Potential-well distortion in barrier Rf Page: 2 of 9
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Factories 2003, Stanford, California, October 13-16, 2003
Notice that the first term in the exponent of
Eq. (1), Urf(r) -2/(2a.), represents the lin-
earized rf potential and is an even function of the
particle arrival time r, while the second term,
~ aRNpOr, the cause of the asymmetric solution, is
an odd function of r, representing the perturbation
to the rf potential arising from the interaction of the
beam with the resistive impedance. The reactive
part of the coupling impedance, on the other hand,
distorts the rf potential symmetrically and therefore
contributes only to the lengthening and shortening of
the beam. The longitudinal asymmetry of the beam
is only significant, however, when the second term
is comparable to the first term. Proton beams are
usually long and the bunch spectra roll off before
they reach the broadband resonance. In other words,
proton beams can hardly see the real part of the
impedance. Thus the effective aRN/ 2 is usually
very small and, as a result, no significant head-tail
asymmetry has ever been reported. However, at
the Fermilab Recycler Ring where rf barriers are
used , the rf potential experienced by most part of
the beam is essentially zero. For this reason, head-
tail asymmetry of the beam profile has been observed.
2. THE RECYCLER RF
Broadband cavities are employed to create barriers
of opposite polarities to confine antiprotons . Some
of the merits are:
1. The beam can spread out uniformly, as indicated
in the top plot of Figure 3, so that the space-
charge force becomes smaller.
2. Two batches can be merged easily by moving
them in two separate barrier buckets close to-
gether and then annihilating the two central bar-
riers, as indicated in the lower plot of Figure 3.
3. The length of a batch can be compressed by
moving the two barriers closer together slowly.
4. The whole batch can be moved from one location
to another by moving the two confining barriers
slowly in the same direction.
The are four 50 Q broadband ferrite-loaded rf sta-
tions . The amplifiers are of 3.5 kW from 10 kHz to
100 MHz, capable of supplying a total of 2 kV. The
rf waveform generated is determined by the amplitude
and phase of each of the 588 revolution harmonics.
When the baseline between the two barrier pulses is
nonzero, as shown in the top plot of Figure 4, the bar-
rier potential becomes slanting, as shown in the lower
plot. Such nonzero baseline can come from either rf
errors or the coupling impedance of the vacuum cham-
ber. Here, we are talking about a deviation of ~ 10 V
from zero, out of a total barrier voltage of 2 kV, or
-- - -- - E - -- - -
Figure 3: Top: The trajectory of a proton inside a
barrier bucket set up by two barrier waves with equal
and opposite polarities. Bottom: Two barrier buckets set
up by 4 barrier waves are prepared side-by-side. When
the two central barrier waves annihilate each other, the
two buckets will be merged into one.
Figure 4: When the baseline voltage (top) inside a
barrier bucket set up by two barrier waves of opposite
polarities is different from zero (dashed), the rf potential
well (bottom) no longer has a flat bottom. A beam inside
the potential well therefore no longer has a head-tail
symmetric linear density.
2.1. Nonlinearity in High-Level RF
In the top plot of Figure 5, head-tail asymmetry
is evident for the proton beam in a barrier bucket at
the low intensity of ~ 1 x 1011 with the beam pro-
file leaning towards the head, which is the left in the
display. The Recycler Ring is a permanent-magnet
storage ring at 8.9383 GeV operating below transition.
Since the beam leans forwards, it cannot be because of
the interaction with the resistive coupling impedance,
because particles losing energy will lag behind below
After some investigation, a small parasitic 90 kHz
(revolution harmonic component) sinusoidal compo-
nent was found imposed on the rf vector sum of all
four rf stations . This is equivalent to having a
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Ng, King. Potential-well distortion in barrier Rf, article, April 29, 2004; Batavia, Illinois. (digital.library.unt.edu/ark:/67531/metadc781171/m1/2/: accessed February 19, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.