Space Charge Resonances and Instabilities in Rings
I. Hofmann, G. Franchetti* and A. Fedotov1
*GS7, 64291 Darmstadt, Germany
, Upton NY 11973, USA
Abstract. We show that the influence of space charge on resonances of coasting beams in circular machines
can be discussed in terms of a coherent frequency shift leading to a "coherent advantage", and of space charge
induced resonant instabilities absent in the zero space charge limit. These mechanisms are exemplified for the
SNS ring as well as the SIS 18 of GSI.
INTRODUCTION
While resonances in circular machines driven by systematic or imperfection lattice terms have been widely explored since the early times of accelerators, the role of
space charge has found comparable attention only in recent years [1, 2, 3, 4, 5, 6, 7]. The most obvious effect
of space charge is the incoherent tune shift and spread,
which leads to an extended foot print of single particle
tunes in the tune diagram. Less obvious is the observation that the response on a resonance may be modified
by the coherent motion of all or a large fraction of particles. This leads to an additional oscillating force, which
must be added to the external forces and may cause a
coherent shift of the resonance condition; furthermore a
different response is obtained, depending on whether the
resonance is crossed from above or below, which was
first studied in Ref. [1]. The main point of this paper is
to show that a systematic study of space charge effects
needs to consider the existence of both, resonances and
instabilities. Such structure instabilities have been studied earlier in the context of linear transport systems [8],
but these findings can be applied also to circular accelerators. Our model is that of a coasting beam or, equivalently, a constant line current beam in an ideal barrier
bucket. In usual bunched beams in an rf bucket the combined effect of synchrotron motion and line current variation leads to additional effects due to resonance crossing
discussed elsewhere in this conference [9].
COHERENT SHIFT OF RESONANCE
The analytical basis for calculating coherent frequency
shifts is the calculation of eigenfrequencies for charge
density oscillations with arbitrary tune and emittance ratios, hence fully anisotropic beams, by using the dispersion relations derived in Ref. [3] for KV beams and in
smooth approximation. These eigenmodes can be excited
resonantly in a circular accelerator by appropriate external field perturbations (for round isotropic beams see
also Ref. [5]); furthermore, they can be excited by space
charge itself, or a combination of both. Hence, the coherent space charge contribution leads to a shifted condition
of resonance of the coherent frequency co with a field
perturbation at harmonic n according to
co = mvx + / vy + Aco = n.
(1)
Here we express the "mode tune" co in units of the revolution frequency similar to single particle tunes, and introduce Aco as coherent shift due to the resonant density oscillations. The importance of such a shift was first
pointed out by Smith [10] for second order, and extended to any order by Sacherer [11]. Numerically, it was
demonstrated in Ref. [1] - using particle-in-cell simulation - for an example of fourth order resonance crossing.
In order to more easily quantify the effect of the coherent shift, a coherent mode coefficient Cmk defined by
the equation n = m(v0 —C m ^Av), thus including the incoherent and coherent space charge effect, was proposed in
Ref. [4] for isotropic beams and uncoupled modes, based
on the dispersion equations of Ref. [3]. Here m is denoting an azimuthal, and k a radial mode number, while
Av = V0 — v. The radial mode number reflects the order of the perturbed space charge potential; the azimuthal
one gives the multiple of V0 for vanishing space charge.
Note that for non-KV beams v and Av must be understood as rms values. Cmk can thus be interpreted as coefficient of attenuation of the single particle space charge
shift Av by the coherent motion, provided that C < 1.
The intensity can thus be increased by the factor 1/C
compared with the single particle based resonance condition, which expresses the "coherent advantage". For the
symmetric breathing mode the result is C = 1/2, which
reflects the strong density perturbation by the cross sectional breathing. This allows increasing Av by a factor
2 compared with the corresponding single particle resonance condition n = 2 v; for the quadrupolar mode the coefficient is only 3/4, hence an allowed intensity increase
by the factor 4/3. The general case for split horizontal
and vertical tunes (also unequal emittances) leads to a
coefficient C close to the arithmetic mean, e.g. 5/8, for
both modes, hence a "coherent advantage" by the fac-
CP642, High Intensity and High Brightness Hadron Beams: 20th ICFA Advanced Beam Dynamics Workshop on
High Intensity and High Brightness Hadron Beams, edited by W. Chou, Y. Mori, D. Neuffer, and J.-F. Ostiguy
© 2002 American Institute of Physics 0-7354-0097-0/02/$ 19.00
248
tor
1.6.
In
fourth
order
we
find
that
C
≈
0.87
is
good
tor 1.6.
1.6. In
In fourth
fourth order
order we
we find
find that
that C
C≈
w 0.87
0.87 is
is aaa good
good
tor
approximation
for
a
broad
variation
of
tune
splitting
and
approximation for a broad variation of tune splitting and
emittance
ratios.
This
implies
a
weakening
of
the
“coheremittance
ratios.
This
implies
a
weakening
of
the
"coheremittance ratios. This implies a weakening of the “coherent
advantage”
if
fourth
order
resonances
are
expected
to
ent advantage”
advantage" if
if fourth
fourth order
order resonances
resonances are
are expected
expected to
to
ent
play
role.
We
note
here
that
modes
with
C
>
may
play aaa role.
role. We
We note
note here
here that
that modes
modes with
with C
C>
> 111 may
may
play
appear,
in
principle,
as
solutions
of
the
analytical
KVappear, in
in principle,
principle, as
as solutions
solutions of
of the
the analytical
analytical KVKVappear,
based
dispersion
relations.
Following
the
discussion
in
based dispersion relations. Following the discussion in
Ref.
[12]
they
should,
however,
be
discarded
as
artifact
[12] they should,
should, however,
however, be
be discarded
discarded as
as artifact
artifact
Ref. [12]
of KV-distributions
KV-distributions associated
associated
with
“negative
energy”.
associated with
with “negative
"negative energy”.
energy".
of
SECOND ORDER
ORDER RESONANCES
RESONANCES
SECOND
APPLIED TO
TO SNS
SNS
APPLIED
We
first
appearance
of
an
imperfection
driven
We first
first discuss
discuss the
the appearance
appearance of
of an
an imperfection
imperfection driven
driven
second order
order resonance
resonance for
for
the fictitious
fictitious
un-split working
working
for the
fictitious un-split
working
second
point (((v
, νV0y
)=
(4.6,
4.6) driven
driven
by
harmonic
=
= (4.6,
(4.6,4.6)
driven by
by aaa harmonic
harmonic nnn =
= 999
0x
0jc, ν
0 )) =
point
4.6)
νν0x
0y
gradient
error.
The
effect
of
the
error
resonance
depends
depends
gradient error. The effect of the error resonance depends
primarily on
on the
the tune
tune and
and not
not the
the details
details of
of the
the focusing
focusing
primarily
lattice, hence
hence we
we can
can assume
assume aaa constant
constant focusing
focusing for
for
for
lattice,
simplicity.
The
expected
resonance
condition
is
condition is
is
simplicity. The expected resonance condition
(2)
(2)
(2)
An anti-symmetric
anti-symmetric error
error in
in xx and
and
(as
would
be
proand yy (as
(as would
would be
be proproAn
duced
by
a
single
quadrupole)
then
drives
a
quadrupolar
single quadrupole) then drives
drives aa quadrupolar
quadrupolar
duced by a single
mode resonance,
resonance, whereas
whereas aaa symmetric
symmetric
error
is
needed
to
symmetric error
error is
is needed
needed to
to
mode
drive
the
breathing
mode
resonance.
Note
that
for
sufresonance. Note
Note that
that for
for sufsufdrive the breathing mode resonance.
ficiently split
split
tunes aaa single
single
quadrupole
error
will
drive
split tunes
single quadrupole
quadrupole error
error will
will drive
drive
ficiently
both
We
assume
equal
emittances
in
and
y,
and
both modes.
modes. We
We assume
assume equal
equal emittances
emittances in
in xxx and
and y,
y,and
and
solve
symmetric
solve the
the envelope
envelope equations
equations with
with aaa symmetric
symmetric relative
relative
gradient error
error Fourier
Fourier harmonic
harmonic of
of
0.001
at
=
as
well
of 0.001
0.001 at
at nnn =
= 999 as
as well
well
gradient
as aaa small
small
initial
envelope
mismatch
of
2%.
If
the
sinsmall initial
initial envelope
envelope mismatch
mismatch of
of 2%.
2%. If
If the
the sinsinas
gle particle
particle tune
tune is
is exactly
exactly on
on
resonance
there
is
only
on resonance
resonance there
there is
is only
only aaa
gle
small beating
beating of
of the
the envelope
envelope due
due to
to proximity
proximity of
of the
the cocosmall
herent
resonance
(Fig.
1
top).
The
resonance
of
the
in(Fig. 11 top). The
The resonance of
of the inherent resonance (Fig.
phase
mode
should
be
expected
at
ν
=
4.4
based
on
x,y
should be
be expected at
at νvx,y
= 4.4 based on
on
phase mode should
xy =
the
the calculated
calculated value
value for
for C.
C. At
At this
this value,
value, in
in fact,
fact, the
the enenvelope
significant
though
not
yet
maximum
velope response
response is
is significant
significant though
though not
not yet
yet maximum
maximum
(Fig. 111 bottom).
bottom). The
The complete
complete picture
picture is
is
shown
in
Fig. 222
is shown
shown in
in Fig.
(Fig.
for both,
both, symmetric
symmetric and
and anti-symmetric
anti-symmetric gradient
gradient
errors.
gradient errors.
errors.
for
Due to
to the
the nonlinear
nonlinear nature
nature of
of the
the envelope
envelope response
response
the
response the
Due
maximum is
is reached
reached for
for
even slightly
slightly
stronger
tune dedefor even
slightly stronger
stronger tune
maximum
pression, with
with aa sharply
sharply dropping
dropping response
response beyond.
beyond. As
As
pression,
expected,
the
zero-current
case
has
infinite
response
at
at
expected, the zero-current case has infinite response at
ν
=
4.5.
For
the
actual
simulation
the
particle-in-cell
x,y
= 4.5. For
For the
the actual
actual simulation
simulation the
the particle-in-cell
particle-in-cell
νVj^
x,y =
ORBIT
was
ORBIT code
code was
was applied
applied to
to the
the SNS
SNS lattice
lattice with
with aaa workwork,
)
=
(6.45,
4.6).
The
response
to
an
erν
ν
ing
ing point
point (((v
,
V
)
=
(6.45,4.6).
The
response
to
an ererν0x
0x
0y
0jc ν0y
0y = (6.45, 4.6). The response to an
ror in
in aa single
single quadrupole
quadrupole leads
leads to
to aaa similar
similar
behavior of
of
similar behavior
of aaa
ror
shifted curve
curve and
and
steeply
dropping
maximum
envelope
and steeply
steeply dropping
dropping maximum
maximum envelope
envelope
shifted
slightly beyond
beyond the
the envelope
envelope resonance
resonance condition.
condition.
Recondition. ReReslightly
sults for
for the
the out-of-phase
out-of-phase mode
mode excited
excited by
by crossing
crossing the
the
sults
resonance
condition
in
the
y-plane
are
shown
in
Fig.
3,
j-plane are
are shown
shown in Fig. 3,
3,
resonance condition in the y-plane
where
intensity
is
plotted
in
terms
of
a
normalized
tune
in terms
terms of
of aa normalized
normalized tune
tune
where intensity is plotted in
shift defined
defined as
as
actual
space
charge
tune
shift
relative
to
as actual
actual space
space charge
charge tune
tune shift
shift relative
relative to
to
shift
the
distance
of
the
bare
tune
from
the
half-integer,
i.e.
from the half-integer, i.e.
i.e.
the distance of the bare tune from
249
FIGURE
1.
In-phase
mode
envelope
evolution
at
the
single
FIGURE 1.
1. In-phase
In-phase mode
mode envelope
envelope evolution
evolution at
at the
the single
single
FIGURE
particle
resonance
condition
=4.5
(top),
and
at
the
point
of
x,y
particle resonance
resonance condition
condition ννvx,y
=4.5 (top),
(top), and
and at
at the
the point
point of
of
particle
JCJ=4.5
ν
=4.4
(bottom)
for
ν
=4.6.
coherent
resonance,
x,y
coherent
resonance,
v
y=4A
(bottom)
for
v
=4.6.
0,x,y
coherent resonance, νx,y
X =4.4 (bottom) for ν0,x,y
Qxy =4.6.
1.8
1.8
normalized
normalizedenvelope
envelope
+∆
∆ω
ω=
= 9.
9.
22ννxx +
1.6
1.6
out-of-phase
out-of-phase
in-phase
in-phase
zero-current
zero-current
1.4
1.4
1.2
1.2
1.0
1.0
1.0
4.3
4.3
4.4
4.4
4.5
4.5
4.6
4.6
tune
tune
FIGURE
2.
Maximum
envelope
response
(normalized
to
iniFIGURE 2.
2. Maximum
Maximum envelope
envelope response
response (normalized
(normalized to
toiniiniFIGURE
=4.6 as
as
tial
envelopes)
for
both
eigenmodes
and
bare
tune
tial envelopes)
envelopes) for
for both
both eigenmodes
eigenmodes and
and bare
bare tune
tune ννV0,x,y
as
=4.6
tial
0,x,y
0 ^=4.6
function
of
depressed
incoherent
tune.
function of
of depressed
depressed incoherent
incoherent tune.
tune.
function
∆
. The
solid
vertical
line
indicates
the
coher/∆
sc
Av
/Av
The solid
solid vertical
vertical line
line indicates
indicates the
the cohercoher∆
ννsc
ννinc
/∆
inc
5C
/nc.. The
ent
resonance
condition,
where
the
normalized
tune
shift
ent resonance
resonance condition,
condition, where
where the
the normalized
normalized tune
tune shift
shift
ent
equals
1/C,
the
long-dashed
the
r.m.s.
incoherent
resoequals 1/C,
1/C, the
the long-dashed
long-dashed the
the r.m.s.
r.m.s. incoherent
incoherent resoresoequals
nance
condition,
and
the
short-dashed
line
the
incoherent
nance condition,
condition, and
andthe
the short-dashed
short-dashed line
line the
the incoherent
incoherent
nance
limit
due
to
the
small-amplitude
particles
in
waterbag
limit due
due to
to the
the small-amplitude
small-amplitude particles
particles in
in aaa waterbag
waterbag
limit
distribution.
As
with
the
simple
envelope
model
above,
distribution. As
As with
with the
the simple
simple envelope
envelope model
model above,
above,
distribution.
we
find
again
that
the
resonance
happens
at
significantly
we find
find again
again that
that the
the resonance
resonance happens
happens at
at significantly
significantly
we
higher
particles in
in good
goodagreeagreehigherintensity
intensitythan
higher
intensity
than for
for single
single particles
ment
with
the
theoretical
expectations.
In
this
context
the
ment
withintensity
the theoretical
theoretical
In this
the
ment
with
the
expectations.
higher
than for
single particles
in context
good agree-
44
ν4
νyy y
νy
ment with
the theoretical expectations. In this context the
33
1
2.75
2.75 3
3.5
3.5H
3.5
3.5
normalized envelope
normalized envelope
normalized envelope
2.75
2.5
2.5
2.5
2.25
2.25
2.25
22
3 3 I •••th•••••••••
th th
4order
order
3 44 thorder
1.75 2
1.75
order
24nd order
nd
1.75
1.5
2 order
1.5
2nd order
1.5
1.25
2.5
2.5
2.5
2.53.5
3.5
3.5
3.5
1.25
1.25
1.25
1.5
1.75
2
1 normalized
1.25
1.5tune1.75
shift 2
0.75
1 normalized
1.25
1.5tune
1.75
2
shift
normalized
tune 1.75
shift
0.75
1
1.25
1.5
2
normalized tune shift
2.25
2.5
2.25
2.5
2.25
2.5
FIGURE 3.
3. Maximum
Maximum envelope
envelope response
response for
SNS
lattice
as
FIGURE
for
SNS
lattice
asas
FIGURE
3. normalized
envelope
response
for
SNS
lattice
function
of
tune
shift
for
,νfor
)=(6.45,4.6).
FIGURE
3.Maximum
Maximum
envelope
SNS
lattice
as
0x,v
0y
function
of
normalized
tune
shift and
andresponse
for ((vνQx
)=(6.45,4.6).
0y
function
of normalized
tune
shift
function
of normalized
tune
shiftand
andfor
for(ν
(ν0x ,ν,ν0y )=(6.45,4.6).
)=(6.45,4.6).
0x
0y
question arises,
arises, if
if fourth
fourth order
order resonances
resonances might
question
might play
play aa
question
arises,
if33 fourth
order
resonances
might
play aa
question
arises,
if
fourth
order
resonances
might
role
over
the
10
turns
required
for
filling
the
ring.
role over the 103 turns
required for filling the ring. For
For
3 turns
role
over
thevalue
1010of
turns
required
for
For
role
over
the
required
forfilling
filling the
the ring.
ring. For
the
typical
C
=
0.87
the
corresponding
normalthe typical value of C = 0.87 the corresponding normalthe
typical
value
of
C
=
0.87
thecorresponding
corresponding
normalthe
typical
value
of
C
=
0.87
the
normalized
tune
shift
would
only
be
1.15,
hence
nearly
cancelized tune shift would only be 1.15, hence nearly cancelized
shift
would
onlybebe1.15,
1.15,hence
hencenearly
nearly cancelcanceling
thetune
second
order
“coherent
advantage”.
ized
tune
shift
would
only
ing
the
second
order
"coherent
advantage".
ing
the
second
order
“coherent
advantage”.
ing the second order “coherent advantage”.
RESONANCE AND
AND INSTABILITY
INSTABILITY FOR
RESONANCE
RESONANCEAND
ANDSIS18
INSTABILITYFOR
FOR
RESONANCE
INSTABILITY
FOR
THE
THE
SIS18
THESIS18
SIS18
THE
We next consider the working diagram of the SIS18 latWe
next
consider
thetheworking
diagram
ofofthe
18 latWe
next
consider
working
diagram
theSIS
SIS18
tice
with
12
superthe
periods
and diagram
triplet
focusing.
In Fig.lat4
We
next
consider
working
of
the
SIS18
lattice
with
12
super
periods
and
triplet
focusing.
In
Fig.
44
with 12the
super
periods
andorder
tripletsystematic
focusing. In
Fig.
wetice
indicate
usual
fourth
singletice
with
12
super
periods
and
triplet
focusing.
In
Fig.
wewe
indicate
the
usual
fourth
order
systematic
singleindicate
the line
usual
systematic
single-4
particle
resonance
4νyfourth
= 12, order
its coherent
counterpart
we
indicate
the
usual
fourth
order
systematic
singleparticle
resonance
line
4v
=
12,
its
coherent
counterpart
y
particle
νy = 12,which
its coherent
ν + ∆ω =line
12 4(dotted),
nearlycounterpart
coincides
given
by 4resonance
given
by
4v4yyν+y +
Aco
which
nearly
coincides
particle
resonance
412
ν(dotted),
=
12, its
coherent
counterpart
y(dotted),
∆line
ω= =12
which
nearly
coincides
given
with
thebylower
harmonic
of the fourth
order
coherent
with
thethe
lower
harmonic
ofofthe
fourth
order
coherent
+
∆
ν
ω
=
12
(dotted),
which
nearly
coincides
given
by
4
with
lower
harmonic
the
fourth
order
coherent
y
resonance 2νy + ∆ω = 12/2; furthermore the coherent
resonance
2v2yν+yharmonic
Aco
= =12/2;
furthermore
the
coherent
with
the
lower
of
the
fourth
order
coherent
resonance
ω
12/2;
furthermore
the
+
∆
second order resonance condition 2νy + ∆ω = 6,coherent
which
second
order
condition
2v2yνy++Aco
==6,coherent
which
second
order
condition
6,gradiwhich
∆ωthe
resonance
2νyresonance
ω a=6-th
12/2;
furthermore
+resonance
∆by
could
be excited
harmonic
imperfection
could
be
excited
by
a
6-th
harmonic
imperfection
gradicould
be
excited
by
a
6-th
harmonic
imperfection
gradisecond
order
resonance
condition
2
ν
ω
=
6,
which
+
∆
y shifts are calcuent error (not studied here). The coherent
ent
error
(not
studied
here).
The
coherent
shifts
are
calcuent
error
(not
studied
here).
The
coherent
shifts
are
calcucould
be
excited
by
a
6-th
harmonic
imperfection
gradilated for an incoherent space charge tune shift ∆νy = 0.4
lated
forfor
anpractically
incoherent
space
charge
shift
lated
an
incoherent
space
charge
shiftAv-y
∆
νy=calcu=0.4
0.4
ent
error
(not
studied
here).
The
coherent
shifts
are
νtune
and
found
independent
oftune
0x .
and
found
practically
independent
of
V
.
ν
.
and
found
practically
independent
of
0jc0x shift
latedInfor
an incoherent
space charge
tune
∆νy =co0.4
Table
1 we summarize
analytically
calculated
In
Table
1 1coefficients
wewesummarize
analytically
calculated
In
Table
summarize
analytically
calculatedcocoherent
mode
relevant
for
resonances
in
.
and
found
practically
independent
of νthe
0x
modecoefficients
coefficients
relevant
forthe
theresonances
resonances
herent
relevant
for
inin
Fig.
4.mode
They
are
practically
independent
of
both tunes
Inherent
Table
1 we
summarize
analytically
calculated
coFig.
4.
They
are
practically
independent
of
both
tunes
Fig.
4.
They
are
practically
independent
of
both
tunes
provided
thatcoefficients
the incoherent
shift isfor
notthe
comparable
within
herent
mode
relevant
resonances
that
the
incoherent
shift
notxcomparable
comparable
with
provided
that
the
incoherent
shift
isisnot
with
orprovided
larger
than
the
tune
splitting
between
and
y,both
in which
Fig.
4.
They
are
practically
independent
of
tunes
or
larger
than
the
tunesplitting
splittingbetween
betweenx xand
andy,y,ininwhich
which
or
larger
than
the
tune
case corrections
are
needed. shift is not comparable with
provided
that the are
incoherent
case
corrections
areneeded.
needed.
case
corrections
TABLE
1.
Some
coherent mode
coefficients
for
or larger
than
the
tune
splitting
between
x cients
and y,CCin
which
TABLE
1.
Some
coherentmode
modecoeffi
coefficients
Cfor
for
TABLE
Some ratios.
coherent
different 1.
emittance
case corrections
are
needed.
different
emittance
ratios.
different
emittance
ratios.
4.0
4.0
4.04.0
4.5
5.0
4.5 ν
5.0
v
4.54.5νvxxνx5.0 5.0
x
FIGURE
4.
and
coherent
shifted
resonance
lines
FIGURE
4. 4.Incoherent
Incoherent
andand
coherent
shifted
resonance
lines
FIGURE
Incoherent
coherent
shifted
resonance
ν
=3.4
variable
ν
.
of
SIS18
for
fixed
FIGURE
4.
Incoherent
and
coherent
shifted
resonance
lines lines
0y0y=3.4 and variable V
0x0jc.
of SIS18 for fi xed V
variable
ofSIS18
SIS18for
forfixed
fixed
ν0yν=3.4
andand
variable
ν0x .ν0x .
of
0y =3.4
conducting
conducting boundary
boundaryconditions
conditions[13].
[13].We
Wehave
haveused
usedrms
rms
conducting
boundary
conditions
[13].
We
have
used
conducting
boundary
conditions
[13].
We
have
used
rms
matched
waterbag
distributions
generated
by
filling
aarms
matched waterbag distributions generated by filling
matched
waterbag
distributions
generated
byit filling
a
matched
waterbag
distributions
generated
by
filling
4D
hyper-ellipsoid
uniformly
and
deforming
accord4D hyper-ellipsoid uniformly and deforming it accord- a
4D
hyper-ellipsoid
uniformly
andand
deforming
it accord4D
hyper-ellipsoid
uniformly
deforming
it
according
to
the
rms
matching.
Carrying
out
the
self-consistent
ing to the rms matching. Carrying out the self-consistent
ing
Carrying
out
the the
self-consistent
simulation
with
νmatching.
isisrelatively
arbitrary
ingtotothe
therms
rmsmatching.
Carrying
out
self-consistent
simulation
with
V0x0jc == 4.9
4.9 (which
(which
relatively
arbitrary
simulation
with
ν
=
4.9
(which
is
relatively
arbitrary
=
3.4
we
find
in
Fig.
that
ν
in
this
context)
and
simulation
with
ν
=
4.9
(which
is
relatively
0x
0y
in this context) and0xV0 = 3.4 we find in Fig.55arbitrary
that
3.43.4
we we
findfind
iny Fig.
5 that
ν0yν = =
inin this
context)
and
there
is
a
significant
emittance
growth
in
in
the
reFig.
5 that
this
context)
and
there
is
a
significant
emittance
growth
in
y
in
the
re0y
there
isis<
a asignificant
emittance
growth
in two
yininydistinct
the
region
2.7
ν
<
2.95.
In
fact,
we
expect
y
there
significant
emittance
growth
in
the
gion
< vy <
2.95. In
fact, we
we expect
expect two
two distinct
distinct region 2.7
In and
fact,
y play
mechanisms
a role
cause
broadtwo
a stopgion2.7
2.7<<νto
νy<
<2.95.
2.95.
In
fact,
wethis
expect
distinct
mechanisms
to
play
a
role
and
cause
this
broad
stopmechanisms
to play
a roleenvelope
and cause
this broad
aastopband.
Firstly, the
instability
mechanisms
tostructure
play a role
and cause
this described
broad
a stopband.
Firstly,
the
structure
envelope
instability
described
band.
Firstly, the
∆ω =envelope
νstructure
12/2
is instability
expected.
Itdescribed
is described
charby
the equation
2the
y+
band.
Firstly,
structure
envelope
instability
by
the
equation
2v
+
Aco
=
12/2
is
expected.
It
is
char∆ω = 12/2180
is 0expected.
It is charby the equation
2νyy +advance
acterized
by a phase
of the underlying
+ ∆ω =of
It is charby the equation
2νyadvance
acterized
by
aa phase
of12/2
180°0isof
ofexpected.
the underlying
underlying
acterized
by
phase
advance
of
180
the
envelope mode per focusing period, hence
1:2 (half0 ofaathe
envelope
mode
per
focusing
period,
hence
1:2
(halfacterized
by
a
phase
advance
of
180
underlying
envelope
mode per between
focusing mode
period,
hence
a 1:2
(halfinteger)
relationship
and
lattice
periodicinteger)
relationship
between
mode
and
lattice
periodicenvelope
mode
per
focusing
period,
hence
a
1:2
(halfinteger)
relationship
between
mode and
lattice
periodicity;
secondly
we expect
a structural
fourth
order
resoity;
secondly
we
expect
a structural
structural
fourth
order resoresointeger)
relationship
between
mode
and
lattice
periodicity;
secondly
we
expect
a
fourth
order
nance or instability. Such structure instabilities for secnance
or
Such
structure
instabilities
forsecsec-resoity; secondly
we expect
a structural
fourth for
order
nance
or instability.
instability.
Such structure
instabilities
nance
2.4or instability. Such structure instabilities for sec-
∆ε/ε
1
2.4
2
2.4
2
1.6
1.6 2
1.2
1.2
1.6
0.8
0.8
1.2
0.4
0.4
00.8
0
∆ε∆ε
/ε /ε
0.75
0.4
0
2.7
2.7
2.8
2.8
2.9
2.9
ννy
y
3
3
ν
FIGURE 5. Saturated
emittance
effect after
turns 3
2.7 r.m.s.
2.8
2.9 200
FIGURE
5.linear
Saturated
r.m.s.
effect
turns
y
FIGURE
Saturated
emittance
effect after
after
200
turns
ν0x ,ν0y )=(4.9,3.4)
as 200
function
in
an ideal5.
lattice r.m.s.
with (emittance
ν0x
,,v
ν0y
)=(4.9,3.4)
as
function
in an
an ideal
ideal linear
linear lattice
lattice with
((v
in
with
)=(4.9,3-4)
as
function
0jc
0};
of νy (εx /εy =4). Also shown is 10x(λ -1) of the theoretical
νy fe/£y=4).
(εinstability
=4). Saturated
Also
shown
is
the
r.m.s.
emittance
effect
after
200 turns
x /εy5.
ofofFIGURE
Vy
Also
shown
is λ10x(
lOx(A-l)
of
the theoretical
theoretical
. λ -1) of
envelope
growth
factor
envelope
instability
growth
in an ideal
lineargrowth
lattice factor
with λ(A.ν. ,ν )=(4.9,3.4) as function
envelope
instability
factor
εx /εy 1.
TABLE
ε x /ε y
4 1 resonance 4 1 instability
Sc/ey
different
emittance
ratios.
4
0.66
0.88
0.87
0.66
0.88
0.87
41 4
0.66
0.88
0.87
0.61
0.85
0.89
nd
th
th
εx1/ε1y
20.61
4 / 0.89
instability
0.61 4 / resonance
0.85
0.89
0.85
0.66
0.88we have employed
0.87 the MIFor4 the simulation
study
For
simulation
study
wehave
haveemployed
employed
MI1 thethesimulation
0.61
0.85we
0.89 theMIFor
study
CROMAP 2D particle-in-cell code employingthe
5x10444
CROMAP
2D
particle-in-cell
code
employing
5x10
CROMAP
2D particle-in-cell
code
employing
5;d0
simulation
particles
onona a128x128
rectangular
grid
with
simulation
particleson
128x128
rectangular
grid
with
simulation
particles
a 128x128
rectangular
grid
For
the simulation
study
we have
employed
thewith
MI-
ond
higher
order
have
beenissystematically
explored
of and
νand
Also
shown
10x(λ -1) of the
theoretical
y (εhigher
x /εy =4).order
ond
have
been
systematically
explored
ond
and
higher
order
have
been
systematically
explored
for
periodic
axi-symmetric
transport
systems
in
Ref.
[8].
λ
.
envelope
instability
growth
factor
forperiodic
periodic axi-symmetric
axi-symmetric transport
transport systems
in
for
inRef.
Ref.[8].
[8].
These
findings apply equally
to ringssystems
with different
foThese
findings
apply
equally
to
rings
with
different
foond
and
higher
order
have
been
systematically
explored
These
findings
apply
equally
to
rings
with
different
focusing
ininxxand
yywith
modified
resonance
cusing
and
withappropriately
appropriately
modified
resonance
cusing
in x and
y with
appropriately
modified
resonance
for periodic
axi-symmetric
transport
systems
in Ref. [8].
CROMAP 2D particle-in-cell code employing 5x104
simulation particles on a 128x128 rectangular grid with
These findings apply equally to rings with different focusing in x and y with appropriately modified resonance
2nd
4thth th/ resonance
Some
mode
2nd
2nd coherent
4 / resonance
4th
/ instability
coefficients
C for
th
4th / instability
250
0x
0y
conditions. Thus, it is sufficient for an instability of the
envelopes to have the zero current phase advance per period sufficiently close to, but above 90° in at least one
of the directions. In order to confirm the appearance of
an envelope instability we have used the envelope eigenvalue solver KVXYG [14] and found that the present
linear lattice with a zero-current phase advance per cell
(T0 = 110° is subject to envelope instability in the region
2.68 < vy < 2.84, with an associated envelope growth
per cell by a factor A . This permits us to associate the
first part of the simulation stop-band with this mode,
while the rest of the simulation stop-band can be only
due to the structural fourth order resonance or instability
as predicted analytically in Fig. 4 and Table 1. To verify
the fourth order nature in this part of the stop-band we
show in Fig. 6 the associated structure in phase space for
Vy = 2.86 at an early and a more advanced stage. Note
^0,003
^0.003
0.002
0.002
0.001
0.001
0
0
-0.001
-0.001
-0.002
-0.002
driven by a n = 6 imperfection gradient term would not
be distinguishable from the envelope instability in our
example, but it would be observable in the zero current
limit, where the envelope instability vanishes. A more
complete study of second order resonances and the envelope instability in rings (including a weak imperfection
driven envelope instability) is found in Ref. [7]. For the
fourth order mode above we postulate the simultaneous
presence of resonance driven by the structural fourth order term in the waterbag space charge potential, and of
structural instability - which may exist near (T0 = 90°
following Ref. [8] - similar to the self-consistent "Montague resonance" of the next section.
THE "MONTAGUE RESONANT
INSTABILITY"
FIGURE 6. Phase space projections in y for fourth order
structural response and Vy=2.86 after turn 2 and 30.
that the location of the coherent fourth order resonance
was calculated as Vy w 2.95 in Fig. 4, hence close to the
right edge of the simulation stop-band . The asymmetry
of the latter is a general nonlinear feature of resonances
with significant space charge already identified in the
nonlinear envelope simulation of Fig. 1. It is a combined
effect of the tune spread existent in a waterbag simulation
as well as the space charge detuning effect associated
with emittance growth of the fraction of particles on resonance. There is still an effective "coherent advantage" of
about 50% compared with the single-particle resonance
picture, where one expects the small amplitude particles
of a waterbag to be on resonance for v^ < 3.1.
In our case of strongly split tunes we find that the instability has almost no effect in the ^-direction. In order to distinguish between resonances and instabilities
we point out that in case of a resonance the driving term
is assumed to be given by the lattice or by the space
charge of the matched beam; for instabilities the driving term initially exists only on the noise level, but rises
- under resonant conditions - exponentially in time. A
characteristic feature of such instabilities is their complete absence in the zero-current case. In contrast, the
second order resonance 2vy + Aco = n, which could be
251
An important case of coupling resonance driven by
space charge only is the well-known "Montague resonance", a fourth order difference resonance first studied
in Ref. [15], It can be seen in a simplified way as a resonance driven by the zero-th harmonic of a quartic term
in the space charge potential of a non-uniform beam.
In machines with noticeably larger horizontal than vertical emittance it may lead to an exchange of emittances
and possibly loss due to vertical acceptance limitation.
It has been found in a number of synchrotrons and usually avoided by choosing a large enough tune split. For a
recent detailed experimental study see Ref. [16].
The original analysis by Montague was based on a single particle approach using a "frozen-in" space charge
potential defined by the initial Gaussian distribution,
which neglects the effect of the induced time-varying
collective space charge force. It is, however, adequately
described only by a coherent resonance condition of the
type 2vx — 2vy + Aco = 0. The self-consistent simulation
explored in detail in Ref. [17] shows, however, that this
case is in reality a combination of instability and resonance: it develops as a pure instability for a KV distribution, which has no initial driving nonlinearity as is
shown in Fig. 7 (top) for vy/vQ = 0.8 and £x/£y = 2. In
this simulation the focusing ratio was re-defined for each
data point such as to obtain a scan over a range of tune
ratios. Note that for equal emittances the KV distribution leads to a symmetric loop around vx/vy = 1 as was
recently shown in a study related to the SNS [18]. The
initial growth for the KV-case is found to be exponential,
starting from noise level (see Ref. [17]).
An initial waterbag distribution, instead, has all features of a space charge driven resonance due to the presence of a finite nonlinearity from the beginning (bottom of Fig. 7). The instability is, however, still superimposed on the resonance phenomenon. We conclude
this by comparing it with a case, where the initial wa-
16.016.0
terbag
space
isis “frozen-in":
"frozen-in":
the
maxterbagspace
spacecharge
chargedistribution
distributionis
“frozen-in": the
the maximum
imumemittance
emittanceexchange
exchangereaches
reaches only
only half
half the
the amount
amount
shown
shownininFig.
Fig.77 due
dueto
to absence
absence of
of aa coherent
coherent response.
response.
Effective
exchange
occurs
for
specific
shifted
Effective exchange occurs for specific tune
tune ratios
ratios shifted
shifted
Effective
11
-0.5
-0.5
11
∆ε
∆ε/ε/ε
111
/νyy
ννxx/ν
1.05
1.05
1.05
1.1
1.1
-0.5
-0.5
0.95
0.95
0.45
0.45
0.55
0.55
15.5
15.5
ννV0x
Ox
0x
15.0
15.0
15.0
16.0
16.0
16.0
and analytical
analytical theory
confirmedthe
existence
and
theory results
results and
and confirmed
confirmed
the existence
existence
of
resonant
instabilities
in
second
and
fourth
second and
and fourth
fourth order.
Both
of resonant instabilities in second
order. Both
Both
are shown
shown to
to give
significant “coherent
"coherentadvanadvanare
give rise
rise to
to aaa significant
significant
“coherent
advantage”, which
tage",
which should
should be
be useful
useful for
for bunch
bunch compression
compressioninin
tage”,
=
3.
These
studies
also
increase
ν
SIS18
near
y
SIS
18
near
Vy
=
3.
These
studies
also
increase
ourconficonfiour
confiSIS18 near νy = 3. These studies also increase our
dence in
in the
the simulation
should be
dence
simulation codes
codes and
and should
should
be considered
considered
as aa basis
basis for
for further
as
further more
more systematic
systematic studies.
studies.
00
0.9
0.9
-15.5
15.5
15.5
FIGURE 8.
8. Stop-band
FIGURE
Stop-band of
of self-consistent
self-consistent Montague
Montague resoresocolor scales
scales indicate
growth
of
nance: color
indicate saturated
saturated
growth
of the
theoriginally
originally
saturated
smaller emittance
emittance in
εεyy//εεyy..
Ae-y/e-y.
smaller
in units
units of
of ∆
∆
εεxx
εεyy
0.5
0.5
0.15
0.15
0.35
0.35
15.0
15.0
15.0
15.0
0.95
0.95
0.05
0.05
0.25
0.25
00
0.9
0.9
∆∆εεyy/ε
/εyy::
15.5
εεxx
εεyy
0.5
0.5
∆ε
∆ε/ε
/ε
ν0y
16.0
16.0
16.0
∆νy=0.5
εx/εy=2
11
/νy
ννxx/ν
y
1.05
1.05
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et al., these
ProceedingsNo.
16. I.
I. Sakai
Sakai et
et al.,
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these
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these
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Hofmann
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O. Boine-Frankenheim,
Phys. Rev. Lett.
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Rev. ST Accel. Beams 4, 084202 (2001)
(2001)
FIGURE 7.7. Saturated emittance exchange for KV (top)
FIGURE
Saturated emittance exchange for KV (top)
and waterbag (bottom) initial distributions, with vνy/v/ν0y
)=0.8,
and
waterbag (bottom) initial distributions, with νyy /ν0y0y)=Q.S,
)=0.8,
ε
/
ε
=2,
as
function
of
ν
/
ν
.
x
y
x
y
£εxx/£
/εyy=2,
=2, as function of νvxx/v
/νyy..
from the single particle condition 2νx —
− 2ν = 0 due to
from
from the single particle condition 22v
ν − 22vνyy = 0 due to
spacecharge.
charge.ItItcan
canbe
be shown [17] thatx maximum growth
space
shown [17] that maximum growth
ratesasaswell
well as the
the width
width of
of the
the stop-band are roughly
roughly
rates
as
stop-band are
proportional to ∆ν , while the detailed shape of exchange
proportional to Av,
while the
the detailed
detailed shape of
of exchange
∆ν , while
profiles and maximum emittance exchange are nearly inprofiles and maximum emittance exchange are nearly independent of ∆ν . This enables us to construct a stopdependent of Av.
∆ν . This enables us to construct a stopband within the usual tune diagram of a circular machine,
band within the usual tune diagram of a circular machine,
which is shown in Fig. 8 for weakly split working points
which is shown in Fig. 8 for weakly split working points
(arbitrary integer parts), using the data of the waterbag
(arbitrary integer parts), using the data of the waterbag
simulation in Fig. 7. The asymmetry of the stop-band
in Fig. 7. The asymmetry of the stop-band
simulation
with respect to the single
particle condition 2νx −2νy = 0
with
respect
to the 8) is due
particle
2v
— 2v
condition
νx −2
νy = 0
(dotted line in Fig.single
to the
coherent2space
charge
line
in
Fig.
to
the
coherent
charge
(dotted
8)
is
due
space
effect and the fact that εx > εy . Note that the width of
effect
and the fact
Noteshrinks
that thetowidth
of
εexx > εeyy. case
effect
thatwaterbag
the stop-band
for the
zero for
the
stop-band
for
the
waterbag
case
shrinks
to
zero
for
equal emittances; in this limit there is no energy to exequal
emittances;
in thisgrowth
limit there
is actually
no energy
to exchange.
The strongest
of εy is
closer
to
change.
The
growth
e
strongest
of
ε
is
actually
closer
to
y
2ν0x − 2ν0y = 0 than to 2νx − 2νy = 0.
2v
- 2v
=0
2ν0y
0 than to
to 22v
νxx−- 22v
νyy ==0.0.
2ν0x
0x −
0y =
CONCLUSION
CONCLUSION
We have found good agreement of space charge induced
We
have
good agreement
of space charge
induced
shifts
offound
resonances
between self-consistent
simulation
shifts of resonances between self-consistent simulation
shifts
252