Collective Instabilities in the Tevatron Complex

Valeri Lebedev and Alexey Burov

FNAL, P.O. Box 500, Batavia, IL 60510, USA

**Abstract.** Although the luminosity growth for Tevatron Run II was
slower than expected, steady growth of luminosity has been demonstrated
during the last three years with the peak luminosity of 1.02×10^32^
cm^-2^s^-1^ achieved in July 2004. Suppression of instabilities has been
a valuable contributor to the luminosity growth. This report discusses
the transverse instabilities in the Tevatron and Recycler.

# 1. Introduction {#introduction .INTRODUCTION}

The luminosity growth demonstrated during Tevatron Run II has been
achieved through large number of improvements for each of circular
machines, as well as for the linac and multiple transfer lines. Although
each improvement separately did not produce significant luminosity
growth, altogether they contributed more than tenfold luminosity growth
during Run II started in the middle of 2001. Five of six rings (except
Debuncher) have sufficiently large beam intensity and related problems
with beam stability. The instabilities of almost every type are present
there: single and multi-bunch, transverse and longitudinal, due to
electromagnetic interaction with vacuum chamber and due to interaction
with ions stored in the beam. In this report, we discuss only transverse
instabilities in Tevatron and Recycler.

# 2. Head-tail instability in tevatron and its suppression

Transverse head-tail instability has been a long-standing problem for
the Tevatron. It forced to keep high chromaticities, yielding reduced
beam lifetime. A detailed experimental study was presented in Ref.
\[1\]. The instability was diagnosed as the single-bunch instability,
although some couple-bunch contributions were not being excluded.
Experimentally, it was not clear whether it was a weak or strong
head-tail case. Calculations showed that it was at the margin. The
problem has been alleviated after two step reduction of Tevatron
impedance in 2002 and 2003.

## 2.1. Source of the Impedance

Instability growth rate was measured at beam injection. To make it more
pronounced, the chromaticities were set to small negative values.
Monte-Carlo single-bunch code with resistive wake and Gaussian beam
distribution was used to compute the impedance from the measurements. It
yielded the transverse resistive impedance $Z \geq 4\ M\frac{\Omega}{m}$
at 100 MHz. It was expected that the vacuum chamber resistivity should
make the main contribution. The measured value was about 4 times higher.
Consequent analysis has shown that this difference was related to the
contribution of laminated Lambertson magnets. Their impedance can be
calculated by integrating the resistance over the low frequency current
passing along the laminations, leading to \[2\]:

$Z = (1 - i)\frac{\text{gZ}_{0}L}{2{\pi a}^{3}}\frac{c}{\sqrt{2\text{πσω}}}\frac{2a\sqrt{\mu}}{d}$
. (1)

Here *a* is the chamber radius, or a half-gap for a flat chamber, *L* is
the magnet length, *d* is the lamination thickness, *μ* is the magnetic
permeability, *g* is the geometrical Yokoya factor (*g* = 1 for round
chambers), and $Z_{0} = \text{377}\ \Omega$.

When a reason of the instability was understood, one Lambertson magnet
(unused) was removed, and another (injection magnet) was shielded.
Presently, the transverse impedance is dominated by wall resistivity of
the main vacuum chamber (stainless steel), so the total impedance was
reduced by a factor of 3-4.

## 2.2. Single and Coupled Bunch Contributions

Within the air-bag (hollow-bunch) approximation, beam transverse modes
are characterized by single-bunch, or head-tail, number *l*, and a
coupled-bunch number *m*. The head-tail motion is driven by
high-frequency impedance, at $\omega \geq \frac{c}{\sigma_{z}}$, or
$f \geq \text{50}$MHz for the Tevatron, while the coupled-bunch motion
is related to much lower frequencies, approximately between the
revolution frequency and bunch-to-bunch frequency,
$\omega_{0} \leq \omega \leq {N\omega}_{0}$. A coherent tune shift of a
given mode linearly depends on the impedance. Because the single- and
coupled-bunch contributions are driven by the well separated frequency
ranges of the impedance, the total growth rate is a sum of single and
multi bunch rates:
$\Lambda_{\text{lm}} = \Lambda_{l}^{s} + \Lambda_{\text{lm}}^{c}$.
Following \[3\],

$\begin{array}{r}
\Lambda_{l}^{s} = - \frac{N_{b}r_{0}}{2{\pi Z}_{0}\text{γν}_{\beta}}\int_{- \infty}^{\infty}{d\omega}\text{Re}Z(\omega)J_{l}^{2}(\frac{{\omega z}_{0}}{c} - \chi)\ , \\
\Lambda_{\text{lm}}^{c} = - \frac{\text{NN}_{b}r_{0}\omega_{0}}{2{\pi Z}_{0}\text{γν}_{\beta}}J_{l}^{2}(\chi)\sum_{p = - \infty}^{\infty}{\text{Re}Z\left( \omega_{0}\left( \nu_{\beta} + \text{pN} + m \right) \right)}\text{.}
\end{array}$ (2)

where is the head-tail phase,$z_{0}$ is the longitudinal amplitude, and
all other notations are conventional. The two contributions are
comparable for the Tevatron. Their behavior with the chromaticity is
shown in Figure (1), assuming the resistive wall impedance. At low
chromaticity end, single-bunch contributions clearly dominate.

![](media/image1.wmf){width="3.066666666666667in"
height="1.6409722222222223in"}

Figure 1. Single-bunch (solid) and the most unstable coupled-bunch
(dashed) growth rate contributions to *l*=1 (red), *l*=2 (green) and
*l*=3 (blue) head-tail modes, shown versus the chromaticity. Units for
the rates are arbitrary.

## 2.3. Damper Optimization

A transverse bunch-by-bunch damper was built before the problem with
Lambertson magnets was resolved. It is still used during beam injection
and acceleration. Its scheme is sketched in Fig. 2. Use of this damper
allowed decreasing the chromaticity from

![](media/image2.jpeg){width="3.067361111111111in"
height="0.8541666666666666in"}

Figure 2. Transverse damper layout; signal (blue) is mixed with the 53
MHz RF frequency, integrated by the low-pass filter, delayed 1 turn,
mixed again with 53 MHz, and applied to the kicker.

![](media/image3.wmf){width="3.0347222222222223in"
height="1.70625in"}Figure 3. Instability rates (dashed) and damping
rates (solid) versus chromaticity for the same modes as Fig.1. The mode
*l*=1 gets unstable at the chromaticity \~ 4. The mixing phase shift
$\theta = 0$.

\~6 to \~4 units (un-normalized), but at lower chromaticity the beam was
still unstable.

For air-bag distribution, the damping rates are:

$\Gamma_{l} = g\ \left| J_{l}(\chi + q)\ e^{i\theta} + J_{l}(\chi - q)e^{- i\theta} \right|^{2},$
(3)

where *g* is a constant related to the amplification coefficient,
$q = \omega_{\text{RF}}\frac{z_{0}}{c}$. The parameter $\theta$ is the
phase shift of mixing frequency determined to be
$\ \text{sin}(\frac{\text{qz}}{z_{0}} + \theta)$ at the pickup, and
$\ \text{cos}(\frac{\text{qz}}{z_{0}} + \theta)$ at the kicker. Such
phase correlation guarantees positive sign of the introduced damping
rate for all the modes. Choice of the phase shift $\theta$ is a matter
of optimization. Until now the damper has been operated for only
$\theta = 0$ and as one can see from Eq. (3) it damps both the dipole
and head-tail modes. The second was not anticipated at the design stage
and came as a pleasant surprise. A comparative behavior of the
instability rate $\Lambda_{\text{lm}}$and the damping rate $\Gamma_{l}$
for the first 3 head-tail modes is shown in Fig. 3. Note that at low
chromaticity, $\chi \leq 1$, the instability dominates over damping:
$\Lambda_{l} \propto \chi$, while $\Gamma_{l} \propto \chi^{2}$for odd
modes and $\theta = 0$; and the main stopper is *l*=1 mode. Thus, for
$\theta = 0$ the threshold chromaticity can be reduced but the
instability cannot be suppressed at low chromaticities. Both statements
agree with the observations. The opposite
choice,$\theta = \text{90}{^\circ}$, would be about as bad, killing
damping of dipole mode and making *l*=2 mode the low chromaticity
stopper. The optimum should be expected somewhere in the middle. The
same comparison is shown in Fig. 4 for $\theta = \text{30}{^\circ}$.
Such choice damps dipole and all head tail modes and can drastically
improve beam stability.

![](media/image4.wmf){width="3.063888888888889in"
height="1.7229166666666667in"}

Figure 4. The same as Fig. 3, except $\theta = \text{30}{^\circ}$.

# 3. Resistive wall instability in Recycler

Recycler is 8 GeV storage ring which is presently under commissioning in
Fermilab. It was built as additional antiproton accumulator to boost the
antiproton production and their total intensity. Both the stochastic and
electron cooling are planned to be used. Fully operational Recycler
should increase both the antiproton production rate and the maximum beam
intensity by about 4 times so that 610^12^ antiprotons would be
accumulated after \~15 hours storage. The ring circumference is 3328 m,
and the betatron tunes are *~x~*/*~y~*=25.58/24.42. To prevent the
accumulation of ions produced by the beam from the residual gas and to
decrease the beam longitudinal emittance the Recycler beam is stored in
one long (\~600-1500 m) bunch kept by the barrier bucket RF. It creates
the potential well with rigid walls at bunch ends making the rectangular
longitudinal density distribution.

## 3.1. Experimental measurements

Recent measurements, performed with beam intensity of about 10^12^,
exhibited that if the machine chromaticity is close to zero and the beam
is sufficiently cold there appears a transverse instability. This
instability causes particle loss but is followed by a comparatively
little emittance growth; so that after beam stabilization the rms beam
emittance is more than an order of magnitude smaller than the vacuum
chamber acceptance. This and a few other peculiarities forced us to
carry out detailed studies of the instability.

![](media/image5.wmf)

Figure 5. Signals from longitudinal and transverse pickups for three
consecutive turns; number of particles -- 10^12^, time of one full
revolution is presented.

A fast digital oscilloscope connected to the sum and differential
outputs of vertical pickups and triggered by the beam instability
recorded a continuous signal for almost 90,000 turns with 128 ns
sampling time. Figure 5 presents data for three consecutive turns. One
can clearly see the betatron motion wave running along the bunch. To
separate the betatron and head-tail motions the betatron phases and
amplitudes along the bunch were computed from each three consecutive
turns. The results are shown in Figure 6. The amplitude of the motion
grows from the head to the tail, reaches its maximum at \~2/3 of the
bunch length, and then decreases. The shape of the amplitude waveform
and the betatron phase advance along the bunch are not changed in the
course of instability development pointing out that only one unstable
mode dominates the instability. Figure 7 presents changes in the
amplitude and phase for fixed point in the bunch. The accuracy of phase
calculations is so good that the phase changes due to tune modulation at
power line frequencies are clearly seen. Fitting yields amplitudes of
tune modulation to be 4.610^-4^ and 3.210^-4^ for the first and third
power line harmonics, correspondingly.

## 3.2. Theoretical model

Because of large bunch length and small momentum spread the synchrotron
motion is much slower than the instability. For Recycler the synchrotron
motion period is \~1 s, while the instability growth time less than 20
ms. In this case one can neglect the longitudinal particle displacement
in the course of instability development. The instability would not
develop in the absence of the tail-to-head feedback. It is created by
the wake-fields delayed by one turn. The Recycler transverse impedance
is dominated by the resistive wall impedance with slowly decaying wake,
$\sqrt{s}$. Neglecting the particle longitudinal motion and the spread
of betatron tunes one obtains the motion equation in the beam frame:

$\frac{d^{2}x(s,t)}{\text{dt}^{2}} + \left( \omega_{b^{2}} - \frac{e^{2}N_{b}}{M\gamma}\overset{\sim}{D}(s) \right)x(s,t) = \frac{e^{2}N_{b}}{M\gamma}\overset{\sim}{W}(s)\ \text{.}$(4)

Here $\overset{\sim}{W}(s)$ and *D*(*s*) represent the integral effect
of the normal and detuning (quadrupole) wakes:

![](media/image6.wmf)

Figure 6. Amplitudes and betatron phases (referenced to the bunch head)
along the bunch for turns 2000, 4000 and 6000.

![](media/image7.wmf)

Figure 7. Dependence of betatron amplitude and phase on the turn number.

$\begin{array}{r}
\overset{\sim}{D}(s) = \int_{s}^{L}{D(s^{'} - s)\rho(s^{'})ds^{'}} + \sum_{n = 1}^{\infty}{\int_{0}^{L}{D(s^{'} + \text{nC} - s)\rho(s^{'})ds^{'}}}\ , \\
\overset{\sim}{W}(s) = \int_{s}^{L}{W(s^{'} - s)\rho(s^{'})x(s^{'},t - \frac{s^{'} - s}{v_{0}})ds^{'}} + \\
\ \sum_{n = 1}^{\infty}{\int_{0}^{L}{W(s^{'} + \text{nC} - s)\rho(s^{'})x(s^{'},t - \frac{s^{'} + \text{nC} - s}{v_{0}})ds^{'}\ \text{.}}}
\end{array}$Recycler has an elliptical vacuum chamber with large aspect
ratio, which impedance is close to the impedance of flat chamber. For
frequencies of interest, the impedance is dominated by wall resistivity
yielding horizontal and vertical wakes:

$\begin{array}{r}
W_{v}(s) = 2W_{h}(s) = \frac{\pi C}{6a^{3}}\sqrt{\frac{v_{0}}{\sigma\ s}}\ , \\
D_{v}(s) = - D_{h}(s) = \frac{\pi C}{\text{12}a^{3}}\sqrt{\frac{v_{0}}{\sigma\ s}}\ \text{.}
\end{array}$ , (5)

The solution of Eq. (4) was carried out numerically for 600
macro-particles and the vertical resistive wall impedance[^1] of Eq.
(5). The results are presented in Figure 8. There is good agreement with
analytical model for continuous beam (*B*=1) and small mode numbers (n
10). Divergence for large mode numbers (two bottom curves) is related to
insufficient number of particles per oscillation length. For fixed
number of particles, bunching of the beam increases the growth rate. For
small bunching factor, *B*, the rate is $\frac{1}{B^{\frac{1}{4}}}$, in
agreement with a two-particle model. Figure 9 presents structure of the
first three unstable modes. The structure of the lowest mode is close to
the measurements presented in Figure 6. The measured and computed growth
rates for this most unstable mode coincide within 10-20%. That verifies
that at low frequencies the impedance is dominated by the

Figure 8. Instability growth rates of unstable modes for different
bunching factors, *B*; *N~b~* = 610^12^*.*

![](media/image8.wmf) Figure 9. Amplitudes, \|(*s*)\|, and phases,
(*s*), along the bunch for first 3 unstable modes.

Figure 10. Structure, Re((*s*)), and spectrum of mode *m* = 101, B =
0.3.

wall resistivity of stainless steel vacuum chamber. Note that the
detuning wake makes the betatron tunes dependent on particle position
along the bunch. That results in achieving the peak in the amplitude at
about 2/3 of bunch length. For the round vacuum chamber the detuning
wake is zero and simulations yield that the maximum amplitude is
achieved at the bunch tail. Every next mode has the head to tail
betatron phase advance increased by 2 and the more peaked amplitude.
Figure 10 shows the mode 101 and its Fourier spectrum. One can see that
the frequency of oscillations grows from head to tail bringing
additional increase of the spectrum width. For *B*\<\<1 and *n*\>\>1 the
maximum of spectral density is achieved at frequency

$f_{n} \approx \frac{2\text{.}5n}{B}f_{0}$ . (6)

For a given mode number, comparing to the continuous beam the bunched
beam has an increased instability growth rate and an increased
oscillation frequency (measured at pickup). Both of them do not benefit
the instability damping.

## 3.3. Requirements for instability damping

The betatron tune spread

${\Delta\nu}_{n} \approx \left( \eta\ n - \nu^{'} \right)\frac{\Delta p}{p}$
(7)

gives rise to beam stabilization, Landau damping. In the absence of the
space charge, all modes with n 3 would be stabilized. However, the
required phase space of the antiproton beam is so small that the space
charge (incoherent) tune shift is expected to be quite large *~sc~*
0.03 - 0.1. It will suppress the Landau damping up to very high
frequencies

${\Delta\nu}_{n} \approx \frac{{\Delta\nu}_{\text{sc}}}{F_{\text{sc}}}\ ,\ F_{\text{sc}} \approx 3\  - \ 6$
. (8)

As a result, the required frequency band of the instability damper goes
to well above 100 MHz.

#  Discussion

Presently, instabilities do not produce severe limitations on the
collider luminosity. Transverse instabilities in Tevatron and Recycler
are well understood. We plan further reduction of the Tevatron
chromaticity. Introducing cubic (octupole) non-linearity is the main
direction. Further improvements of head-tail damper may be required if
operational difficulties with octupoles will be encountered. To suppress
Recycler instability, the two band transverse damper: 10 kHz - 10 MHz
and 10 - 300 MHz is planned to be built.

# Acknowledgments

The authors thank V. Shiltsev, P. Ivanov, S. Nagaitsev, J. Crisp, V.
Scarpine, E. Lorman for discussions and sharing experimental data.

# References

1\. P. M. Ivanov et al., *Head-Tail instability at Tevatron,* PAC 2003,
p. 3062.

2\. A. Burov and V. Danilov, *private communication,* 2002.

3\. A. W. Chao, *Physics of Collective Beam Instabilities*, J. Wiley &
Sons, Inc, 1993.

[^1]: The vertical impedance is larger and, consequently, the vertical
    degree of freedom is more unstable.