Simulation of the Fermilab Booster using Synergia Page: 2 of 5
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For high precision 3D simulations, large numbers of macroparticles, on the order of 106-7, and
fine space-charge grids, of typical size 33 x 33 x 257, are required. In order to obtain the necessary
computing power for such simulations, we have ported our code to different parallel machines,
including commodity PC clusters, as well as specialized parallel computers. A detailed discussion
of the performance of the Synergia code for FNAL Booster modeling can be found in . The
performance depends on an interplay of networking speed, latency, and processing speed. For
our Booster simulations, peak performance of - 100 Booster turns per hour is obtained running
on 512 processors on the NERSC supercomputer.
2. Fermilab Booster Simulations
The Fermilab Booster  is a rapid-cycling, 15 Hz, alternating gradient synchrotron with a
radius of 75.47 meters. The lattice consists of 96 combined function magnets in 24 periods,
with nominal horizontal and vertical tunes of 6.9 and 6.7, respectively. The Booster accelerates
protons from a kinetic energy of 400 MeV to 8 GeV, using 17 rf cavities with frequency that
slews from 37.8 MHz at injection to 52.8 MHz at extraction. Typically, the injection process
lasts for ten Booster turns, resulting in a total average current of 420 mA. The injected beam is
a stream of bunches equally spaced at the linac RF frequency of 201.2 MHz.
In this section we study how space-charge affects the Booster beam during the first 500 turns
of the cycle (injection, capture, and bunching). For the simulations we use an idealized Booster
lattice without any non-linear elements, but we do employ second order maps and a beam with
realistic energy spread.
2.1. Emittance dilution
First we investigate how space-charge affects the emittance of the Booster without including
rf. In Fig. 1 we plot the normalized 4-D transverse emittancel for five different initial beam
conditions: matched beam without space-charge and with and without a momentum spread
of 0.0003, beam of 0.420 Amps total current and momentum spread of 0.0003, matched and
mismatched. Multi-turn injection of 11 turns was used in all cases, except one (see figure). As
expected, in the cases where the beam was matched there is no emittance growth. (Our matching
procedure takes into account space-charge effects on the second moments of the beam). In the
mismatched cases, with a 20% mismatch, we observe a 12% increase of the beam emittance
during the first 10 to 15 turns after injection. The effect is a combination of chromatic and
space-charge effects and is very similar for both the single- and multi-turn injection cases. The
total current is the same, 0.420 Amps, in both cases. The emittance growth can be related to
the conversion of beam free energy from mismatch oscillations into thermal energy of the beam,
due to the effect of the non-linear space-charge forces . With a mismatch parameter of 1.2, as
in the case of our simulation, the free energy model predicts a 4-D transverse emittance growth
of 13%, in good agreement with the Synergia result.
Including rf in the simulation increases the space-charge effects through bunching and
introduces a stronger coupling between the horizontal (bending) and the longitudinal planes.
The simulation of a typical Booster cycle consists of 10 injection turns (linac beam with 200 MHz
structure), followed by 20 turns of debunching (no rf), then 200 turns of capture. In the capture
process, cavity pairs start paraphased and are brought linearly in phase. In Fig. 2 we show a
longitudinal phase-space slice, 27r wide in the 37.8 Mhz rf phase, after the capture process. The
bunch is "s-shaped", due to space charge and it maintains some structure (uneven density) due
to the "folding" of the injected linac bunches. Fig. 3 shows the effects of the rf on the normalized
4-D transverse emittance: the emittance growth is - 2.5 times larger (30% versus 12%) than in
1 Defined as the square root of the determinant of the covariance matrix of the transverse phase space.
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Spentzouris, Panagiotis & Amundson, James. Simulation of the Fermilab Booster using Synergia, article, June 1, 2005; Batavia, Illinois. (digital.library.unt.edu/ark:/67531/metadc881069/m1/2/: accessed February 18, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.