# Lattice optimization for the stochastic cooling in the Accumulator ring at Fermilab.

V.Lebedev, V.Nagaslaev\*, S.Werkema\
Fermilab, PO Box 500, Batavia IL 60510, USA

*Abstract*

New efforts are under way at Fermilab to increase the rate of antiproton
production. This program includes optimization of machine optics in the
Antiproton Accumulator to improve stochastic cooling. The new lattice
was implemented in May of this year. Results are discussed, as well as
some aspects of model development and lattice measurements.

## Introduction

A broad effort to increase antiproton production for the Tevatron
accelerator complex at Fermilab was initiated in 2005. The goal was to
optimize the performance of all machines in the production chain:
Booster, Main Injector, Debuncher, Accumulator and beam lines, in order
to maximize the flux of antiprotons to the Accumulator. This effort
succeeded in reaching the peak rate of 20 mA/hr in February 2006.

Further increase of the stacking rate was limited by the capability of
the stacktail stochastic cooling system in the Accumulator such that any
further increase in the incoming flux would not result in an appreciable
increase in the antiproton accumulation rate.

The new effort that started after the shutdown of 2006 concentrated
primarily on the stacktail cooling system. Subsequently, there was a
further increase in the peak rate (23mA/hr in April, 2007), but more
importantly, also the average stacking rate. This progress, combined
with very successful improvements in the Fast Transfer Protocol
\[1\]**,** resulted in nearly doubled average weekly production of
antiprotons for the Tevatron in March, 2007.

A significant outcome of this effort was the development of an
integrated physics model of Accumulator stochastic cooling \[2\] that
identified physical and technological limitations of the system, as well
as the way to improve its performance. Here we discuss the optimization
of the Accumulator lattice as suggested by this model, the
implementation of the optimized lattice, and first results.

## Accumulator lattice

The Accumulator has a periodicity of 3, and mirror symmetry in each of 3
sectors. It has 3 straight sections and 3 arcs. The Accumulator lattice
functions are shown in Figure 1. Continuous injection of the antiproton
beam from the Debuncher is maintained using stochastic stacking. Beam
arrives at the injection orbit at an energy that is approximately 140
MeV higher than that of the circulating core beam. 100 msec later the
injected beam is adiabatically bunched and RF displaced to the
deposition orbit, which is approximately at the center of aperture. From
this point it falls under the action of the stochastic cooling force
(Stacktail system) that starts pushing it towards the main core beam (60
MeV below the central orbit energy). A 6D-cooling of the main core beam
is performed by separate core stochastic cooling systems.

Large dispersion in the arcs (10m) separates the beam according to
energy, whereas in straight sections beams of all energies are merged
together and compressed in order to fit into the very narrow aperture of
the stochastic cooling tanks. Beam focusing and flattop dispersion in
the arcs are maintained by the quad quadruplets on each side of the
small straight sections inside the arcs. These high dispersion sections
house extraction/injection kickers and the momentum stochastic cooling
pickups. In the long straight sections the dispersion is cancelled at
the small bend magnets on each side**.** These low dispersion sections
accommodate stochastic cooling kickers, RF cavities, a DCCT
transformer**,** and dampers.

![](media/image1.jpeg){width="2.89375in" height="1.8993055555555556in"}

Figure 1. Accumulator Twiss-functions. Red and green traces show the
horizontal and vertical beta-functions**.** The horizontal dispersion is
shown with the blue trace.

It is important to keep dispersion as low as possible in the long
straight sections**.** Any residual dispersion here would couple the
longitudinal kicks of the stacktail kickers into the transverse
dimensions causing transverse heating of the beam.

## lattice optimization

### Objectives

The main objective for the lattice optimization was to increase the slip
factor (η). This would directly help the stack tail cooling as the
maximum flux is proportional to η \[2\]:

$$J_{\text{max}} = |\eta|T_{0}W^{2}x_{d}$$

The maximum useful value of  η is limited by overlapping schottky bands
at high harmonics ("bad mixing"), but there is still some room (15-20%)
available, according to the model of the stacktail cooling \[2\].

Another benefit of raising the slip factor is additional separation of
the stacktail and core revolution frequencies**.** This would mitigate
the effects of resonant heating of the core. It has been found recently
that the electrical center of the core momentum kickers depends on
frequency and shifts to about 2mm at 3.3GHz \[2\]. This heats particles
in the core whose transverse sidebands overlap with the stacktail
harmonics at 3.3 GHz. Figure 2 shows the calculated emittance growth
rate for a particle with a given revolution frequency. The vertical dash
marker shows the center position of the core particles. Separation of
the core and stacktail would better center the core between the two
sideband peaks, thus reducing the heating of the core particles.

Great attention has been paid to the dispersion reduction in the long
straight sections. As the standard mults affect the slip factor, they
are only used for the fine

![](media/image2.wmf){width="2.94375in" height="2.182638888888889in"}

Figure 2. Emittance growth rate versus the particle frequency due to
resonant heating at 3.3GHz. Vertical dash line shows the location of the
core. Horizontal dash line shows average core heating rate due to this
effect.

tuning, and the main corrections were made within the lattice change.

It was very desirable also to increase the design machine aperture to at
least 14-15π mm-mrad. Presently the machine admittance is limited at
8-10π mm-mrad and is not very well understood. Compressing the beam size
at most narrow locations may improve the admittance.

### Lattice model

For optics calculations in the Accumulator ring we utilized a model
based on the OptiM program \[3\]. The model parameters were fit to the
measured data taken in May, 2007. The method is similar to that we used
earlier for the Debuncher ring \[4\].

Model calibration data was taken as difference between positive and
negative closed orbit bumps produced by each corrector magnet in the
machine. In this case small orbit drifts during the measurement cancel
out. As the orbit measurement precision is crucial for the calculations,
each orbit was sampled 15-20 times. The combined response at each Beam
Position Monitor (BPM) to the excitation of each corrector determines
the Response Matrix (RM). For the dispersion measurement a revolution
frequency scan of 5 steps around the equilibrium orbit was taken making
the maximum variation of momentum equal to 0.1%. Figure 3 shows these
measurements for the BPMs located at low dispersion regions. The
estimated sensitivity of this method is about 1mm in dispersion units.
The whole process of measurement is automated and takes about 1-1.5
hours.

The data have been analyzed using the *SRLOCOFitting* program. This
package has been developed at ANL \[5\] and recently adopted at FNAL
\[6,4\]. It fits the model parameters to minimize the differences
between the calculated and the measured RMs.

The main difficulties of lattice fitting are the limited precision of
the BPMs and a very limited number of the correction elements in the
Accumulator. In this case the Singular Value Decomposition (SVD)
algorithm used by *SRLOCOFitting* turned out to be very efficient.

![](media/image3.jpeg){width="2.7888888888888888in" height="2.15625in"}

Figure 3. Dispersion measurement as a slope of orbit displacement with
momentum varied within ±0.1%.

The average residual rms on the response matrix fit is 10-15 micron.
From this we estimate the accuracy of the beta-function calculations to
be about 5% or better. The method was used not only to find out the quad
errors, but also to determine separately instrumentation errors (like
BPM calibration and trim magnet strength corrections). Also, many other
variables can be included in the fit, as long as they can serve as model
parameters.

### Optimization

In order simulate the desired solution for the new lattice we have
started with the existing model with the new fitted parameters in the
beam line mode. In this case one can model sequential changes that shape
only the downstream lattice functions. At the end of the process, when
reaching the end of the line, final functions have to be matched to
functions at the beam line beginning to satisfy the closure condition.
This procedure is repeated iteratively until an acceptable solution is
reached. Flattop dispersion in the arcs is constrained to be kept at the
same level. In order to increase the slip factor we varied the
dispersion function in the negative wells, see Figure 4.

![](media/image4.png){width="2.8534722222222224in" height="1.90625in"}

Figure 4. Dispersion corrections made to increase the slip factor. Red
trace shows the new dispersion and blue data points correspond to the
old lattice. Dispersion is changed in the negative wells and preserved
at the flattop.

The slip factor has been increased by 15% in the final lattice design
(0.012  0.015). Also we reduced the beam size at 6 most narrow
locations (4 of them in horizontal plane and 2 -- in vertical plane).
This brought the minimum design aperture from 11 π up to 15 π. The new
design also has smaller average beta-functions which reduces the IBS
heating term by 12%. Stochastic cooling pickup to kicker phase advances
were corrected also, although those were already fairly close to
required values.

### Results

Direct implementation of the new lattice design was complicated due to
an additional operational step that we had to do in parallel. We had to
change the bus cycling protocol, because it contained the historically
obsolete procedure of double lattice ramping. So, the hysteresis
protocol was not the same as for the old lattice. However it turned out
that the difference was limited to just tune corrections, so we
proceeded smoothly after this was taken into account.

The new slip factor was measured to be very close to the design one.
This was proved by using two independent methods of slip factor
measurement. One method was the direct measurement of gamma-t factor
from the measurement of revolution frequency response to bend bus
variations**;** and another method extracted the slip factor from the
synchrotron frequency dependence on the RF voltage.

Because the requirement on dispersion suppression in the straight
sections is so tight (\<5cm, compared to the flattop dispersion of
10m!), it took 2 iterations of dispersion corrections. In the final
measurement this dispersion was made lower than 2.5 cm. Final
measurements have also shown that the minimum design aperture in the
Accumulator ring has increased to 15 π.

The new lattice was implemented in the Accumulator and made operational
as of May 16 this year. The observations of its immediate effect on
stacking are not

![](media/image5.png){width="3.017361111111111in"
height="2.0569444444444445in"}

Figure 5. Accumulator transverse emittances. The break point corresponds
to the day of the new lattice implementation.

very conclusive because it was made concurrently with other changes in
the stack tail system as well as temporary degradation of the incoming
flux to the Debuncher. A better evaluation will be available later this
year at the completion of the whole project. However a very important
immediate effect of new lattice was in the substantial reduction of the
transverse heating. Figure 5 shows average core emittances within a
period of one month before and one month after the optics change.

High transverse emittances affect the efficiency of antiprotons
transport to Recycler and, therefore, the Tevatron operations. Stacktail
performance is also constrained by the core heating, so reduction of
emittances gave more freedom for the stacktail optimization.

### Conclusions

The new model of the Accumulator lattice has been developed using the
OptiM and *SRLOCOFitting* software. Based on this model lattice
optimization has been performed in order to help stochastic cooling. The
new lattice design was implemented and subsequent measurements have
shown that design parameters were achieved. Substantial reduction in the
core transverse heating has been observed.

## References

\[1\] K.Gollwitzer, Antiproton Source. DOE Review report, March 2007.

\[2\] V.Lebedev, Antiproton Production and Accumulation. COOL'07, Bad
Kreuznach.

\[3\] [http://www-bdnew.fnal.gov/pbar/organizationalchart/
lebedev/OptiM/optim.htm](http://www-bdnew.fnal.gov/pbar/organizationalchart/ lebedev/OptiM/optim.htm)

\[4\] V.Nagaslaev et al., Measurement and optimization of the lattice
functions in the Debuncher ring at Fermilab.

EPAC'06, Edinburgh.

\[5\] V.Sajaev and L. Emery, "Determination and correction of the linear
lattice of the APS Storage Ring", EPAC2002, Paris, France. P742-744

\[6\] V.Lebedev et al., Measurement and correction of linear optics and
coupling at Tevatron complex. ICAP-04, St. Petersburg; NIM A558 (2006),
p299.