# Measurement and optimization of the lattice functions in the Debuncher ring at Fermilab\*.

V. Nagaslaev^\#^, K. Gollwitzer, V. Lebedev, A. Valishev -Fermilab, PO
Box 500, Batavia IL 60510, USA

V. Sajaev - ANL, Argonne, IL 60439, USA

Abstract

Tevatron Run-II upgrade requires substantial increase of antiproton
production. The central step towards this goal is maximizing the
Debuncher ring admittance which necessitates detailed understanding of
the Debuncher optics and aperture limitations. The method of the
response matrix optimization has been used to determine quadrupole
errors and to build a model of machine optics. We estimate that the
model predicts beta-functions with accuracy of about 5% mainly limited
by Beam Position Monitor system resolution and small number of steering
elements in the machine. The improvements of optics model were used to
redesign Debuncher optics so that the beam envelopes would be minimized
at regions with small aperture.

## Introduction.

The Debuncher is the first of two Antiproton source. The goal of the
Debuncher is to reduce a very large phase space of the beam in all 3
dimensions in order to fit acceptance of the second ring, the
Accumulator. First, the momentum spread is reduced using RF bunch
rotation and adiabatic debunching. During the rest of Debuncher cycle
(about 2 sec) the beam is cooled using transverse and longitudinal
stochastic cooling.

The best production rate of the Antiproton source in Run-I was 7∙10^10^
hour^-1^. In 2005 peak stacking rate reached 17∙10^10^ hour^-1^, which
is still well below of 30∙10^10^ hour^-1^ anticipated by the Run II
upgrade plan. Therefore, massive efforts were taken in the end of 2005
to raise the stacking rate and the work described below was part of
these efforts. The acceptance of the Debuncher ring was one of few
critical places to be improved. The best measured horizontal/vertical
ring acceptances in 2005 were 30/25 mmmrad whereas the acceptances
corresponding to the physical aperture were 34/31mmmrad. The goal of the
Debuncher acceptance upgrade was to reach 35mmmrad in both planes.
Earlier simulations \[1\] have shown that together with an upgrade of
Lithium lens this would gain about 60% for antiproton yield. Initially,
the Debuncher upgrade plan anticipated an increase of physical aperture
of two stochastic cooling tanks that have had major aperture limitation.
After lattice measurements and analysis were performed we found that
Debuncher optics modification would be an easier way to achieve the
required acceptance. It also eliminated risks of deterioration of
stochastic cooling systems.

## Debuncher lattice

The Debuncher ring has a periodicity of 3, and mirror symmetry in each
of 3 sectors. It has 3 straight regions and 3 arcs. The magnet structure
consists of 57 FODO cells each with ≈60^o^ phase advance. Straight
sections accommodate stochastic cooling tanks, RF cavities and
injection/extraction septa. The regularity of FODO structure is only
slightly perturbed in straight sections. This is important for
maximizing the dynamic aperture and acceptance of the ring. Figure 1
shows the beta-functions and the dispersion in the ring.

![](media/image1.wmf){width="3.040277777777778in"
height="1.417361111111111in"}

Figure 1. Beta-functions (red line -- horizontal, green line --
vertical) and dispersion (blue line) in Debuncher. The plot starts at
antiproton injection kicker. Horizontal units are meters.

Natural chromaticity is compensated by two families of sextupoles
bracketing all arc quads. There is also a considerable non-linearity
introduced by the dipoles. In order to include those nonlinearities into
optics model, we fit the model predictions to the measured tune
dependence on momentum and found that the sextupole and decapole
components make major contributions (B~2~/B~0~=-3.6∙ 10^-5^ cm^-2^,
B~4~/B~0~=1.6∙10^-5^ cm^-4^). Figure 2 presents these data and the fit.
Sextupole correction has been applied to make the tunes flat in the
central part of the region shown. Figure 3 shows calculated variation of
beta-functions and dispersion with momentum. As one can see they have
moderate dependence on momentum and, consequently, have little effect on
the Debuncher momentum acceptance. Direct tracking simulations have
shown negligible effect of machine non-linearities on the Debuncher
acceptance for both on- and off-momentum particles.

The free drift space in the straight sectors is very tightly packed with
stochastic cooling devices. In order to accommodate them, some of the
dipole correctors were removed from the ring. There remain 10 horizontal
and 7 vertical correctors in the entire Debuncher.

![](media/image2.wmf)

Figure 2. Dependence of tunes on momentum computed using Debuncher
optics model (solid lines) and measured (crosses).

Major aperture limitations are the stochastic cooling pickups and
kickers for bands 3 and 4, and the injection kicker. The maximum
momentum spread is limited by scraping in arcs at high dispersion
regions (D2.2 m) and is about 0.03. Additionally, the extraction kicker
limits the momentum spread for particles with large betatron amplitude
resulting in 7% decrease in number of captured antiprotons.

![](media/image3.png)

### Figure 3. Calculated variation of beta-functions and dispersion with momentum; curves present data for Dp/p = -0.029, 0 and +0.029. Only the first third of Debuncher ring is shown. 

### Lattice measurement method

Lattice measurement and calculation are made via fitting of the response
matrix (RM) -- the Beam Position Monitor (BPM) vector response to closed
orbit excitation produced by each of the corrector elements \[2\].
Finding the best model for the lattice description is equivalent to
finding the best fit between the calculated and measured response
matrices. For the model calculations we used the OptiM program \[3\].
This program supports 6D calculations for the coupled motion and the
nonlinear transport. The *SRLOCOFitting* program was used for the fit.
This program was developed at ANL and then successfully extended to the
Tevatron \[4\] and to the Debuncher. The parameters of the model that
were varied included individual BPM and trim gains, rolls, quadrupole
gradient errors and power supply error factors. The difficulties of
accurate lattice measurements in the Debuncher are related to the
limited accuracy of BPMs and the small number of correctors.
*SRLOCOFitting* is using the robust method of Singular Value
Decomposition (SVD), which is beneficial in our case of low data
redundancy.

### Measurement, data analysis, accuracy

Orbit measurements in the Debuncher were done using reverse protons
injected from the Main Injector via Accumulator. As efficiency and
stability of the fit critically depends on the measurement accuracy, the
precision was the main focus of data taking. Typical BPM resolution in
the Debuncher is about 50 m with a sampling time a little above 1 sec.
To get better resolution we took large number of samples (20-25). BPM
response had been recorded as a difference of positions between positive
and negative kicks for each corrector. This approach was used to
minimize such effects as slow machine drifts, BPM nonlinearities and
hysteresis-like effects. For the dispersion measurements, the RF
frequency was scanned in 5 steps equivalent to Dp/p=0.1% between steps,
and orbits recorded for each step. The dispersion was then computed as
the slope of a straight line fitting the data. As the data taking
application runs on the controls network and analysis is done on the
LINUX farm, application creates the data in a format suitable for
*SRLOCOFitting* and then that data are transported to the farm. The
difference between the measured and calculated orbits before and after
the fit is shown in Figure 4. Typical residual errors of the final
response matrices are 10m.

![](media/image4.png){width="2.6527777777777777in"
height="2.0430555555555556in"}

Figure 4. Measured and calculated orbit comparison before and after the
RM fitting. Green line shows the residual error of the fit at each BPM.

The key point of the SVD method is to exclude degenerate combinations by
means of removing (zeroing) small Singular Values (SV). One has to
decide on the threshold of SV rejection. Cutting out too many
eigenvectors will make the solution space incomplete and resolution will
suffer. Leaving in too small SVs will make the solution unstable. Figure
5 shows the plot of SVs arranged in the descending order. On the same
plot the residual RMS in X and Y planes and the RMS of the quad errors
are also shown versus the number of SVs left in. This shows that the
solution is surprisingly stable for a wide range of selected thresholds.
We chose to set the number of selected SVs at 450 which corresponds to
minimal SV\~1.

The accuracy of resulting beta-functions was estimated using simple and
naive calculations from the residual RMS to be 3%. A more solid way to
do this, although still an estimate of the lower limit was to compare
computed new functions with those measured the same way after the
lattice change. This also results in \~3% in RMS difference.

![](media/image5.png){width="2.4381944444444446in"
height="1.667361111111111in"}

Figure 5. Singular values (in descending order). Shown are also residual
RMS and quad errors RMS depending on the number of SV selected.

![](media/image6.wmf){width="2.9694444444444446in"
height="0.9270833333333334in"}

![](media/image7.wmf){width="2.952777777777778in"
height="0.9680555555555556in"}

Figure 6. Beam envelopes in the AP-10 (top) and AP-30 (bottom) straight
sectors for acceptances of 40.5 and 37.5 mmmrad for horizontal and
vertical planes, correspondingly; red line -- horizontal beam envelope,
green line -- vertical beam envelope, and blue line -- the contribution
to beam size due to momentum spread of 0.03. Vertical lines present
aperture limitations with color corresponding to the curve of the same
plane.

### Lattice optimization

Original optics design had regular behavior for beta-functions, and the
stochastic cooling tanks with smallest aperture were the major aperture
limitation. We modified the optics so that to minimize the beam size in
these tanks. The acceptance increase was limited by the beam size growth
in larger aperture tanks. Lattice changes were also constrained by
preservation of the tunes (9.764, 9.785) and the slip factor (0.006).
This optics modification should result in Debuncher acceptances of 40.5
and 37.5 mmmrad for horizontal and vertical planes, correspondingly.
These numbers are greater than the apertures of the Run II design.
Additionally, it also resulted in better conditions for the stochastic
cooling due to beam size increase in large aperture tanks. Figure 6
presents the beam envelopes and physical apertures in the AP-10 and
AP-30 straight sections where the stochastic cooling pickup and kickers
are located. One can see that beam envelope nicely fits within the
aperture limitations.

The implementation of new optics was strongly supported by our ability
to perform high precision optics measurements. It also required
reconfiguration of quadrupole shunts so that the independent current
regulation would be possible for each quad.

### Aperture increase

The new lattice has been implemented in the Debuncher in the end of the
last winter study period in January 2006. This resulted in the best
achieved acceptance of 35.3/34.6mmmrad, which reaches the design goals.
Therefore we do not need to do costly stochastic cooling tanks upgrade
with potentially negative impact on their performance. Increasing the
Debuncher admittance was only a part of a big effort that resulted in a
new stacking rate record of 20.110^10^ hr^-1^ set in February 2006. Its
major significance is opening the road for new substantial production
improvements. A number of issues need to be solved in order to get the
full benefit of the larger Debuncher aperture, such as beam line and
injection channel acceptance, larger momentum aperture in the Debuncher
extraction kicker and stochastic cooling performance. All those issues
were planned to be addressed during the 2006 Tevatron complex shutdown
and the year after that.

## Conclusions and future plans

A successful Debuncher lattice measurement and optimization has been
performed which resulted in a substantial opening of the Debuncher
physical aperture. It brings a potential of 60% increase in the
antiproton yield after corresponding optimization in the beam lines and
injection channel. It also eliminated the need to make an upgrade of the
stochastic cooling tanks. Still new degrees of freedom were added to the
lattice design after adding more quad shunts during the 2006 shutdown.
We will try to use it to compensate the insufficient aperture for
off-momentum particles in the extraction kicker which upgrade has been
postponed beyond the shutdown.

## References

1\. V. Lebedev. Production and collection of antiprotons. Fermilab-Pbar
note 666.

2\. W.J. Corbett, M. Lee and V. Ziemann. Proceedings of the PAC'93,
Washington, DC, p. 108 (1993).

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

V. Sajaev, *et al.*, "Fully Coupled Analysis of orbit response matrices
at the FNAL Tevatron". 2005 Particle Accelerator Conference, Knoxville,
Tennessee, p3662-3664.