Progress of Space Charge Calculation in the Code ORBIT
Progress of Space Charge Calculation in the Code ORBIT l
1
A. U. Luccio and N. L. D’Imperio
A. U. Luccio and N. L. D'lmperio
Brookhaven National Laboratory, Upton,NY
Brookhaven National Laboratory, UptonJSfY
Abstract.
Abstract. The
The code
code ORBIT
ORBIT [1]
[I] has
has been
been designed
designed for
for PIC
PIC tracking
tracking of
of beam
beam in
in aa high
high intensity
intensity hadron
hadron
synchrotron.
synchrotron.InInthe
thecode,
code,space
spacecharge
chargeforces
forces are
are continuously
continuously calculated
calculated and
and applied
applied to
to the
the individual macro
particles
particlesasasmomentum
momentumand
andenergy
energy kicks.
kicks. Space
Space charge
charge solvers
solvers in
in the
the code,
code, as
as developed at Brookhaven are
described.
described.
PIC
PICTRACKING
TRACKINGWITH
WITHSPACE
SPACECHARGE
CHARGE
AAPIC
PIC[2]
[2]herd
herdofofmacro
macroparticles
particles isis pushed
pushed through
through
a alattice
lattice represented
represented by
by maps
maps[3]
[3] (external
(external fields).
fields). At
At
’space
'spacecharge
chargenodes’
nodes'ininthe
thelattice
lattice (internal)
(internal) forces
forces are
are
calculatedand
andapplied
appliedtotothe
themacros
macrosas
askicks.
kicks. We
Weuse
use aa
calculated
SplitOperator
Operatortechnique
techniquewith
withindependent
independent treatment
treatment of
of
Split
mapsand
andkicks.
kicks.For
Forspace
spacecharge
chargekicks
kicksthe
theherd
herdisisbinned
binned
maps
ona agrid
gridaccording
accordingtoto(x,
(jc,y,
cA^)totofind
findthe
thecharge
chargedensity
density
on
y, c∆t)
andaccording
according toto(p(px ,x,p
find the
the current
current
ρp,
, and
pyy,,Ap/p),
∆p/p), toto find
~
densityj.j.Then,
Then,we
wesolve
solvethe
thepartial
partial elliptic
elliptic differential
differential
density
equationsfor
forQ(source
^(sourcepoint)
point)→
—>P(field
P(field point)
point)
equations
∇2 Φ(P) = − ρ ε(Q)
0
~
; ∇2~A(P) = − j(Q)
µ
(1)
(1)
0
findthe
thescalar
scalarΦOand
andthe
thevector
vectorpotential
potential~A.
A.
totofind
For
long
bunches,
the
beam
current
is
parallel
walls,
For long bunches, the beam current is parallel totowalls,
electricrepulsion
repulsionand
andmagnetic
magneticattraction
attractionpartially
partially comcomelectric
pensateand
andwe
wemay
mayapprox.
approx.only
onlysolve
solvefor
forΦ
O(multiplied
(multiplied
pensate
2
byaafactor
factor1/I//
thefollowing
followingwe
wewill
will refer
refer to
to this
this
γ 2).).InInthe
by
approximation.
approximation.
Independentvariable
variablecan
canbe
betime
time t t or
or the
the longitudilongitudiIndependent
nalcoordinate
coordinates.s.Time
Timeisisattractive
attractivebecause
because Eqs.(1)
Eqs.(l) are
are
nal
solvedwith
withall
allmacros
macrosatatthe
the same
same t.t. Space
Space isis conveconvesolved
nientininperiodic
periodicaccelerators
acceleratorsasasparticles
particles cyclically
cyclically pass
pass
nient
through
the
same
positions
and
maps
are
referred
to
s.
Eithrough the same positions and maps are referred to s. Either
way,
we
must
apply
relativistic
transformations
bether way, we must apply relativistic transformations betweenspace
spaceand
andtime.
time.ORBIT
ORBITuses
usesspace
spaceasasthe
theindepenindepentween
dent
variable.
dent variable.
3DTREATMENT
TREATMENTOF
OFSPACE
SPACECHARGE
CHARGE
3D
Useaatransverse
transversegrid
gridterminated
terminated atat wall
wall boundary,
boundary,and
and
Use
a
longitudinal
grid
over
the
length
of
the
beam
bunch.
a longitudinal grid over the length of the beam bunch.
Forlong
longbunches
buncheswe
wecan
cancomfortably
comfortablymake
makessgrid
gridsteps
steps
For
larger
than
transverse,
since
(i)
the
space
charge
distribularger than transverse, since (i) the space charge distributionvaries
variessmoothly
smoothlyalong
alongthe
thebeam,
beam,and
and(ii)
(ii)the
themotion
motion
tion
alongthe
thebeam
beamisismuch
muchslower
slowerthan
thanininthe
thetransverse.
transverse.
along
1
1
Work performed under the auspices of the U.S.Department of Energy
Work performed under the auspices of the U.S.Department of Energy
FIGURE 1.1. Slicing
Slicing aa beam.
beam. Wavy
Wavy lines:
lines: envelope
envelope of the
FIGURE
beam ((/3-wave).
Dashed vertical
vertical lines:
lines: planes where to solve
β -wave). Dashed
beam
Poisson
Poisson
For the
the longitudinal
longitudinal grid
grid we
we cut
cut the
the beam in slices,
slices,
For
long enough
enough that
that in
in each
each the
the average
average density, the translong
verseaspect
aspect ratio,
ratio, and
and the
the surrounding
surrounding wall configuration
configuration
verse
can be
be considered
considered constant
constant (Fig.
(Fig. 1).
1).
can
ρ (x, y, z) = ρ⊥ (x.y) ρk (z),
(2)
and solve
solve the
the 2D
2D transverse
transverse problem
problem simultaneously
simultaneously in
and
each slice,
slice, by
by parallel
parallel computing
computing .. As
As aa slice length we
each
use aa fraction
fraction of
of the
the beam
beam envelope
envelope wavelength. A slice
use
associated with
with the
the local
local wall
wall configuration
configuration -A similar
isisassociated
approach has
has been
been used
used by
by L.G.Vorobiev
L.G.Vorobiev et al [4],[5]approach
Each macro
macro particle
particle reach
reach the
the same
same node at different
different
Each
time. To
Toreconstruct
reconstruct the
the beam
beam at
at aa fixed
fixed time we project
time.
macros forward
forward and
and backward
backward using
using the
the maps
maps for
for the
the
macros
bare
lattice
-more
accurately,
we
should
use
the
selfbare lattice -more accurately, we should use the selfconsistent lattice
lattice including
including extra
extra focusing
focusing due
due to
to SC
SC
consistent
forces.
It
is
an
approximation,
however
consistent
with
forces. It is an approximation, however consistent with
thesplit
split operator
operator techniquetechniquethe
POISSON SOLVERS
SOLVERS
POISSON
Inits
itsintegral
integral form,
form, Poisson
Poisson Eq.(1a)
Eq.(la) is
is
In
φ (P) =
1i
4πε0 γ 2
Z
rp(Q)
ρ (Q) dQ,
dQ,
r
(3)
(3)
with rr =
= |P
\P−
—Q|
Q\ +
+ εe22,, and
and ε£ aa smoothing
smoothing parameter
parameter
with
to
avoid
poles.
The
equation
can
be
solved
by direct
direct
to avoid poles. The equation can be solved by
integration
(BF),
a
slow
but
very
transparent
method
integration (BF), a slow but very transparent method
good as
as aa check,
check, or
or by
by FFT,
FFT, with
with the
the integral
integral reduced
good
reduced
to
a
convolution
to a convolution
While in
in the
the integral
integral form
form the
the image
image on
on walls
walls is
is part
part
While
of
the
input,
in
differential
Poisson
solvers
with
boundary
of the input, in differential Poisson solvers with boundary
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
253
0.10.1
0.08
0.08
0.06
0.06
BF BF
SOR
SOR
0.04
0.04
FIGURE
FIGURE2.2.
2. Solving
Solvingwith
with
perfectly
conducting
walls
FIGURE
Solving
withperfectly
perfectlyconducting
conductingwalls
walls
0.02
0.02
64 64
0
(4)
(4)
(4)
Twodifferential
differential2D
2Dsolvers
solverswere
were
implemented
ORTwo
Two
differential
2D
solvers
wereimplemented
implementedinin
inORORBIT:
(i)
LU
Decomposition
plus
matrix
multiplication,
BIT:
(i)
LU
Decomposition
plus
matrix
multiplication,
BIT: (i) LU Decomposition plus matrix multiplication,
and(ii)
(ii)Successive
SuccessiveOver
OverRelaxation
Relaxation
(SOR).
and
and
(ii)
Successive
Over
Relaxation(SOR).
(SOR). 2
2
For
(i),
express
the
Laplacian
operator
discrete
For
(i),
express
the
Laplacian
operator
For (i), express the Laplacian operator∇V
∇2inin
indiscrete
discrete
form
on
a
M
x
N
grid
that
extends
to
wall
form
on
a
M
×
N
grid
that
extends
to
wall
form on a M × N grid that extends to wall
1 1 −1−1
−4
πρ
ρ (Q)
Liklj kl
Φkl ; ; Φ(P)
i j i=
−4
πρ
ρ (Q) (5)(5)
Φ(P)==−−
4π4L
ij Φ
j=L
kl
πL
2
2 2×
2 22 band-sparse
J£ isis
isanan
anMM
M
xNN
N
band-sparsematrix
matrix
KroLL
×
band-sparse
matrix(δ(8
(δ isis
isKroKronecker's),whose
inverse
is
unfortunately
not
sparse
necker’s),whose
inverse
is
unfortunately
not
sparse
necker’s),whose inverse is unfortunately not sparse
kl
k ^8'j
k kl l
l l
+δi8?
l δ k k δ jl +
k δ jl +
LL
= −4δ k δk l +
δikδδkj−1
i j ikl
δi+1 δ jl +δi−1
δi−1
δ jl +
δδi j+1
δ j+1++
j = −4iδi jδ j +i+1
i δ j−1
and multiplythe
the inversebybythe
the M xNN~ρp.
and
andmultiply
multiply theinverse
inverse by theMM××
N ~.ρ .
For(ii),
(ii),solve
solvebybyiteration,
iteration,starting
startingwith
with a guess.At
At
For
For (ii), solve by iteration, starting witha aguess.
guess. At
stepk +
k +1 the
1 thediscretized
discretizedPoisson’s
Poisson'sisis
step
step k + 1 the discretized Poisson’s is
k+1 1 1 1 k
k
k
k
k+1
k
k
k
k
ΦΦ
ρ
Φ
+
Φ
+
Φ
+
Φ
−
=
^j
=
\
i−1, j + Φ
i, j+1 + Φ
i+1, j + Φ
i, j−1 −i,ρj . .
i, j
Φ
i−1, j
i, j+1
i+1, j
i, j−1
i, j
i, j =4
4
Since
the
beam
density
evolves
slowly
from
one
space
Since
the
beam
density
evolves
slowly
from
Since
the
beam
density
evolves
slowly
fromone
onespace
space
chargenode
node
tothe
the
next,iterative
iterativetechniques
techniques
show
rapid
charge
to
next,
show
rapid
charge
node
to
the
next,
iterative
techniques
show
rapid
convergence.Iterative
Iterativeprocedures
proceduresused
usedwere:
were:Basic
BasicSOR
SOR
convergence.
convergence.
Iterative
procedures
used
were:
Basic
SOR
(mostefficient
efficientfor
forsmall
smallgrids,
grids,M,
M,N
128),SOR
SORwith
with
(most
NN<<<128),
(most
efficient
for
small
grids,
M,
128),
SOR
with
Chebychevacceleration
acceleration(large
(largegrid),
grid),and
andConjugate
ConjugateGraGraChebychev
Chebychev
acceleration
(large
grid),
and
Conjugate
Gradient,that
thatshowed
showedthe
themost
mostrapid
rapid convergence.
dient,
dient,
thatare
showed
theintroduced
most rapidconvergence.
convergence.
Walls
naturally
differentialPPsolvers.
solvers.
Walls
are
naturally
introduced
inin
differential
Walls
are
naturally
introduced
in
differential
P solvers.
For
a
perfect
conductive
walls
Ip^
=
^Lp
Asa a
image
For
a aperfect
conductive
walls
ρ ρ .-As
∑ ρbeam
For
perfect
conductive
walls
ρbeam==∑2D
∑image
schematic
example,
examine
the∑
explicit
calculation
image . As a
schematic
example,
examine
the
explicit
2D
calculation
schematic
example,
examine
the are
explicit
2D calculation
onthe
thegeometry
geometry
ofFig.2.
Fig.2.
Walls
represented
bynn◦,o,
onon
ofof
Walls
by
the
geometry
Fig.2.
Wallsare
arerepresented
represented
by n ◦,
the
interior
by
m
•.
The
system
of
equations
is
exactly
the
interior
mm•.•.The
ofofequations
isisexactly
the
interiorbyby
Thesystem
system
equations
exactly
determined:
n
+
m
known
quantities,
i.e.
O
=
0
at
the
determined:
n n++mmknown
quantities,
nnn◦o◦
determined:
known
quantities,i.e.
i.e.Φ
Φ=
=0 0atatthe
the
and
p
at
the
m
•;
m
+
n
unknowns,
i.e.
m
values
of
O
to
and
ρ ρatatthe
mm++n nunknowns,
i.e.
mmvalues
ofofΦΦtoto
and
themm•;
•;the
unknowns,
i.e.
values
be
calculated
at
*'s
and
n
values
of
p
at
the
o's.
image
bebecalculated
atatthe
•’s and
n nvalues
of ρimage
atatthe
calculated
the
values
ρimage
the◦’s.
◦’s.
Once
we have
O,•’s
theand
space
chargeofforce
and
the
moOnce
we
have
Φ,
the
space
charge
force
and
the
moOnce
we
have
Φ,
the
space
charge
force
and
the
momentum kick on each macro are
mentum
mentumkick
kickononeach
eachmacro
macroare
are
e~
~
F(P)
~φ ; ;
~
F(P)==γ 2γe∇
2 ∇φ
∆~p
1R
p∆~p==p 1
p
p
R~
F dt
~
F dt
128128
160160
192192
224224
256256
FIGURE
3.3.3. Comparison
ofofof
field
between
2-D
BFBF
and
SOR,
FIGURE
Comparison
field
SOR,,
FIGURE
Comparison
field
between
2-D
and
SOR,
with
walls
bin
256
with
walls
atatat
bin
256
with
walls
bin
256
conditions
conditionsthe
theimage
imageisis
part
the
answer
conditions
the
image
ispart
partofof
ofthe
theanswer
answer
ρ (P)
2 Φ(P)
ρ (P)
2 Φ(P)
)=0
∇∇
==−−
Φ(P
wall
ε0ε . . ; ; Φ(P
wall ) = 0
96 96
(6)
(6)(6)
With
kt=
L//3c,
the
transverse
momentum
kick
and
β c,
the
transverse
momentum
kick
and
With
dtdtdt===
∆t
L/
β c,
the
transverse
momentum
kick
and
With
∆t==
L/
the
longitudinal
energy
kick
are
the
longitudinal
energy
kick
are
the longitudinal energy kick are
δ pδ⊥p
∂φ φ
⊥ ℘ ∂L
℘
p p ==
∂ r∂ r⊥L⊥
; ;
∂ φ∂ φ
δ ∆E
δ ∆E β 2℘
β 2℘
E E ==
∂ z ∂LzkLk
(7)
(7)
(7)
with
the
separation
between
successive
transverse
SC
with
withthe
theseparation
separationbetween
betweensuccessive
successivetransverse
transverseSC
SC
kicks
Lj_,
the
separation
between
longitudinal
kicks
L,,,
kicks
L
,
the
separation
between
longitudinal
kicks
L
kicks ⊥L⊥ , the separation between longitudinal kicks kL, k ,
4πλ
qhrqhr
0 »the charge per unit length A
4πλ
the
perveance
^
℘£?
===∆x^02
λλ
charge
perper
unit
length
the
2 γ 3 m ,0 the
℘
, the
charge
unit
length
theperveance
perveance
β
∆xβ 2 γ 30m0
and
the
size
of
a
square
grid
cell
Ax.
and
a square
grid
cell
∆x.
andthe
thesize
sizeofof
a square
grid
cell
∆x.
COMPARISON
OF
2D
SOLVERS
COMPARISON
COMPARISONOF
OF2D
2DSOLVERS
SOLVERS
Wecompared
compared the SC field calculated
calculated with
with aa BF
BF integral
integral
We
We comparedthetheSCSCfield
field calculated
with a BF
integral
and
a
SOR
differential
solver
on
a
256
x
256
grid,
conand
××
256
grid,
conanda aSOR
SORdifferential
differentialsolver
solveronona 256
a 256
256
grid,
conductive
walls,
and
a
Gaussian
random
beam
in
the
chamductive
walls,
and
a Gaussian
random
beam
inin
thethe
chamductive
walls,
and
a Gaussian
random
beam
chambercenter
center (Fig.3).The
TheBF
BFfield
field goes
goes to
to zero
zero at
at large
large disdisber
ber center(Fig.3).
(Fig.3). The
BF field
goes to
zero at
large distance,
while
SOR
ends
at
the
walls
with
a
finite
value,
tance,
SOR
ends
with
a finite
value,
tance,while
while
SOR
endsatdensity
atthethewalls
walls
with
a finite
where
the
image
charge
is equal
equal
to the
the
field value,
and
where
the
image
charge
density
is
to
field
and
where
the
image
charge
density The
is equal
toofthe
field
and
the
field
lines
are
perpendicular.
sum
the
image
the
field
lines
areareperpendicular.
The
sum
ofofthetheimage
the
field
lines
perpendicular.
The
sum
image
chargesequals
equalsthe
thetotal
total of the
the real charges.
charges. In
In this
this case
case
charges
charges
equals
the totalofdistributed.
of thereal
real charges.
In this
case
the
images
areuniformly
uniformly
the
images
are
distributed.
the
images
are
uniformly
distributed.
Fig.4shows
showsthe
the imagecharge
chargeon
on the
the walls
walls for
for an
an offoffFig.4
Fig.4
shows theimage
image charge
on the
walls for
an offset
beam.
setsetbeam.
beam.
3DFORCES
FORCES IN A LONG BEAM
BEAM
3D
3D FORCESININAALONG
LONG BEAM
Transversekicks
kicksdepend
dependon
onthe
thetransverse
transverse aspect
aspect ratio
ratio of
of
Transverse
Transverse
kicks
depend
onforce
the transverse
aspect larger
ratio of
a
slice.
For
the
same
&
the
is
on
the
average
a slice.
thethe
same
℘, the
force
is is
onon
thethe
average
larger
a slice.For
For
same
, the
force
average
where
the
valueof
of the
the℘
Twiss
/3 function
function
is smaller,
smaller, larger
and
where
the
value
Twiss
β
is
and
where the This
valueisofshown
the Twiss
β function
isfrom
smaller,
and
vice-versa.
in
Fig.5
and
Fig.6
a
SOR
vice-versa.
This
is isshown
inin
Fig.5
and
Fig.6
from
a SOR
vice-versa.
This
shown
Fig.5
and
Fig.6
from
a SOR
simulation
of
a
FODO
lattice.
simulation
ofof
a FODO
lattice.
simulation
athe
FODO
lattice. kick calculated from the
Fig.77shows
shows
longitudinal
Fig.
the
longitudinal
kick
calculated
from
thethe
Fig.
7
shows
the
longitudinal
kick
calculated
from
difference of potential between analogous
(jc,y) points
in
difference
of
potential
between
analogous
(x,
y)
points
in
difference
potentialplane
between
analogousslices
(x, y)ispoints
the
median of
transverse
of successive
com-in
the
transverse
plane
ofof
successive
slices
is is
comthemedian
median
transverse
plane
successive
slices
compared
with the
traditional
formula
for a beam
of radius
pared
with
the
traditional
formula
for
a
beam
of
radius
with the
a beam
of radius
apared
in a round
pipetraditional
of radius bformula
[6]. Z0 for
is the
impedance
of
a ainina round
b [6].
Z0Zis is
thethe
impedance
ofof
a roundpipe
pipeofofradius
radius
b [6].
impedance
254
0
7 0 the charge gradient along the beam.
free space
spaceand
andA
free
charge gradient along the beam.
free
λλ0 the
free space and λ 0 the chargeλ0 gradient
along
the beam.
0
bb + f (r) .
λ
(∆E)
∝
Z
1
+
2
ln
(∆E)SC
(8)(8)
SC ∝ Z000 22 1 + 2 ln + f(r) .
b aa
λ 22γγ
256
256
256
128128
(∆E)SC ∝ Z0
128
2γ 2
1 + 2 ln + f (r) .
a
(8)
0 0
0
−128
−128
0.0004
0.0004
−128
0.0004
I from
from
fromlong.
long.SCSCkick
kickformula
formula
−256
−256
−256
−256
−128
−128
00
128
128
0.0002
0.0002
256
256
−256
−256
−128
0
128
256
FIGURE
Beam
offset
in
and
in
square
FIGURE4.4.
4. Beam
Beamoffset
offsetin
inxxxand
and yyy in
in aaa square
square chamber,
chamber. pρ
FIGURE
chamber.
and
o
.
Calculated
by
differential
SOR
.
Calculated
by
differential
SOR
and
σ
FIGURE
Beam offset
in x and y SOR
in a square chamber. ρ
image
.4.Calculated
by differential
and
σimage
image
and σimage . Calculated by differential SOR
0O
O
O
*«/*
F
B
O
B
O
O
B
B
F
F
DD
O
O
D
F
DD
D
0O
OO
00
0
-0.0002
-0.0002
D
D
0O
O
F
F
B
BB
B B
B
B
B
O
0
O O
F
-10
BB
B
Thu Apr409:46:182002
4 09:46:18 2002
ThuApr
Thu Apr 4 09:46:18 2002
A A rvA~;
<=*
macro/slice
-••macro/slice
macro/slice
max[F
x]/(macro/slice)
Bmax[F
max[FJ/(macro/slice)
x]/(macro/slice)
1/2 1/2
βmacro/slice
yβ
C
y
y
185
190
195
0
2
2
2
4
4
4
6
6
6
8
8
8
10
10
10
High Energy Accelerators. Wiley, New York, 1993.
max[F ]/(macro/slice)
180
0
J.D.G
ALAMBOS , J.A.H
OLMES , D.K.O
LSEN A.L
UCCIO
J.D.GALAMBOS,
J.A.HOLMES,
D.K.OLSEN
A.Luccio
1. 1.
ALAMBOS , J.A.H
OLMES , D.K.O
LSEN A.L UCCIO
1.J.D.G
J.D.G
ALAMBOS
, J.A.H
OLMES
, D.K.O
LSEN
A.L
UCCIO
and
J.B
EEBE -WANG
:
Orbit
User’s
Manual
Vers.
1.10.
and
J.BEEBE-WANG:
User's
Manual
andand
J.BJ.B
EEBE -WANG : Orbit User’s Manual Vers. 1.10.
EEBE
-W
ANG
:
Orbit
User’s
Manual
Vers.
1.10.
Technical
Report
SNS/ORNL/AP
011,
Rev.1,
1999.
Technical
Rev.l,
Technical
Report SNS/ORNL/AP 011,Oil,
Rev.1,
1999.
Technical
Reportand
SNS/ORNL/AP
011,: Rev.1,
1999.
2.R.W.H
R.W.H
OCKNEY
J.W.E
ASTWOOD
Computer
R.W.HOCKNEY
J.W.EASTWOOD:
2. 2.
OCKNEY
and J.W.E
ASTWOOD
: Computer
2.
R.W.H
OCKNEY
and
J.W.E
ASTWOOD
:
Computer
Simulation
Using
Particles.
Adam
Hilger,
IOP
Publishing,
Simulation
Using
Simulation
Using
Particles. Adam Hilger, IOP Publishing,
Simulation
Using Particles. Adam Hilger, IOP Publishing,
New
York,
New
York,
1988.
New
York,
1988.
New
York,
1988.
3.H.G
H.G
ROTE
F.C
H .I SELIN
MAD
program,
Vers.8.19.
H.GROTE<mdF.Cll.ISELlN:The
MAD
program,
3. 3.
ROTE
andand
F.C
H .I SELIN
: The: The
MAD
program,
Vers.8.19.
3.
H.G ROTE
andCERN/SL/90-13,
F.C
H .I SELIN : The
MAD
program,
Vers.8.19.
Technical
Switzerland,
Technical
Report
CERN/SL/90-13,
Geneva,
Technical
Report
Geneva,
Switzerland,
Technical Report CERN/SL/90-13, Geneva, Switzerland,
1996.
1996.
4.L.G.V
L.G.V
OROBIEV
R.C.Y
: Beam
1996.
L.G.VOROBIEV
R.C.YORK:
Dynamics
4. 4.
OROBIEV
andand
R.C.Y
ORKORK
: Beam
Dynamics
Modeling
By
Sub-Three-Dimensional
Particle-In-Cell
4.
L.G.VBy
OROBIEV
and R.C.YORKParticle-In-Cell
: Beam
Dynamics
Modeling
Sub-Three-Dimensional
Modeling
By
Sub-Three-Dimensional
Conference
2001,
Modeling
By Particle
Sub-Three-Dimensional
Particle-In-Cell
Code.
In:In:
Proc.
Accelerator
Conference
2001,
Code.
Proc.
Particle
Accelerator
Conference
2001,
2001.
Paper
RPAAH097.
2001.
Paper
Code.
In:RPAAH097.
Proc.
Particle Accelerator Conference 2001,
5.L.G.V
L.G.V
OROBIEV
R.C.Y
: Method
Of Template
5. 5.
OROBIEV
andand
R.C.Y
ORKORK
: Method
Of Template
2001.
Paper
RPAAH097.
L.G.VOROBIEV
R.C.YORK:
Potentials
Find
Space
Charge
Forces
High-Current
Potentials
To To
Find
Space
Charge
Forces
For For
High-Current
Potentials
Find
Space
Charge
5.
L.G.VOROBIEV
and
R.C.Y
ORK
: Method
Of
Template
Beam
Dynamics
Proc.
Particle
Accelerator
Beam
Dynamics
Beam
Dynamics
Simulation.
In: Proc.
Particle
Accelerator
Potentials
To Simulation.
Find
SpaceIn:
Charge
Forces
For
High-Current
Conference
2001,
2001.
Paper
RPAAH098.
Conference
2001,
2001.
Paper
RPAAH098.
Conference
2001,
Beam Dynamics
Simulation.
In:
Proc. Particle Accelerator
6. 6.
HAO
: Physics
of
Collective
Instabilities
in in
6.A.W.C
A.W.C
HAO
:2001,
A.W.CHAO:
Physics
of Collective
Beam
Instabilities
in
Conference
2001.
Paper Beam
RPAAH098.
High
Energy
Accelerators.
Wiley,
New
York,
1993.
High
Energy
Accelerators.
Wiley,
New
York,
1993.
Energy
Wiley, New
York,
1993.
6. High
A.W.C
HAO :Accelerators.
Physics of Collective
Beam
Instabilities
in
x
1/2
non−freezing
comparison
non−freezing
comparison
non-freezing
comparison
β
Mon Apr 22 12:02:09 2002
Mon Apr 22 12:02:09 2002
Mon Apr 22 12:02:09 2002
-2
-2
0
REFERENCES
REFERENCES
REFERENCES
AUL Fx−vs−x.D1
non−freezing 195
comparison
190
190
195
190
195
s [m]
s [m]
s[m]
-4
-4
-2
O
AUL Fx−vs−x.D1
AUL Fx−vs−x.D1
185
185
185
-6
-6
-4
OO
O
FIGURE 5.
SCforce
forceFFFxxvs.
vs.xxxinin
ineach
eachofof
ofa aa40-slice
40-slicebeam
beam
FIGURE
5.5. SC
SC
force
vs.
each
40-slice
beam
FIGURE
x
FIGURE
5. SC
force
F
vs. x in each
oflocation
a 40-slice beam
x
whose
central
slice
is
in
a
defocusing
lattice
location
whose
central
slice
is
in
a
defocusing
lattice
whose central slice is in a defocusing lattice location
whose central slice is in a defocusing lattice location
180
180
180
-8
-8
-6
FIGURE 7. Longitudinal SC energy kick in a 9-slice beam
FIGURE
FIGURE 7.Longitudinal
Longitudinal
SC energy
kick
in a 9-slice
beam
FIGURE
SC energy
kickfor
in
a 9-slice
x,beam
y. Thick
(AGS). 7.
Each line: distribution
of kick
various
x,y.
(AGS).
Each
line:
distribution
of
kick
for
various
x,
y.
Thick
(AGS).
Each
line:
distribution
of
kick
for
various
x,
y.
Thick
line: simulation that uses Eq.(8)
line:
line:
simulation
thatthat
usesuses
Eq.(8)
line:
simulation
Eq.(8)
D
/+
Thu Apr 4 09:46:18 2002
-10
-10
-8
TueApr
Apr 22 16:37:24
16:37:242002
2002
Tue
Apr 2 16:37:24
Tue AprTue
2 16:37:24
2002 2002
DD
D
F
F
-0.0004
-0.0004
-0.0004
O
O
O
F
F F
-0.0002
D
F
O
OO
BB
O
B
BB
B
B
B
B
F
F
O
O
D
D
O
B
O
D
F
O
DD
B
B
FF
FF
B
B
B
B
B
B
O
0O
O
O
B
0O
OO
O
B
O
B
B
F
F
O
D
B
F
'/
D
D
O
0.0002
777 7 7 /
7 7 '/ 7 7 '! 7 7
77 7 / / /
//// 7 / /
/ 7 7 7 7 / 7
F
F
from long. SC kick formula
200200
200
205205
205
200
205
s [m]
FIGURE
FIGURE 6.
6. Maximum
Maximumtransverse
transversekick
kickinin
ina aa9-slice
9-slicebeam
beam
FIGURE
6.
Maximum
transverse
kick
9-slice
beam
(AGS)
compared
with
2D
calculation
(AGS)
compared
with
2D
calculation
(AGS)
compared
with
2D
calculation
FIGURE 6. Maximum transverse kick in a 9-slice beam
(AGS) compared with 2D calculation
Mon Apr 22 12:02:09 2002
255