# Coherent instabilities at the fnal booster\*

V. Lebedev^\#^, A. Burov, W. Pellico, X. Yang, FNAL, Batavia, IL 60510,
U.S.A.

Abstract

This paper presents results of experimental and theoretical
investigations of transverse beam stability at injection to Fermilab
Booster and discuses a novel scheme for transition crossing allowing to
avoid the longitudinal emittance growth related to the transition. At
reduced chromaticity a multibunch high order head-tail mode develops
with growth time of 12 turns at fractional part of tune close to zero.
An estimate of the growth rate based on known sources of impedance
results in significantly smaller value and cannot explain observed
instability growth rate.

## Introduction

Booster \[1\] is a fast cycling proton synchrotron operating at 15 Hz.
To exclude the eddy currents excited in the vacuum chamber by fast
changing magnetic field its vacuum chamber is formed by poles of
laminated combined function dipoles. That creates large contribu-tions
to transverse and longitudinal impedances affecting both transverse and
longitudinal beam stability. Presently, transverse instabilities are
suppressed by large chromatici-ty, which negatively affects the dynamic
aperture and the beam lifetime. Earlier attempts to stabilize the
instability by transverse feedback system were unsuccessful. This paper
presents results of studies aimed to understand mechanisms of
instability at injection. It also discusses the beam dynamics at the
transition crossing and considers a novel scheme allowing one to
suppress the longitudinal emittance growth at transition. Main Booster
parameters are presented in Table 1.

## 1. Beam dynamics at injection

The instability was evoked by reducing both vertical and horizontal
chromaticities. This caused instability in both planes and partial beam
intensity loss started at turn 150 after injection. Four channel digital
scope running with sampling time of 0.4 ns recorded beam signals of
horizontal and vertical beam position monitors (BPM) with 15 cm long
plates. Total recording time of 1.6 ms represents \~700 turns. To
maximize the resolution for transverse beam motion the signals of two
BPM plates for each BPM were combined at a wide band hybrid making the
sum and difference signals. The measurements were performed for nominal
and half of nominal beam intensities. Off-line treatment of the data
included marking boundaries for each bunch, subtracting bunch offsets
from the differential signal, and integrating bunch signals to obtain
the density and dipole moment distribu-tions. The following bunch
parameters were computed for each bunch: AC bunch intensity,
longitudinal and transverse centers of gravity, rms bunch length and
transverse dipole moments. Note that the relative revolution frequency
change during the first 200 turns is about 10^-3^, and needs to be taken
into account when bunch boundaries are marked. Beam positions (offsets)
are also changing fast and they were corrected for each turn.

Table 1: Booster parameters

  ------------------------------------------------ ----------------------
  Energy                                           0.4 -- 8 GeV

  Transition energy                                5.1 GeV

  Total number of particles                        4.5∙10^12^

  Circumference                                    474.2 m

  Harmonic number, *q*                             84

  Betatron tunes, Q~x~/Q~y~                        6.82 /6.81

  RF voltage                                       0.7- 0.9 MV

  Injection type                                   H^-^, 10 turns
  ------------------------------------------------ ----------------------

![](media/image1.wmf){width="3.2020833333333334in"
height="2.283333333333333in"}

Figure 1: Changes of beam parameters during first 150 turns; red line --
AC beam intensity, blue line -- rms bunch length, vertical (brown line)
and horizontal (green line) rms transverse dipole moments. Dotted lines
present exponential fits to the rms dipole moments inside a bunch. All
signals are averaged over all 84 bunches; beam intensity - 4.5∙10^12^.

The following sequence of actions happens during injection (see Figure
1). First, injection orbit bump is created just before injection. Then,
the linac beam is injected during 5 or 10 turns for half or full beam
intensity, respectively. When the injection is finished the orbit bump
is switched off (it takes 10-20 turns) and RF voltage is adiabatically
increased causing beam to be bunched at turn 70 with RF voltage
continuing to grow. The first sign of the instability appears at turn 80
causing the beam intensity drop at turn 150.

An accurate knowledge of betatron and synchrotron tunes is important to
understand the nature of the instability. Small oscillations excited by
injection process were helpful to determine the tunes. Figure 2 presents
turn-by-turn vertical positions for all 84 bunches and corresponding
spectrum. Although betatron motion decays fast the direct fitting by
exponentially decaying sinusoid allowed us to determine the fractional
part of tune with three digits accuracy. The decay time is used to
estimate the lattice chromaticity (see below). To determine the
synchrotron tune we analyzed the frequency of quadrupole synchrotron
oscillations excited in the course of beam bunching. They are clearly
visible on the rms bunch length presented in Figure 1. The synchrotron
tune at turns 50 to 120 grows from 0.03 to 0.06 in accordance with
expectations. Because of beam loading and beam space charge the
synchrotron tune dependence on time varies with beam intensity. Tables 2
and 3 summarize results of measurements.

![](media/image2.wmf){width="3.329861111111111in"
height="1.4902777777777778in"}![](media/image3.wmf){width="3.3180555555555555in"
height="1.1270833333333334in"}

Figure 2: Dependence of vertical bunch-by-bunch positions excited by
injection process on turn number and corresponding spectrum; beam
intensity 4.5∙10^12^.

Table 2: Results of measurements of tunes and damping times at turns 30
to 60

  -------------------------------------------- ------------- ------------
  Number of particles                          2.3∙10^12^    4.5∙10^12^

  Horizontal tune, Q~x~                        6.830         6.825

  Vertical tune, Q~y~                          6.831         6.819

  X-plane decoherence time, turns              15            13

  Y-plane decoherence time, turns              18            16
  -------------------------------------------- ------------- ------------

Knowing synchrotron tune and longitudinal density distribution one can
obtain the longitudinal distribution function. Fitting to the data of
turn 69 corresponding to the end of bunching process yields (see Figure
3):

$f(I) = \frac{2}{C}\left( 1 - \frac{E(I)}{E_{\text{max}}} \right)$ , (1)

where

$\begin{array}{r}
E = \frac{{\overset{˙}{\varphi}}^{2}}{2} + \Omega_{s^{2}}\left( 1 - \text{cos}\varphi \right)\ ,\  \\
E_{\text{max}} = 2\Omega_{s^{2}}\ ,\ C = \int_{0}^{I\left( E_{\text{max}} \right)}{f\left( E(I) \right)\text{dI}}\ ,
\end{array}$ (2)

*φ* ∈\[-*π*, *π*\] is the longitudinal coordinate inside RF bucket, and
*Ω~s~* is the synchrotron frequency. Corresponding longitudinal
emittance (area in the phase space including 100% particles) is 0.06
eV⋅s per bunch.

The measurement results clearly demonstrate that the instability does
not develop before bunches are formed; but when it finally starts it
looks rather unusual: the dipole moment distribution for all bunches on
nearby turns exhibits almost the same pattern only slowly moving in the
direction opposite to the beam direction as shown in Figure 4. Note that
although the data are presented for every 5-th turn the distributions
for missed turns are monotonically changing between the data for
presented turns. Such behavior corresponds to the head-tail instability
developing at tunes equal to *Q~x,,y~* −7 + 3*Q~s~*. Their values are
close to zero (see Table 3). At full beam intensity the horizontal
motion in the bunch tail is larger than in the head pointing out that
the horizontal instability is the strong head-tail instability.
Therefore data for half of nominal intensity, where the head-tail is
weak, were used for impedance estimates considered below.

Table 3: Results of measurements of instability growth times and
synchrotron tune

  -------------------------------------------- ------------- ------------
  Number of particles                          2.3∙10^12^    4.5∙10^12^

  Synchrotron tune, Q~s~ at turn 100           0.047         0.057

  ∆Q~x~ = Q~x~ −7 + 3Q~s~                      −0.029        −0.004

  ∆Q~y~ = Q~y~ −7 + 3Q~s~                      −0.028        −0.010

  X instability growth time, turns             18            12

  Y instability growth time, turns             14            14
  -------------------------------------------- ------------- ------------

![](media/image4.wmf){width="3.40625in" height="2.120833333333333in"}

Figure 3: Measured (red dots) and fitted (blue dashed line) longitudinal
density distribution at turn 69; beam intensity 4.5∙10^12^.

## 2. Chromaticity estimate

Beam decoherence at turns 30 to 60 allows one to make an estimate of
chromaticities. Neglecting the effect of impedance on the beam stability
and that the beam is partially bunched one can write the damping rate
as:

$n\text{.}_{\text{decoh}^{- 1}} \approx |\xi\  - n\eta|\sigma_{\frac{\Delta p}{p}}$
, (3)

where *η* = *α* -- 1/*γ*^2^ is the slip factor (*η* = −0.458 at
injection), *ξ* is the chromaticity and *n* is the mode number
determined so that the lab-frame betatron frequency is:

$\omega_{n} \approx \omega_{0}(Q + n)$ . (4)

For data presented in Table 2, *n* = −7 and one can neglect *nη* in
comparison with *ξ*. This yields for absolute values of chromaticities:
\|*ξ~x\ ~*\| ≈ 85 and \|*ξ~y\ ~*\| ≈ 70. As will be seen later the real
part of transverse impedance at this frequency is sufficiently small and
does not make significant correction for chromaticities.

![](media/image5.wmf){width="3.357638888888889in"
height="2.011111111111111in"}

![](media/image6.wmf){width="3.3645833333333335in"
height="2.0069444444444446in"}

Figure 4: Measured dipole moment distributions for *x-* (top) and
*y*-planes (bottom) over bunch length for every 5-th turn (turns from 93
to 113); beam intensity 4.5∙10^12^.

The signs of chromaticities can be determined from the direction of
head-tail wave propagation (see Figure 4). In the case of weak head-tail
effect the "air-bag" model yields that the dependence of average
transverse position along the bunch is \[2\]:

$\overline{x}(t,s) = x_{0}\text{cos}\left( {\Delta Q\omega}_{0}t - \kappa\frac{s}{L_{b}} + \psi \right)\text{cos}\left( m\ \text{acos}\frac{s}{L_{b}} \right)\ ,$
(5)

where $\Delta Q = Q + n + \text{mQ}_{s}$ is the mode tune,
$\kappa = \frac{{\xi L}_{b}}{{\eta R}_{0}}$ is the head-tail phase,
*L~b~* is the bunch length, *R~0~* is the machine radius, *m* is the
synchro-betatron mode number and *ψ* is an arbitrary phase. Eq. (5)
yields that the velocity of the head-tail wave is:

$\frac{d}{\text{dt}}\left( \frac{s}{L_{b}} \right) = \frac{{\Delta Q\omega}_{0}}{\kappa}$
. (6)

The mode tunes for parameters of our experiment are presented in Table
2. Data for 2.3∙10^12^ particles (where the weak head-tail approximation
is better applicable) were used to estimate the chromaticity. The
betatron wave velocities are d(∆*s*/*L~b~*)/dt ≈ -0.022 and -0.025
turn^-1^, correspondingly, for *x* and *y*-planes. That yields the
head-tail phases: *κ~x~* ≈ 8 and *κ~y~* ≈ 7 and the negative signs of
both chromaticities. Described above chromaticity measurements yields
\~2 times smaller values for *κ~x,y~* if *L~b~* equal to one forth of
full bucket length is used. Discrepancy is apparently related to the
poor accuracy of ∆*Q*, which small value is obtained as difference of
two large numbers, and insufficient accuracy of Eq. (5) ignoring
particle interaction and bunch structure.

## 3. Instability growth rate

To estimate the instability growth rate for bunched beam we use the
"air-bag" model \[2\]. For weak head-tail interaction it yields that the
complex betatron tune shift is:

${\delta Q}_{n,m} = \frac{\text{iI}_{0}R_{0}Z_{\text{eff}_{n,m}}}{4\pi\left( \frac{\text{mc}^{2}}{e} \right)\text{γβ}^{2}Q}$
, (7)

where

$Z_{\text{eff}_{n,m}} = \sum_{k = - \infty}^{\infty}{Z_{}(\omega_{k})\ }\left( J_{m}\left( \frac{\omega_{k}L_{b}}{\beta c} - \kappa \right) \right)^{2}$
, (8)

where *I*~0~ is the beam current, *c* is the speed of light, *m* and *e*
are the particle mass and charge, *γ* and *β* are its relativistic
factors, *Z*~⊥~(*ω*) and *R*~0~ are the ring impedance and average
radius,
$\omega_{k} = \omega_{0}\left( Q + n + \text{mQ}_{s} + \text{kN}_{b} \right)$,
and *N~b~* is the number of bunches which for Booster coincides with
harmonic number *q*. Taking into account that the instability growth
rate per turn is $\lambda_{n,m} = - 2\pi\text{Im}(\delta Q)$, and that
for observed instability *n* = -7 and *m* = 3, and using instability
growth rates for 2.3∙10^12^ particles from Table 3 one obtains the
effective impedances to be: Re(*Z~Xeff\ -7,3~*) = 43 MΩ/m,
Re(*Z~Yeff\ -7,3~*) = 55 MΩ/m.

The imaginary part of impedance can be found from tune dependence on the
beam intensity presented in Table 2. At that time the beam is not
completely bunched and one can use the continuous beam approximation to
estimate the impedance. In this case the tune shifts for horizontal and
vertical betatron motions are \[3\]:

$\begin{bmatrix}
{\delta Q}_{x} \\
{\delta Q}_{y}
\end{bmatrix}_{n} = - \frac{I_{0}R_{0}}{4\pi\left( \frac{\text{mc}^{2}}{e} \right)\text{γβ}^{2}Q}\left\{ \begin{matrix}
\text{Im}\left( Z_{X}(\omega_{n}) - Z_{D}(0) \right)\ , \\
\text{Im}\left( Z_{Y}(\omega_{n}) + Z_{D}(0) \right)\ ,
\end{matrix} \right.\ $ (9)

where *Z~D~*(*ω*) is the detuning impedance (Fourier transform of the
detuning wake). For *n* = -7 that yields: Im(*Z~X~*(*ω*-~7~)-*Z~D~*(0))
= 24 MΩ/m and Im(*Z~Y~*(*ω*-~7~)+ *Z~D~*(0)) = 58 MΩ/m, where *ω*-~7~=76
kHz. Earlier measurements \[4\] with beam motion excited by the vertical
kicker yielded very close value for the vertical tune shift. The
horizontal tune shift was not measured. Note that in distinction from
measurements presented here, the kicker measurements excite many
multi-bunch modes while BPMs report a single beam position per turn
combined from unknown weighted sum of bunch signals. Thus, the frequency
at which the response is measured is not well determined.

## 4. Transverse booster impedance

In the first approximation we can consider that the Booster impedance is
formed from two major contributions coming from the round stainless
steel vacuum chamber and the laminated dipoles.

The impedance per unit length of round vacuum chamber with thin wall was
computed in Ref. \[5\]. The result is:

$Z_{L\bot = \frac{\text{iZ}_{0}}{2{\pi a}^{2}\beta}\left\lbrack \frac{2\beta^{2}\left( e^{\text{kd}}\left( 1 + \text{kb} \right) - e^{- \text{kd}}\left( 1 - \text{kb} \right) \right)}{e^{\text{kd}}\left( 1 + \text{ka} \right)\left( 1 + \text{kb} \right) - e^{- \text{kd}}\left( 1 - \text{ka} \right)\left( 1 - \text{kb} \right)} + \frac{1}{\gamma^{2}} \right\rbrack}$.
(10)

where *a* and *b* are the inner and outer radii of vacuum chamber,
*d*≡*b*-*a*\<\<*a* is its thickness, $k = \frac{(1 + i)}{\delta}$,
$\delta = \frac{c}{\sqrt{2\text{πσω}}}$ is the skin depth, and
$Z_{0} = 4\frac{\pi}{c} \approx \text{377}\ \Omega$ is the impedance of
free space. The total length of such vacuum chamber is 197 m, *a*=43 mm,
and *d*~0~=1.6 mm. Figure 5 shows corresponding contribution to the
impedance. As one can see its real part achieves maximum at 3 kHz
where$\delta \approx \sqrt{\text{ad}}$. Its value does not depend on the
vacuum chamber conductivity and is:

$\text{Re}$ . (11)

The imaginary part of impedance achieves its maximum at zero frequency
where it is determined by pure electrostatic beam interaction with
vacuum chamber.

Computation of the impedance for laminated vacuum chamber in the entire
frequency range is a complicated problem and as far as we know has not
been solved. Before we consider it let us discuss the impedance of flat
vacuum chamber manufactured from non-laminated steel. As follows from
results of Ref. \[6\] the horizontal and vertical impedances per unit
length are:

$\begin{array}{r}
Z_{\text{Lx}} = i\frac{Z_{0}}{2{\pi a}^{2}\beta}\left\lbrack \frac{\pi^{2}}{\text{24}} + \beta^{2}\int_{0}^{\infty}\frac{\left( u - \frac{\text{ka}}{\mu} \right)\text{exp}( - u)\text{udu}}{k\text{cosh}(u) + u\text{sinh}(u)} \right\rbrack\ , \\
Z_{\text{Ly}} = i\frac{Z_{0}}{2{\pi a}^{2}\beta}\left\lbrack \frac{\pi^{2}}{\text{12}} + \beta^{2}\int_{0}^{\infty}\frac{\left( u - \frac{\text{ka}}{\mu} \right)\text{exp}( - u)\text{udu}}{k\text{sinh}(u) + u\text{cosh}(u)} \right\rbrack\ ,
\end{array}$ (12)

where the vacuum chamber is assumed to be confined by two infinite
magnetic blocks filling upper and lower half spaces and separated by
distance 2*a*, *μ* is the magnetic permeability, and similarly to the
round vacuum chamber $k = \frac{(1 + i)}{\delta}$ and
$\delta = \frac{c}{\sqrt{2\text{πσωμ}}}$. There are two types of
dipoles: focusing and defocusing. The summed length of each type is 139
m, and the half-gaps are 20.8 and 28.5 mm, respectively. Figure 5 shows
corresponding contributions to the impedances. As one can see the real
parts of both impedances achieve maximum at \~100 kHz (where *μδ *≈
*a*). Their values do not depend on vacuum chamber conductivity and *μ*.
They are:

$\text{Re}\begin{bmatrix}
Z_{\text{Lx}} \\
Z_{\text{Ly}}
\end{bmatrix}|_{\text{max}} \approx \frac{\text{iZ}_{0}\beta}{4{\pi a}^{2}}\left\{ \begin{matrix}
0\text{.}\text{336}\ ,\ \frac{a}{\text{μδ}} \approx 0\text{.}\text{372}, \\
0\text{.}5\ ,\ \frac{a}{\text{μδ}} \approx 0\text{.}\text{89}\text{.}
\end{matrix} \right.\ $ (13)

With frequency decrease, both impedances decrease and approach zero at
zero frequency. The reason of impedance decrease at small frequencies is
related to the increase of skin depth with subsequent magnetic flux
"shortening" by the wall magnetism; so that the tangential component of
beam magnetic field is diminishing. That causes decrease of currents
excited in the vacuum chamber, and, consequently, decreases power loss
and impedance. Note that the real parts of impedances for the round thin
wall and flat magnetic vacuum chambers achieve almost the same maximum
value. Actually, the smaller value for flat vacuum chamber is related to
its more open geometry than to its magnetism. This is a manifestation of
more general law which limits the absolute value of wide-band transverse
impedance so that $$ (see "careless limit" in Ref. \[2\]).

To make an estimate of impedance introduced by laminated dipoles we take
into account that for the given skin depth both the resistance and the
magnetic conductivity of vacuum chamber per unit length are increased by
factor 2*a*/*d~lam~* where *d~lam~* is the lamination thickness. That
results in that the horizontal and vertical impedances can be
approximated by Eq. (12) where *ka*/*μ* need to be replaced by
*kd~lam~*/2*μ*. At high frequencies this result coincides with the
result obtained in Ref. \[7\].

![](media/image7.wmf){width="3.2527777777777778in"
height="2.682638888888889in"}

![](media/image8.wmf){width="3.2527777777777778in"
height="2.676388888888889in"}

Figure 5: Contributions to the real (top) and imaginary (bottom) parts
of impedance from different pieces of vacuum chamber; *Z~R~* -- round
vacuum chamber, *Z~xL~* and *Z~yL~* -- laminated dipoles with *μ*=1000,
*Z~xnL~* and *Z~ynL~* -- mark contributions corresponding to
non-laminated dipoles.

To get the total ring impedances one needs to sum contributions of round
vacuum chamber and focusing and defocusing laminated dipoles with fudge
factors accounting the beta-function variation. They are equal to:

$k = \frac{Q}{2{\pi R}_{0^{2}}}\oint_{\text{over}\ \text{given}\ \text{type}}^{}{\beta(s)\text{ds}}$
. (14)

Integration over Booster design lattice yields: for focusing dipoles --
*k~x~* = 2.64, *k~y~* = 0.65; for defocusing dipoles -- *k~x~* =1.00,
*k~y~* =1.62; for round vacuum chamber -- *k~x~* = 0.64, *k~y~* = 1.82.
Figure 6 presents resulting total impedances.

Computation of effective impedances responsible for the tune shift
yields: Im(*Z~X~*(*ω*-~7~)-*Z~D~*(0)) = 26 MΩ/m and Im(*Z~Y~*(*ω*-~7~)+
*Z~D~*(0)) = 85 MΩ/m. Taking into account that the estimate of impedance
for laminated dipoles is rough at small frequencies and the effects of
bunching and momentum spread are completely ignored one can conclude
that there is a satisfactory agreement with the measurement result
presented at the end of Section 2.

![](media/image9.wmf){width="3.2527777777777778in"
height="2.2597222222222224in"}

![](media/image10.wmf){width="3.2527777777777778in"
height="2.3409722222222222in"}

Figure 6: Real (top) and imaginary (bottom) parts of Booster total
impedances and imaginary part of detuning impedance.

There is large disagreement in the estimate of real part of impedances.
From the experiment we know multi-bunch and head-tail modes (*n* = −7,
*m* = 3). That sets frequencies in the sum of Eq. (8) to be *ω~k~*/2*π*
= −0.012 ± 37.9*k* MHz. For *ξ *≈ −70 the head-tail phase is *κ* = 3.4
and the Bessel function J~3~(*κ*)^2^ is close to its first maximum. That
maximizes the contribution for addend with *k* = 0 (*f*≈12 kHz). Also
note that other contributions coming from positive and negative
frequencies (±*k*) in Eq. (8) partially cancel each other, thus reducing
the sum. The considered above estimate of laminated dipole impedance
ignores the plate-to-plate capacitance which reduces the real part of
impedance at frequencies above \~500 MHz. *μ*=1000 is used in the
impedance estimate in the entire frequency range. This cannot be correct
at high frequencies where *μ* should be decreasing due to skin depth
becoming smaller than the domain size. Therefore a frequency range of 0
to 500 MHz (*k* ∈ \[-13, 13\]) is used to compute the effective
impedances of Eq. (8). That results in 6 and 3.2 MΩ/m for horizontal and
vertical planes, correspondingly, where contributions coming from zero
term (*k*=0) are 0.37 and 0.55 MΩ/m. Although the model correctly points
out that the real part of horizontal impedance is larger than for the
vertical one, it predicts the instability growth rates and the real
parts of impedances by an order of magnitude smaller than the measured
values. The reason of the discrepancy is not clear. It hardly can be
explained by underestimation of the impedance for laminated magnets. It
is already close to its theoretical maximum in the required frequency
range. There is a possibility of unaccounted impedance, but also the
instability can be related to the presence of electrons stored in the
beam. The residual gas pressure in Booster is about 10^-7^ Torr. That
corresponds to the beam space charge compensation time of about 0.1 sec.
Although it is too long to make any effect at 200 turns the multipactor
discharge can greatly accelerate the production of electrons. Further
experimental studies are required to test this possibility.

## 5. Transition crossing with RF voltage jump technique

Measurements of beam dynamics at injection were performed similar to the
considered above transverse instability measurements. The sum signal of
a BPM was digitized with 0.2 ns sampling time. Then, the signal of each
bunch was integrated numerically to obtain the particle density along
the bunch. Figure 7 presents dependence of bunch length (computed using
few different algorithms) on time in vicinity of transition. One can see
that the bunch size starts to shrink approximately 0.2 ms after
transition and after achieving a minimum blows up. After few synchrotron
periods these synchrotron oscillations result in the longitudinal
emittance growth.

![](media/image11.png)

Figure 7: Dependence of bunch length (computed using different methods
from measured data) on time in vicinity of transition crossing.
Transition time is shown by the vertical line.

Observed longitudinal dynamics is consistent with development of the
negative mass instability due to longitudinal space charge force. First,
let us consider a simple linear model with a parabolic density
distribution and linearized RF voltage over bunch length. Standard
procedure results in a system of differential equations describing
evolution of the bunch ellipse in the phase space. The solution was
carried out numerically for sinusoidally changing bending field and
given dependence of accelerating voltage on time. The solid line on the
top picture of Figure 8 presents results of numerical simulation for
nominal parameters of transition crossing. One can see close resemblance
with measurement results presented in Figure 7. Examination of the
solution shows that the sign change of the space charge force from
repulsion to attraction after transition crossing causes large amplitude
quadrupole synchrotron oscillations. More detailed multi-particle
simulations with ESME \[8\] show that after few synchrotron oscillations
they decohere causing longitudinal emittance growth.

![](media/image12.wmf){width="3.251388888888889in"
height="3.4715277777777778in"}

Figure 8: Dependence of bunch length (top picture) on time simulated in
linear model in vicinity of transition crossing: solid line -- no fast
change in RF voltage, dashed line -- RF voltage has two pulses as shown
in the bottom picture. Transition time is shown by the vertical line

The following procedure was suggested to prevent a development of such
oscillations. Approximately quarter synchrotron period before transition
crossing the accelerating voltage is rapidly increased. That causes the
bunch to be overfocused. Actually the bunch length changes comparatively
little due to small particle mobility at transition but momentum spread
in the bunch is strongly amplified. Voltage is returned to its nominal
value at transition. The accumulated momentum spread causing bunch
shortening before transition is now causing the bunch lengthening which
prevents bunch collapse due to space charge force. Using the second
smaller voltage pulse after half synchrotron period allows one to
suppress synchrotron oscillations after transition as shown by dashed
line in Figure 8. Figure 9 shows results of ESME simulations with and
without voltage manipulations clearly demonstrating suppression of
longitudinal emittance growth.

This method was tested in Booster and demonstrated expected reduction of
emittance growth. Its use in operations for a few shifts was also
successful. Two hardware problems are pending its permanent use in
Booster operations. The first is related to insufficient time resolution
of the low level RF timing; and the second one is high sensitivity to RF
station loss because Booster RF does not have sufficient voltage excess.

![](media/image13.png)![](media/image14.png) Figure 9: Particle
distribution over bunch length at the end of Booster cycle simulated
with ESME: top -- no RF manipulations, bottom -- RF manipulations are
on.

## References

\[1\] "Booster Synchrotron", edited by E. L. Hubbard, FNAL TM-405, 1973.

\[2\] A. W. Chao, "Physics of Collective Beam Instabilities," J. Wiley &
Sons, Inc, 1993..

\[3\] A. Burov and V. Danilov, "Suppression of Transverse Bunch
Instabilities by Asymmetries in the Chamber geometry.", Phys. Rev. Lett.
82, 2286 (1999)

\[4\] Huang Xiaobiao, *private communications*, 2005.

\[5\] A. Burov, V. Lebedev, "Transverse Resistive Wall Impedance for
Multi-Layer Round Chambers", p. 1452, EPAC'02

\[6\] A. Burov, V. Lebedev, "Transverse Resistive Wall Impedance for
Multi-Layer Flat Chambers", p. 1452, EPAC'02

\[7\] A. Burov, V. Danilov, *private communications,* 2002.

\[8\] J. MacLachlan, "User's Guide to ESME 2002," FNAL, November 2002