Show min, max, range plots of tune shift vs arc

Not surprising points with the largest ranges tend to be points with
large tune shifts for

zero amp particles. But for points in arc, range (0 to 4 sigma) is less
than size of zero

amp tune shift.

Surprising to us that there is so little range in the xix in the long
helix or in the xiy in the short helix.

A better way to view this is probably that the small separation in the
other plane makes the range of xix greater in the short helix and the
range of xiy greater in the long helix.

Now start looking at all 3 configurations again

We are going to work up to showing graphs of the distributions of
particles in the tune plane for all the pbar bunches. While these graphs
are important in themselves, they are more meaningful if we know how the
different crossing points contribute, particularly in that we are still
trying to use these calculations to understand and to design better
separation schemes. Also because the tune footprint is not the only
parameter of importance. As important is how strongly the beam beam
interactions at the different crossing points drive different
resonances. These resonance driving terms can be calculated (although we
have not yet begun that), But we can start to get a very qualitative
feel for the strengths of some of these resonances from the size of the
tune spreads. That is a small spread in tune shifts with amplitude
\*suggests\* that certain resonances will be weak.

Show 3 A17 footprints from crossings \>28 half buckets from the IP\'s,
No contours shown because they couldn\'t be seen since footprints are so
small.

![](media/image1.emf){width="4.985416666666667in"
height="3.9027777777777777in"}

![](media/image1.emf){width="5.0in" height="3.9027777777777777in"}

![](media/image1.emf){width="4.985416666666667in"
height="3.9027777777777777in"}

A17 is a \"typical\" bunch. With the exception of the bunches within 2
of the ends of the trains or sub-trains, the variation of the tune
shifts with amplitude of the other bunches look very similar to A17.
There are differences in the zero amp. tune shifts.

Main point here is that, despite the few small spikes seen in the range
of the tune shifts in the arcs, the tune spread from arcs is very small.
(Small in relation to the contributions to the tune spreads for this
bunch from other regions, which we will show shortly.) As far as tune
spreads are concerned we\'ve got enough separation in the arcs.

While all 3 cases are small, the 100 urad case has a larger xiy spread
than the others.

100 urad case has poor diag sigma sep at several points in the short
helix.

Problem is that after the \"middle\" separators in the short helix, the
horz and vert separations are too close in phase, rather than being pi/2
out of phase. For the zero amp.

particles, this means that the effect of this point is mainly in the
coupling and not in the tune shifts.

Four points in the short helix with diag sigma sep less than 5. These
points have diagonal sigma separations of 3.3, 3.9, 4.5, and 4.9. By
itself, the point with 3.3 contributes xix and xiy spread of about
.0014, not terribly large, but noticibly larger than the other points in
the arcs. This particular point is not a point where A17 sees a proton
bunch, so the effect of this particular point is not shown in the
figure.

We believe that this problem is not intrinsic to all 100 urad
configurations, but rather is a characteristic of this particular case.

Show 3 A17 footprints from all crossings except main IP, show contours

(This will most clearly show differences in crossing angles.)

![](media/image1.emf){width="5.013194444444444in"
height="3.9027777777777777in"}

![](media/image1.emf){width="5.013194444444444in"
height="3.9027777777777777in"}

![](media/image1.emf){width="5.0in" height="3.9027777777777777in"}

Describe contours. NOT contours of equal population. The heavy, black
contour connects the (0,0), (0,2), (2,0), (2,2) points.

(Mention that contours were not shown in the \>28 half buckets figure
because the tune spread was so small that the contours would not be seen
clearly.)

For all cases, a particle with zero amp. is at the bottom left corner.

For zero horz amp, as the vert amplitude increases, the points move up
the left side.

For zero vert amp, as the horz amplitude increases, the points move
along the bottom. Particles with the largest amplitudes shown in both
planes (4,4) are at the top right.

Make a table of estimated areas enclosed by 2, 3, and 4 contours. ??

For a gaussian distribution,

About 3/4 have both horz and vert amps less than 2 sigma

95% of the particles have both horz and vert amps less than 2.72 sigma

97.8% have both the horz and the vert amp less than 3 sigma.

![](media/image1.emf){width="4.985416666666667in"
height="3.9027777777777777in"}

The nnpp2.136ppnp case has min diag sigma sep of 3.3 and 3 points with
dss \< 4, compared to pnpp3css.136nppp which has min dss of 3.5 and only
2 points with dss \< 4. The point of showing the nnpp2.136 plot is that
0.2 diff in min diag sigma sep does make some difference in the
footprint, especially on the 2 contour.

3css No B0, D0 3c contour has an area of about (.0072)\^2 = 52.e-6

nnpp2 No B0, D0 3c contour has an area of about (.0087)\^2 = 75.e-6

Also start to get some idea of sep neccessary.

For 170 urad case, where min diag sig sep is about 4.4, start to see
points spreading out between 2.5 and 3 sigma. (4.4 - 2.7 = 1.7)

For 136 urad case, where min diag sig sep is about 3.3 or 3.5, start to
see points spreading out between about 1.5 and 2 sigma. (3.4 - 1.7 =
1.7)

For 100 urad case, where min diag sig sep is about 2.4, start to see
points spreading out almost immediately, between about 0.5 and 1 sigma.
(2.4 - 0.7 = 1.7)

Consider 100 urad case. A particle with an amplitude of zhat, spends

about 40% of the time at displacements of 0.8(zhat) or greater, and

about 50% of the time at displacements of 0.7(zhat) or greater.

z beam beam kicks peak near z of about 1.5 sigmaz, displ in other plane
\< about 1 sigma

Strong points are within +-2 of the IP\'s.

At the +-1 points, both the horz and vert sig sep are about 3/sqrt(2) =
2.1

At the +-2 points, one is still about 2.1, the other is about 3.3

At particle amps of between 0.5 and 1 sigma, particles are getting
within about 1 to 2 sigma (in both planes) from the opposing beam\'s
core and are starting to see the strong beam beam kicks associated with
the core of the opposing beam.

At particle amps of about 3.5 sigma, particles are getting to the edge
of the strong beam beam kicks on the far side of the opposing beam\'s
core. This is about as strong an effect as it gets from these \*first
few\* near misses. The effect from these first few near misses will stay
about the same, fall off slowly as the amplitudes continue to increase.

Little change between 3 and 6 sigma.

Around 7 sigma arc crossings are starting to be important.

But as amplitudes increase further, they are getting out to arc
separations. At other crossing points, they will be starting to get near
the core of the opposing beam.

Show the 6 (3angles X 2 bunch lengths) main IP only footprints

![](media/image1.emf){width="5.0in"
height="3.9027777777777777in"}![](media/image1.emf){width="5.0in"
height="3.9027777777777777in"}![](media/image1.emf){width="4.985416666666667in"
height="3.9027777777777777in"}![](media/image1.emf){width="5.0in"
height="3.9027777777777777in"}![](media/image1.emf){width="5.0in"
height="3.9027777777777777in"}

![](media/image1.emf){width="5.0in" height="3.9027777777777777in"}

For 170 urad, sigs = 37.1 cm, upper half of the footprint is twisted.

Tune shifts for small amp particles decreases with increasing crossing
angle and/or decreasing bunch length.

Emphasize that these have the mirror symmetry of the footprints for No
B0, D0.

Small amplitude particles are in the upper right corner (for No B0, D0,
these were in the lower left corner)

\"Upper left\" corner has (4,0) (For No B0, D0 this was on the bottom
right corner.)

\"Lower right\" corner has (0,4) (For No B0, D0, this was on the upper
left corner.)

\"Lower left\" corner has (4,4). (For No B0, D0, this was on the upper
right.)

Because the points corresponding to a given amplitude are generally on
opposite sides of the two footprints, when we add the effects, the
resulting spreads are reduced.

Show the 6 A17, all crossings, footprints with contours

![](media/image1.emf){width="5.0in"
height="3.9027777777777777in"}![](media/image1.emf){width="5.013194444444444in"
height="3.9027777777777777in"}![](media/image1.emf){width="5.013194444444444in"
height="3.9027777777777777in"}![](media/image1.emf){width="5.0in"
height="3.9027777777777777in"}![](media/image1.emf){width="5.0in"
height="3.9027777777777777in"}![](media/image1.emf){width="5.0in"
height="3.9027777777777777in"}

These are just the \"sum\" of the figures for the main IP only and the
No B0, D0 footprints.

The No B0, D0 footprints are generally wider in the (xiy - xix)
direction

and the Main IP footprints are quite narrow in the (xiy - xix)
direction.

The Main IP footprints are generally wider in the (xiy + xix) direction.

Roughly speaking, adding the Main IP tune shifts to the No B0, D0 tune
shifts turns the

No B0, D0 footprint \"inside out\" along the (xix + xiy) direction.

For the sigs=37.1 cm cases, the tune shifts from the Main IP\'s and the
No B0, D0 along the (xix + xiy) direction happen to be similar in size
and opposite in direction. The resulting cancellation produces a
footprint occupying less area in the tune plane than the No B0, D0

footprints.

In particular for the 100 urad case with sigs = 37.1 cm, the tune shifts
from the Main IP\'s and the No B0, D0 along the (xix + xiy) very nearly
cancel. The resulting footprint appears flattened and occupies a
relatively small area in the tune plane. We did not design the 100 urad
case to have this behavior, this just happened to work out this way.

For the sigs = 14.0 cm cases, there is still some cancellation, but the
larger Main IP

tune shifts tend to dominate the other crossings.

For the 170 urad case with sigs = 14.0 cm, the tune shifts from the Main
IP\'s and the No B0, D0 along the (xix - xiy) direction very nearly
cancel. The resulting footprint has a needle appearance and occupies a
relatively small area in the tune plane. Again this was not planned.

Show summary histograms for all 3 crossing angles and for both bunch
lengths

Mention how the histograms were generated. 5000 particles per bunch, cut
off at 4 sigma amplitudes, size of bins for hist. \...

Spread from zero amp tune shifts of different bunches is a significant
contribution to

area in the tune plane occupied. Zero amp tune shifts tend to spread
more along the (xix - xiy) direction.

Can see from here that zero amp bunch spread goes down significantly for
170 urad case.

Most of the bunches have footprints very similar to A17. The bunches
within 2 of the ends of the trains or subtrains have footprints with
somewhat different shapes.

Show tune plane diagram and briefly discuss how well footprint fits.

Shape of the clear area is important.

Mention question of whether we can straddle the crossing resonance.

If we can, we have roughly twice as much space available and most of
these distributions \"fit\".

Because we are close to the coupling resonance, more space along the
direction (xix+xiy). For 135 urad, 100 urad, sigs = 37.1 cm, the
distribution is more along the direction

(xix-xiy). It would be easier to fit these between resonances if the
tunes split the half

integer, e.g.(.42, .59). (This is probably not a good idea. This point
lies close to the more destructive nux + nuy resonance.We don\'t really
have circuits set up to compensate this resonance.)

Continue to play with sep configurations

Work on modifying the lattice with an eye towards improving these beam
beam tune spreads.

Working on several schemes to reduce the number of bunches within 2 of
the ends of the bunch trains or sub-trains.

1\) Fast inj. kicker with rise time of about 132 nsec

2\) Shiltsev\'s fast inj kicker

3\) Forming an entire bunch train in the Main Injector at inj.

8 gev lifetime questions

Multi batch coalescing \-- Concerns with beam loading.

4\) Get rid of one or more of the abort gaps

These schemes will cluster \*most\* of the bunches more tightly, but
there will still be a few on the outskirts.

In an era when we are recycling the pbars at the end of a store and
where the experiments request that we hold down the peak luminosity to
keep the number of interactions per crossing down, the loss in
luminosity from a large crossing angle may not hurt the integrated
luminosity too much.

We believe we are making good progress on the spread in the tune shifts
of the zero amp. particles in the different bunches and on the spread in
tune shifts vs. amplitude.

Once we have solutions (lattices and separator configurations) we are
satisfied with for these considerations, we intend to continue on to
calculations of resonance strengths for

nearby resonances, some particle tracking, etc\...

## Luminosity Leveling

The peak luminosity of 10^33^ cm^-2^sec^-1^ with 100 bunches per beam
results in about 10 interactions per bunch crossing. It is difficult
(and expensive) to build high performance detectors to operate in this
environment. One possible operational scenario involves a technique
known as "luminosity leveling". The idea is that the luminosity is held
at a maximum value--say 0.5×10^33^ cm^-2^sec^-1^--during the initial
part of the store and kept constant for as long as possible. The
luminosity modulation could be accomplished, in principle, by varying
any of the beam parameters on which the luminosity depends such as
crossing-angle, bunch length, β\*, beam separation, etc. A simulation of
luminosity leveling is shown in Figure 37 for the nominal parameters in
Table II. The loss of luminosity from intrabeam scattering and residual
gas effects is the same for both stores, but the luminosity limited
store retains more antiprotons (fewer collisions) and has a higher
luminosity at long times in the store.

The limited store yields 23.4 pb^−1^ in 14.0 hours (a 13 hour store plus
one hour for shot setup) for an average luminosity of 1.67 pb^−1^/hr
compared to 36.2 pb^−1^ in 15.6 hr and an average luminosity of 2.32
pb^−1^/hr for the unlimited store. The average luminosity obtained with
a luminosity leveled store is less sensitive to the initial antiproton
bunch intensity than a unleveled store, but the highest integrated
luminosity is obtained in either case with the highest possible initial
antiproton intensity. The loss of luminosity from leveling depends on
the store parameters: the importance of the antiproton intensity to the
lifetime, the amount of luminosity reduction desired, and the length of
the store.

![](media/image1.emf){width="5.999305555555556in"
height="4.794444444444444in"}

It should not be supposed that luminosity leveling is trivial or even
straightforward. An enormous amount of effort is invested in minimizing
beam loss by adjusting orbits, tunes, and chromaticity when the beams
collide. The techniques that are used to modulate the luminosity may
have adverse effects on the loss rate. One leading candidate to
implement luminosity leveling is to modulate the beta function at the
interaction point. Our experience is that changes to magnet excitation
are likely to result in increased loss rates. It is not clear how well
we will be able to control the beam loss rates when changes are made to
the machine parameters to keep the luminosity constant.

#  Summary of upgrades required

The following tables are intended to provide a convenient summary of the
major technical efforts necessary to implement the TeV33 scenario as
outlined above. Table XVII summarizes the R&D efforts required. All of
these R&D projects represent an extrapolation of existing techniques;
they are not design efforts utilizing proven technology. If the R&D
should prove to be unsuccessful an alternative technology would have to
be identified. Possible alternatives are discussed in section 6.

#  Risks & Alternative approaches

## Risks

The TeV33 plan that has been outlined has some obvious risks in that it
relies on technology that is still being developed. The most difficult
technologies are also the most critical, namely,

1)  High sensitivity stochastic cooling pickups and kickers for the 4-8
    GHz band.

2)  Electron cooling of 8 GeV antiprotons.

The plan to increase the antiproton acceptance to 32π mm-mrad is clearly
feasible with an unlimited budget. However, we have not yet determined
what modifications are required to reach this goal.

Slip stacking has not been demonstrated at Fermilab, but has been
studied at CERN. Other bunch coalescing techniques are in routine use at
both CERN and Fermilab. We expect a more difficult time - at least with
the scenario outlined in this report - because of beam loading and other
collective effects.

The Tevatron has never operated with a non-zero crossing-angle. The size
of crossing angle effects depends on the scaled angle:

$\alpha^{'} = \frac{\text{αs}_{z}}{s_{t}}$ \[18\]

The problem of synchro-betatron resonances was first appreciated when
the DORIS storage ring attempted to run[^39] with α'=0.45. The LHC is
proposed to operate with α'=0.5 and a total beam-beam tune spread of
0.015, while the TeV33 parameters are a more aggressive α'=1.8 with a
total tune spread of about 0.025. More detailed calculations are useful,
but the operational limits of the Tevatron collider with a crossing
angle will not be definitively known until experiments are performed in
Run II.

## Alternatives

It appears that there is no alternative to the requirement for higher
antiproton production rates and higher cooling rates. The development of
4-8 GHz cooling technology also seems essential. The only known
alternative--more cooling steps or parallel paths--seems impractical to
implement at a reasonable cost. It might be possible to upgrade the Run
II Recycler stochastic cooling system as an alternative to the electron
cooling system proposed for TeV33. While an optimized 4-8 GHz stochastic
cooling system should, in principle, be adequate to accumulate the
required antiproton flux, it would have a much slower cooling rate and
provide larger final emittances than even a modest electron cooling
system.

A wide variety of schemes for increasing the antiproton flux are
possible. Any scheme (like slip stacking) that raises the proton
intensity could be used. Alternatives include an increased Linac energy
(say to 800 MeV), a new or rebuilt Booster, betatron stacking and
possible variations on the slip stacking scheme proposed for TeV33. One
particular proposal that has been developed in some detail involves
targeting the full Main Injector circumference.[^40] This scheme is
attractive because it does not require any improvements in the Main
Injector intensity nor in the injector chain. However, it does require
technological advances in kicker technology and, perhaps more
importantly, has an impact on the NUMI project, which plans to use 5/6
of the Main Injector protons for neutrino production.

The bunch loading scheme, crossing angle, and optics in the Tevatron
have a number of options that have not been explored in much detail. In
particular, it seems likely that we will want to minimize bunch-by-bunch
tune shifts by adopting a more uniform loading scheme than the one
displayed in Figure 29. One specific proposal to build a fast strip-line
kicker with a 20 nsec rise-time is being considered. More ambitious
projects aimed at compensating some or all of the beam-beam interaction
have been discussed, but nothing has emerged as a serious candidate for
consideration.

# Conclusion

An upgrade in the luminosity of the Tevatron proton-antiproton collider
to the vicinity of L=10^33^ cm^−2^sec^−1^ appears to be feasible
provided that the technological advances described in this report can be
realized. A necessary, but not sufficient, condition for the increase in
luminosity is a substantial increase in the antiproton accumulation
rate. We have proposed a plan to increase the stacking rate by a factor
of 4. Another necessary condition is a large increase in the number of
bunches colliding in the Tevatron. Our ability to avoid adverse effects
from the beam-beam interaction is a primary concern, and an area that is
actively being studied. However, based on the beam-beam footprints that
have been obtained, colliding large numbers of bunches appears to be
feasible.

**References**

[^1]: TheTeV2000 committee report (unpublished), but currently available
    on the World Wide Web site http://www-theory.fnal.gov/tev33.ps.

[^2]: During the period October 1, 1994 to July 23, 1995 100.5 nb-1 of
    integrated luminosity was delivered to each of the two experiments
    and 1.91x10^10^ antiprotons were produced.

[^3]: Fermi National Accelerator Laboratory, "The Fermilab Main Injector
    Technical Design Handbook," unpublished, but available on the
    world-wide-web at
    http://www-fermi3.fnal.gov/maininjector/main_injector.html.

[^4]: G. Jackson, "The Fermilab Recycler Ring Technical Design Report,",
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[^5]: In Run Ib 71% of the stores were intentionally terminated. The
    others typically ended because of the failure of some critical
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[^6]: D. Herrup and D. McGinnis, private communication. There is a
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[^7]: D. Boussard and Y. Mizumachi, \"Production of Beams with High
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    C. Ankenbrandt, \"Slip-Stacking: A New Method of Momentum Stacking\"
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[^8]: N.V. Mokhov, \"The MARS Code System User\'s Guide, Version
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[^9]: CM Bhat and NV Mokhov, *\"Calculation of Beam Sweeping Effect for
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[^10]: FM Bieniosek, K Anderson and K Fullett, *\"Requirements for a
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    (1993)

[^11]: G. Dugan et al., IEEE Trans. Nucl. Sci. **30**:3660-3662(1983)

[^12]: S.O\'Day and K.Anderson, Proc. 1995 US Particle Accelerator
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[^13]: Gregory I. Silvestrov, "Lithium Lenses for Muon Colliders," AIP
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[^14]: ^®^ Torlon is a registered trademark of Amoco Corporation.

[^15]: P.P. Bagley, et al., Summary of the TeV33 Working Group, Snowmass
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[^16]: G. Dugan, "Shielding Scaling Calculations,", Fermilab Note No.
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[^17]: J.D. Cossairt, Fermilab unpublished ES&H note No. EP Note 8.

[^18]: In 1995 (a collider run) 15×10^18^ protons were targetted and 19
    Ci were released.

[^19]: S O\'Day, FM Bieniosek, *\"8.9 Gev Pbar Yield Measurements at the
    Fermilab Antiproton Source\"*, [Nuclear Instruments and
    Methods]{.underline} ,A343, p343 (1994)

[^20]: S O\'Day, FM Bieniosek, *\"Pbar Production Measurements at the
    Fermilab Antiproton Source\"*, [EPAC 94 Proceedings]{.underline},
    p2438, London (1994)

[^21]: The Debuncher transverse aperture is smaller at values of ∆p≠0.
    In yield calculations it is assumed that if the transverse aperture
    at ∆p=0 is increased by a multiplicative factor, the aperture at
    ∆p≠0 will increase by that same multiplicative factor.

[^22]: Fermilab III Accelerator Upgrade Phase 1 (1990/94), unpublished.

[^23]: D Olivieri, \"A Dynamic Momentum Compaction Factor Lattice for
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[^24]: Fermi National Accelerator Laboratory, "Design Report Tevatron I
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[^25]: S. van der Meer, "Stochastic Stacking in the Antiproton
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[^26]: MEB E-Cool Design Report 1992, Indiana University Cyclotron
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[^27]: ^^Pelletron is a registered trademark of National Electrostatic
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[^28]: National Electrostatics Corporation Newsletter, Fall 1990 (NEC,
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[^29]: J.A. Ferry, et al., "Accelerator terminal voltage stability", NIM
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[^30]: T. Ellison, R. Brown, and B. DeVries, "The IUCF electron cooling
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[^31]: S. Nagaitsev, et al., " The development of the technology for
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    2-20 GeV/nucleon energy regime" in Proceedings of the 1993 Workshop
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    Cooling, CERN Report No. 94-03, pp. 390-394.

[^32]: Ioanis Kourbanis, talk given to the TeV33 working group, :Proton
    bunch formation with variable spacing," May 13, 1996 (unpublished).

[^33]: P.P. Bagley, et al., "Summary of the TeV33 Working Group,"
    Fermilab Note No. Conf-96-392 (1996), to appear in the Snowmass 1996
    proceedings.

[^34]: Transverse head-tail instability

[^35]: Longitudinal head-tail instability

[^36]: 6x6 damper reference

[^37]: C. Crawford, Fermilab Report No. TM-1564, unpublished.

[^38]: S.D. Holmes, et al., "132 nsec Bunch Spacing in the Tevatron
    Proton-Antiproton Collider," Fermilab Report No. TM-1920 (1994),
    unpublished.

[^39]: A. Piwinski, "Limitation of the Luminosity by Satellite
    Resonances", DESY 77/18 (March 1977).

[^40]: W. Foster, "Multibatch P-bar Production via Snap Coalescing: A
    Plan to Produce Approximately 5x More Antiprotons/Cycle Using Mostly
    Existing Equipment," Fermilab Report No. TM-1902 (1994),
    unpublished.