Dynamic aperture evaluation at the current working
point for RHIC polarized proton operation
Y. Luo, M. Bai, J. Beebe-Wang, W. Fischer, A. Jain, C. Montag,
T. Roser, S. Tepikian, D. Trbojevic
Presented at the 22nd Particle Accelerator Conference (PAC)
Albuquerque, New Mexico
June 25 - 29,2007
Brookhaven National Laboratory
Upton, NV 1 1973-5000
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Figure 1: Below 0.7: The tune footprints with and without
fR multipoles corrections for on-momentum particles up to
Figure 2: Above 0.7: The tune footprints with and without
IR multipoles corrections for on-momentum particles up to
6ao. The tune spots on the diagonal are due to the failure
in tune searching.
push the horizontal tune of the beam center to the horizon-
tal third order resonance at Q, = 28.666. And for the
working point (Q,, Q,) = (28.695,29.685), the beam-
beam interactions push the vertical tune of the beam center
to the vertical third order resonance at Q, = 29.666.
Tune space above 0.7
Limited by the resonances at 0.7 and 0.75, and consid-
ering the beam-beam tune shift 0.02, two working points
(28.735,29.745) and (28.745,29.735) are used for tune
footprint calculation above 0.7. Fig. 2 shows the tune foot-
prints for these two working points. From Fig. 2, the IR
multipole correction does help single particle's short-term
stability above 0.7, too.
Without the beam-beam interactions, for the work-
ing point (28.735,29.745), the vertical tunes for large
amplitude particles are close to the vertical fourth or-
der resonance at & , = 29.75. For the working point
(28.745,29.735), the horizontal tunes for large amplitude
particles are close to the horizontal fourth order resonance
at Q, = 28.75.
DYNAMIC APERTURE CALCULATION
In this section, the dynamic apertures calculations are
presented for the tune range around the current pp work-
ing point [%]. The tracking is performed to 105 turns. The
6-D tracking code SixTrack is used. The initial relative
energy deviation is 0.0007. The beam-beam interactions
are calculated with the weak-strong model. For each tune
spot, the dynamic apertures are searched in five angles in
the normalized coordinate frame. For comparison, in the
following we only show the minimum dynamic apertures
among these five angles. The dynamic apertures are given
in units of design rms beam size 00. To speed up the dy-
namic aperture search, a binary searching method is used.
Eflect of nonlinear corrections
As we mentioned earlier, we are able to locally correct
the sextupole, quadrupole and skew sextupole multipole
field errors in IR6 and IR8. Comparing the DAs with and
without IR multipole correction, we conclude that the local
IR multipole corrections do improve the dynamic apertures
slightly along the diagonal in the tune space.
Then we check the effect of the nonlinear chromaticity
correction and 3Q, correction. The strengths for the non-
linear chromaticities are calculated with MADX. Based on
our simulation study, there is no significant change in the
dynamic apertures along the diagonal below in the tune
space between 213 and 0.7 with and without nonlinear
The horizontal third order resonance 3Q, is corrected
with the 12 local sextupole correctors in the interaction re-
gions although by now only 4 sextupole correctors have
their power supplies. From our simulation study, the 362,
correction does increase the dynamic apertures when the
horizontal tune is close to 2 13.
Eflect o f tune ripples and b2 in arc dipoles
Tune ripples were observed in RHIC operation and may
reduce the beam dynamic aperture. In the following simu-
lation, the observed tune ripples are artificially introduced
by modulated quadrupoles at IP6. In the los turn DA calcu-
lation, no clear dynamic aperture reduction due to the tune
ripples is observed. Larger turn numbers may be needed
to see their effect. Also the sextupole components in arc
dipoles are inlcuded in the simulation study. There is no
clear change in the dynamic apertures with and without
these sextupole components.
Fig. 3 and Fig. 4 show the dynamic apertures calculated
with and without nonlinear chromaticity correction. In both
plots, the 3Q, corrections, tune ripples, and sextupole com-
ponents in arc dipoles are included.
Tune scan above 0.7
Fig. 5 shows the dynamic apertures in the tune scan
above 0.7 in the tune space. The chromaticities are cor-
rected with the 2-family sextupole scheme. From Fig. 5,
Figure 3: DAs in the tune scan under condition of (IR- Download full-text
errcorr + BB + b2 + 2fam ).
Figure 4: DAs in the tune scan under condition of (IR-
errcorr + BB + b2 + 8fam )
Figure 5: DAs in the tune scan above 0.7 under condition
of (IRerrcorr + BB + b2 +2fam ).
the dynamic apertures reaches its maximum when the un-
collisional horizontal tune Q, is close to 0.745. When the
non-collisional horizontal tune is around 0.72,'the dynamic
aperture reduction is seen.
Detailed lo5-turn 6-D dynamic apertures have been cal-
culated around the current RHIC polarization proton work-
ing point. The initial relative momentum deviation is
0.0007. The model includes the updated IR multipole field
errors and their corrections, the sextupole components in
the arc dipoles, and tune ripples. The second order chro-
maticities and the horizontal 3Q, resonance can be effec-
tively corrected before tracking. Based on the dynamic
aperture calculations, we conclude:
m With the beam-beam interactions at IP6 and IP8,
where we have 4" = 0.9m, and a bunch intensity
Nb = 2.0 x lo1', the dynamic aperture at good tune
spots is about Goo.
The dynamic apertures of particles with the same mo-
mentum deviation 0.0007 are not affected by the non-
linear chromaticity correction.
A correction of the 3Q, resonance driving term h3000
leads to an increase in the dynamic aperture for tune
spots above the diagonal.
There is only a small effect of the inclusion of sex-
tupole field errors in the arc dipoles on the dynamic
There is only a small effect of tune ripple on the dy-
namic aperture up to lo5 turns.
Between 0.667 and 0.7 in the tune diagram, the tune
space is tight to accommodate the beam-beam tune
spread of 0.02 with Nb = 2.0 x 10".
Between 0.7 and 0.75 in the tune diagram, the max-
imum dynamic aperture is reached when the non-
collisional tunes are chosen to be near 0.745. The
non-collisional tunes should not be smaller than 0.72.
[l] W. Fischer, Beam-beam and BTF, 2006 RHIC Accelera-
tor Physics Experiments Workshop, November 2-3, 2005,
 S. Tepikian, private communications, 2005.
 Y. Luo, W. Fischer, S. Tepikian, D. Trbojevic, Online non-
linear chmmaticiw minimization, BNL C-AD AP-Note 263,
 Y. Luo, Nonlinear chromaticities and 3Q, resonance cor-
rections, 2006 RHIC Accelerator Physics Experiments
Workshop, November 2-3,2005, BNL.
 J. Bengtsson, The satupole scheme for the Swiss Light
Source(SLS): an analytical approach, SLS Note 9/97,
March 7, 1997.
 Y. Luo, M. Bai, J. Bengtsson, IV. Fischer, D. Trbojevic, Sim-
ulation of 3Qx Resonance Driving Term Measurement with
AC Dipole Excitation, BNL C-AD AP Note-265, Jan., 2007.
 Y. Luo, J. Bentsson, W. Fischer, and D. Trbojevic, Simu-
lation ofproposed on-line third order resonance correction
schemes, BNL C-AD AP Note-264, Dec., 2006.
 Y. Luo, M. Bai, J. Beebe-Wang, R. Calaga, W. Fischer, C.
Montag, S. Tepikian, D. Trbojevic, Dynamic Aperture eval-
uation at the cui-~nt ulorkingpoint,for RHIC polarizedpm-
ton run, BNL C-AD AP Note-271, April, 2007.