Transcript Y. Sato: Electron cloud simulation using the ORBIT code

Electron Cloud Simulations
Using ORBIT Code
- Cold Proton Bunch model
March 14, 2007
EP feedback workshop
at Indiana University Bloomington
Yoichi Sato, IUCF
Y. Sato
IU ep workshop 2007
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Outline
ECloud Properties to Cold Proton Bunch in ORBIT
 Proton Bunch Slope Dependence of ECloud Growth and Energy
Distribution of Electrons Hitting Surface (triangular longitudinal line
density profiles)
Proton bunches of same head triangles and the effect of bunch slope
Proton bunches of same tail triangles and the effect of primary electron amount
 Prompt-Swept Electron Simulation and Discussion of Physics
Parameters with Comparing PSR Data
Constant proton loss rate to beam intensity
Assumption of proton loss rate function of beam intensity
 ECloud Recovery Simulation After Swept
Proton loss rate estimation
Inconsistency between the recovery estimation and prompt-swept slope fit
 Conclusions
Y. Sato
IU ep workshop 2007
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PBunch Triangular Profiles
120ns 110ns
PB tail PB tail
100ns
PB tail
90ns
PB tail
Energy band
250eV-300eV
corresponds to
SEY peak
•The longer tail of pBunch causes
the larger eCloud and its higher
growth rate. The steeper tail pBunch
gives the higher energy of hitting
The energy distribution of
electrons (E_0 > 250eV) and the
lower SEY that may lead the lower
surface hitting electrons
growth rate.
of off-peak-band reduces
ECloud growth
Y. Sato
IU ep workshop 2007
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PBunch Triangular Profiles
•The carry-over and primary
electrons before the peak of
pBunch slightly change eCloud
peak and growth rate.
However, the slight differences
imply that the pBunch effect to
drive electrons may be
mitigated by the existence of
inside electrons during the
beginning of pBunch tail slope.
And effect of carry •To have long head in pBunch
Effect of primary
electrons stored in the over electrons
profile is a possible way to
head of proton bunch surviving beam gap accumulate more protons in a
Y. Sato
IU ep workshop 2007
bunch.
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Future Study for Artificial PBunch
 Comparison of ep instability between Long head
profile (short gap) and isosceles triangular profile
(large gap) using ORBIT capability of simulating
proton bunch dynamics. Even a short gap can remove
the ep oscillation of carry-over eCloud in front of the
next bunch.
 Simulation of ECloud development to proton bunches
of triangular + saw profile. We can expect to mitigate
trailing edge multipaction. The saw ratio and
frequency should be optimized.
Y. Sato
IU ep workshop 2007
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Prompt-Swept Electron Data
from Electron Sweeper at PSR
Beam Pulse
HV pulse
 The prompt electrons come out at the tail of
the beam pulse. The Electron Sweeper (ES)
functions as a large area Electron Detector
(ED) until the HV pulse arrives.
 The swept electron signal is narrow during
HV at the end of the gap.
Swept electron signal
Electron Signal (prompt electron)
The two experimental features:
(1) swept electron slope is flatter than
prompt electron slope,
(2) swept electron slope has saturation
in high intensity beam.
Y. Sato
IU ep workshop 2007
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Prompt-Swept Electrons
constant proton loss rate to beam intensity in ORBIT
SEY_max = 1.5 ~ 2.0 & proton loss rate = 4E-6/turn
The ORBIT simulations are
performed for field free section,
where ED/ES locates in PSR.
Prompt e in ORBIT means peak of
surface current after recovery.
Swept e in ORBIT means EC line
density at PB head after recovery.
SEY_max = 2.0 & proton loss rate = 1E-8/turn ~ 4E-6/turn
PSR features are reproduced
qualitatively but not quantitatively.
In ORBIT, if both SEY_max &
proton loss rate are constant
to beam intensity, promptswept slopes are much flatter
than experimental data.
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Prompt-Swept Electrons: Assumption of
Proton Loss Rate Function of Beam Intensity
Now, we assume proton loss rate in drift space, where
ED/ES locates, is an increase function of beam intensity
(reasonable from space charge).
Starting from the prompt-swept values of
(7uC beam, 4e-6/turn proton loss rate), we performed
simultaneous fitting on both prompt swept slopes with
varying a single parameter “proton loss rate” to different
beam intensity, 4uC ~ 8uC beam.
The searched proton loss rate to fit simultaneously
experimental prompt-swept slopes becomes an almost
exponential function of beam intensity. This is another
possible way to estimate proton loss rate.
Y. Sato
IU ep workshop 2007
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Recovery of ECloud: PSR and ORBIT
7.5 uC proton pulse measured in PSR shows 5
turn recovery of EC peak after clearing electrons
in beam gap by HV-applied electron sweeper.
In ORBIT simulation, a set of parameters
[SEY_max=2.0, Proton loss rate ~1E-8/turn, eyield per lost proton=100] provides 5 turn recovery
of EC to 7.5uC proton beam in field free section.
Both surface current and line density of EC show the
same number of recovery turns
Same carry-over electrons in PB head of [2nd turn,
p_loss=2E-8/turn] and [3rd turn, p_loss=1E-8].
Difference of EC peaks comes from new primary
electrons (Trailing Edge Multipaction).
Energy range of electrons hitting surface keeps almost same
in recovery turns. The amount of carry-over electrons has
little effect on the energy range.
The carry over electrons before PB peak
have weaker effect on EC than the new
primary electrons after PB peak.
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Inconsistency of proton loss rate
between recovery estimation and
prompt-swept slope fit
 To perform the simultaneous fit on prompt-swept slopes, we start
from the case of [SEY_max=1.7, 7uC beam, p_loss=4e-6/turn].
However, if we try to start from the case of [SEY_max=1.7, 7.5uC
beam, p_loss=1e-7/turn], which matches the recovery estimation,
we cannot reproduce prompt-swept slopes.
 [SEY_max=2.0, 6.3uC beam] has minimum of eCloud (both prompt
and swept) in lowering p_loss < 1e-8/turn.
 The recovery process may be dominated by other eCloud sources.
Now, in our simulation, only primary electrons from lost proton
and secondary emitted electrons are considered as eCloud
source.
 If electrons traveled from quadrupole area to drift space are
dominant, the number of recovery turns depends on the traveling
time.
Y. Sato
IU ep workshop 2007
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Screening Net/Wire as EC Remedy
Net/Wire in longitudinal
An idea to reduce the ECE in the vacuum chamber.
It is a positive-voltaged screening net on vacuum
chamber surface. The thin net is a set of conducting
wires longitudinally aligned, close to vacuum
chamber surface but apart from it. The geometry
should be optimized.
+V_sc
V=0
The net is positively voltaged, while
the chamber surface is grounded.
Y. Sato
IU ep workshop 2007
A lost proton passes through the net, hits and
scratches the chamber surface, and produces
primary electrons. The primary electrons are attracted
to the net of positive voltage. The following primary
electrons also stay around the net. They are going to
form an electron cloud layer between the net and
chamber surface.
Each electron feels the screening effect from the
electron cloud layer. Therefore, the effect of potential
decrease in the trailing edge of proton bunch is
mitigated for these electrons.
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Conclusions
 Longer head beam profile is a possible way to
accumulate more protons without making electron cloud
larger. Proton bunch ep dynamics should be examined.
 Simulated prompt-swept e of constant p_loss and
SEY_max reproduces qualitative features, but not
quantitative.
 Introducing p_loss function to beam intensity is a
possible way for quantitative discussion. A possible
p_loss function that fits prompt-swept slopes
simultaneously is exponential.
 Comparing the number of turns of recovery process
between experimental data and simulations, proton loss
rate of 7.5uC beam is ~ 1E-8/turn for SEY_max=2.0 and
~1E-7/turn for SEY_max=1.7.
Y. Sato
IU ep workshop 2007
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Conclusions, cont.
 Both of recovery estimation and p_loss function assumption indicates
quite low proton loss rate for low beam intensity in field free straight
section, and it would hard to cause ep-instability. ECloud kick on
proton beam could be mostly non-straight section.
 There is inconsistency between the recovery estimation of proton loss
rate and the assumption of proton loss rate function of beam intensity
is needed. We cannot perform the simultaneous fitting of promptswept electron slopes including the point of the parameters estimated
by recovery process. We may need additional model of ECloud
source.
 The actual PSR proton bunch profiles depend on beam intensity.
However, this effect does not change prompt-swept electron slopes so
much.
 Therefore, we need to have other model of ECloud source as shoot
electrons from quadrupole area.
Y. Sato
IU ep workshop 2007
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Appendices
 Base Classes of Electron Cloud Module (ECM)
 ECM Benchmarks
 Two stream model
 Cold PBunch Simulations
 Electron Cloud Experiments in PSR
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Main Concern of ORBIT Electron
Cloud Module: e-p problem
Lost proton
Secondary
electrons
e- born at wall
from proton loss
released e-
Tertiary
electrons
energy
Secondary gain
electrons
Captured ebefore bunch
Proton beam bunch
PSR: ~60 m in 90 m ring
SNS: ~200 m in 248 m ring
Vacuum Chamber Wall
Using Electron Cloud Module in ORBIT, we can simulate
Two stream instability to captured electrons inside beam bunch (serious source of ECE)
Trailing edge multipactor (big source of electrons)
multipaction = increase of electrons from secondary e- emission
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Electron Cloud Module in
ORBIT Code: Simulation Approach
L=248 m (SNS) and about 1000 turns
Proton Bunch
Electron Cloud Region
Pipe
• Simulate: a building up an electron cloud, its dynamics, and its effect on
a proton bunch during the whole accumulation period (~1000 turns) or at
least for several turns to detect the development of instability.
• PIC method for both p-Bunch and e-Cloud.
Proton beam 3D SC potential grid
Electron Cloud Grids with few (one)
2D slices aligned longitudinally
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Electron Cloud Module in
ORBIT Code: Structure
ORBIT Code
Super Code Interface Modules
Original ORBIT
EP_Node.cc
Original ORBIT
C++ Classes
ECloud.cc
EP_NodeCalculator.cc
E-Cloud Module: Independent C++ Classes
- ORBIT Electron Cloud Module
•The ORBIT E-Cloud Module is a collection of C++ classes. Only three
classes connect the E-Cloud module with the original ORBIT code, so the
module can be easily modified to use in other acceleration code or
independently.
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Base Classes of the Electron Cloud Module
eBunch
EP_Boundary
Grid3D
Surfaces Classes
Keeps 6D – coordinates of the macro-electrons,
provides method to add and to delete macroparticles. It has parallel capabilities.
It is a 2D SC solver and keeps the transverse 2D
grid parameters.
3D grid with references to the EP_Boundary class.
It has parallel capabilities. The subclasses deal
with SC density and potential.
Collection of the classes describing different
surfaces.
Field Source Classes The collection of classes specifying electrostatic
and magnetic fields. Now it includes p-Bunch, eBunch, and uniform fields.
Tracker
Tracks macro-particles by using arbitrary set of
field sources.
EP_Node
These classes connect the base electron-cloud
EP_NodeCalculator classes to the rest of ORBIT code.
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eBunch Class
addParticle(macrosize, r, p)
deleteParticle(index)
Add or delete macroparticle
compress()
Keep just ALIVE particles newly indexed
Information (macrosize, r, p, DEAD/ALIVE ) list
of all indexed macroparticle
readParticles(file1)
print(file2)
Read macro-particles from the ‘file1’ and
register them as indexed macroparticles
(distributed among all CPUs)
Dump the information of every macroparticles
(among all CPUs) to the ‘file2’
MPI: readParticles, print
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eBunch Class
eBunch Class is a resizable container that keeps information about
macro-electrons:
6D coordinates – x, y, z, px, py, pz
macro-size
dead/alive flag
It provides the following methods to operate with macro-electrons:
access to each of the 6D coordinates
add macro-electron
delete macro-electron
print the all information into a file
create all macro-electrons by reading the external file
Its parallel capabilities are used when it reads and writes the
content of the electron bunch into or from the external file.
We tried to do it as faster as possible.
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EP_Boundary Class
EP_Boundary( (x,y)-grids w/ Boundary )
findPotential(ρ(x,y),φ(x,y)) without boundary
Calc φ w/ convolution method φF = ρF * GF
where G(x,y)|grid points = - ln(x^2+y^2)
addBoudaryPotential(φ(x,y) )
Add potential from boundary to the grid
w/ Capacity matrix method: [Qboundary]=[C][DVboundary]
impactPoint_straightMotion(index, eBunch, r, n)
Find impact point on the boundary and calc its normal vector
2D Green’s function is adopted because of our main concern on ECloud
motion in transverse. Longitudinal force comes from potential gradient.
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EP_Boundary Class
EP_Boundary class :
 keeps information about 2D transverse grid and beam-pipe shape
and size in the XY-plane ( the shape could be a circle, ellipse, or
rectangle)
 is a 2D Poisson solver (Convolution method). It has the method that
accept a 3D grid with space charge density and returns another 3D
grid with potential values at the grid points. Each XY-slice of the
potential 3D grid is a solution of the 2D space charge problem for the
XY-slice of the space-charge density grid.
 uses FFTW library and keeps necessary arrays inside
 Can add a boundary conditions (zero potential on the beam-pipe) to
the potential 3D grid by using the Capacity Matrix Method
 for an electron hitting the surface of the beam-pipe it finds an impact
point on the surface and calculates its normal vector by using internal
geometry information
 does not have any parallel capabilities
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EP_Boundary Class
 FindPotential(ρ(x,y),φ(x,y)) without boundary
Calc ρF from ρ(x,y)|binned in grid points ; GF from G(x,y)|grid points with FFTW
where G(x,y)|grid points = - ln(x^2+y^2) spaghetti green function
Convolution method φF = ρF * GF
Calc φ(x,y)|grid points , from φF with FFTW
rfftwnd_one_real_to_complex(planForward_, in_rho_xy, out_rho);
Loop for [index]_2D_points //convolution method
c_re(out_phi_[index]) = c_re(out_rho[index])*c_re(out_green_[index])c_im(out_rho[index])*c_im(out_green_[index]);
c_im(out_phi_[index]) = c_re(out_rho[index])*c_im(out_green_[index]) +
c_im(out_rho[index])*c_re(out_green_[index]);
rfftwnd_one_complex_to_real(planBackward_, out_phi_, in_phi_xy_);
ρ(x,y)|binned in grid points from ρ(x,y) with interpolation weight [Grid3D class]
φ(x,y) from φ(x,y)|grid points
with interpolation weight [Grid3D class]
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EP_Boundary Class
 Boundary Potential w Capacity Matrix Method
Prepare capacity matrix first, which relates charge and potential on
boundary (includes electrode) points, for induced charge.
The solution is obtained by solving Poisson’s eq twice. First, solve Poisson’s
eq without boundary (by convolution method), and record the difference
between the potential without boundary and desired potential on boundary
points. The multiplication of the prepared capacity matrix and the difference
of boundary potential in array gives the induced charge on boundary points.
Then, solve Poisson’s eq again with the induced charge.
The way to find capacity matrix is to place unit charge on each boundary
point in turn with no charge on other points and solve for the potential. The
potential values on boundary points form one column of the inversed
capacity matrix. Repeating this process for each of l-th boundary points fills
in the l columns of the inversed capacity matrix. Then, the capacity matrix is
its inversion.
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Grid3D Classes
Grid3D
RhoGrid3D
PhiGrid3D
The Grid3D class is the parent class for a Grid3D class hierarchy :
• keeps the 3D array of double values inside itself
• provides direct access to the 3D array and to the 2D slices
• calculates a value and gradient at an arbitrary point inside the 3D grid
and bins macro-particles by using 3x3 points scheme
The RhoGrid3D and PhiGrid3D classes are the subclasses of the Grid3D
class and provide methods specific for space charge density and potential
grids
The parallel capabilities: the Grid3D class has methods that transform
it into a distributed 3D grid
Distributed Grid3D
Grid3D1
CPU #1
Grid3D2
CPU #2
....
Grid3DN
CPU #N
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Grid3D
 3*3*3 point interpolation method
binning, weight function
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Field Source Classes
baseFieldSource
void getElectricField(double x, double y, double z, double& f_x, double& f_y, double& f_z);
void getMagneticField(double x, double y, double z, double& f_x, double& f_y, double& f_z);
eCloudFieldSource
pBunchFieldSource
The source of electric fields from e-cloud
The source of electric and magnetic fields
from p-bunch
UniformField
The source of uniform electric and
magnetic fields
We can create additional sources. For instance, It can be a
magnetic field of dipoles, quads etc.
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EP_Node and EP_NodeCalculator
EP_Node – the subclass of the Node class of the ORBIT code:
• it represents a accelerator lattice element through which the proton
bunch should be propagated.
• there could be an arbitrary number of this nodes in the lattice. Each
node has its own set of macro-electrons in the e-bunch.
• it uses EP_NodeCalculator to propagate the proton bunch through the
e-cloud region
EP_NodeCalcularor – the class that actually combines all classes
together and implements our algorithm
• 1. preparation for calculation:
• checking the sizes of the arrays and resizing if necessary
• proton bunch analysis, fields calculation
• 2. simulation of the electron cloud build up and the electron cloud
field having an effect on the proton bunch
• 3. applying accumulated kicks from electrons to the protons
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Algorithm (1)
Stage 1. Preparations
CPU1
…..
CPUn
Proton beam 3D grids (Grid3D)
The space charge density grid and
proton line density are used in the
future calculation if electrons are
produced by residual gas ionization or
by proton losses on the chamber wall
Memory M=Nx*Ny*Nz*5
At this stage we dealing with the proton
bunch only. The macro-particles of this
bunch are distributed between CPUs.
• get information about distribution
of the slices between CPUs. This
data are provided by a special
class – ParticleDistributor
• resize if necessary 3D arrays:
space charge density, space
charge potential, x, y, z kicks – 5
distributed 3D grids
• bin macro-particles of the proton
bunch into the 3D space charge
density grid. Provide necessary
communication between CPUs to
produce distributed 3D grid
• find the proton bunch potential
• calculates proton line density
2000x64x64 x 5
->
200 Mb – 1 Gb
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Algorithm (2)
Stage 2. Propagation p-bunch through e-cloud
During the turn, as time progresses the proton grid is moved through the
electron cloud region at the beam velocity. It is done by three nested loops.
1-st nested loop (upper level)
T1step= Trev/N1step
N1step = 2000 – 10000 for PSR or SNS case
• The number of steps is defined by the requirement adiabatic
changes in the electron cloud potential.
• In the beginning of this iteration, primary electrons are generated by
routines simulating protons grazing the vacuum chamber or residual
gas ionization. The generated macro-electrons are distributed
randomly between CPUs. During the calculations they reside at the
same CPU where they have been generated. No macro-electrons
distributor is needed.
• the space charge potential of the electrons is calculated. This
potential is sum of all potentials trough all CPUs, so communications
between CPUs are needed. This potential is using as one of the field
sources for the Tracker.
• execute the 2-nd nested loop (intermediate level)
• accumulate kicks on the proton bunch into the grids for x,y,z
directions
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Algorithm (3)
Stage 2. Propagation p-bunch through e-cloud
2-st nested loop (intermediate level)
T2step= T1step/N2step
CPU1
…..
CPUn
N2step = 1 – 5
The potential of proton beam at the
specific position is used to update the
proton beam field source for the
Tracker.
The adiabatic changes in the proton
beam potential could be more
important, so there is a possibility to
update the proton beam field source
more frequently comparing to e-cloud
field source. Usually it is enough to
update fields simultaneously (N2 = 1).
This step require communications
between CPUs, because the p-beam
potential grid is distributed.
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Algorithm (4)
Stage 2. Propagation p-bunch through e-cloud (Continue)
3-rd nested loop (lower level)
T3step= T2step/N3step
N3step = 5 – 20
Ntotal = N1xN2xN3 – tens of thousands
Electron Cloud Grids
Inside this loop we integrate the equation of motion of the electrons
and consider hitting the surface of the beam pipe.
•
Each electron must be tested to determine whether it
remains in the electron cloud region
•
If an electron crosses the beam pipe boundary, the PiviFurman electron-wall interaction routines must be run for that
electron.
•
If an electron leaves the longitudinal node end, its
longitudinal velocity is reflected, i.e. it bounces off the end.
Stage 3. Propagation p-bunch through e-cloud
We are applying accumulated kicks from electrons to the protons.
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Tracker Class
baseParticleTracker – the parent class of all trackers
The subclasses should implement three methods: two to add magnetic and electric
fields sources and the method that will move macro-electrons.
We can add as many sources as we want:
addMagneticFieldSource(BaseFieldSource* fs)
addElectricFieldSource(BaseFieldSource* fs)
The main method:
moveParticles(double time, eBunch* eb, Grid3D* rhoGrid3D, baseSurface* surface)
Where “time” - time of tracking, “eb” – macro-electrons bunch
At this moment there are two methods implemented to calculate the
nonrelativistic electron’s motion:
• can be integrated symplectically using a leapfrog method
• can be integrated analytically, using a constant local field approximation
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Surface Classes
baseSurface
void impact(index, eBunch, r, n)
absrbSurface
perfectRflctSurface
No secondary emission
Perfect reflection
CntrlSecEmSurface
implementation of Furman-Pivi algorithm
PRST-AB 5 124404 (2002)
A surface class should implement impact method of the abstract baseSurface class.
It removes the macro-electron with a particular index from the eBunch and adds the
emitted new macro-electrons to the eBunch at the specific point on the beam-pipe
surface. We can create easily new surface classes if we need new models.
Rationale of our implementation of Furman-Pivi algorithm:
Furman-Pivi algorithm of secondary emission assumes that all primary and
secondary macro-electrons have the same macro-size. As result the number of
macro-electrons increases exponentially during the electron cloud build up for
single bunched proton beam passage. Calculation time grows significantly, so we
wanted to avoid this effect and gained control over the macro-electrons population
in the electron cloud.
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Secondary Emission Surface in ORBIT
Removes electron-macroparticle hitting the surface from the electron bunch data
Applies a Flexible Monte Carlo scheme to control the number of macroparticles and
their macrosize (weight of macroparticle) without changing physics.
M in MC  0 if G  f death E 0 
G = random number between 0-1
fdeath(E0) = user defined function, usually 0 if E0> 1 eV
M in
M in MC 
N newborn E 0  1  f death E 0  Nnewborn(E0) = user defined function
Adds Nnewborn(E0) electron-macroparticles using simplified Furman and Pivi’s model
• each macrosize is multiplied by the secondary emission yield.
 ( E0 ,  0 ) 
secondary current
  e   r   ts
incident electron beam current
• each energy is determined by 3 type model spectrum with transformation
e
 ts
r
method. The type is determined in the probability of
M in , E0
pe 
(nx,ny)
0 
M out , E
M out  0

; pr 

; pts 

if G  f death E 0 
M out  M in MC  
Nnewborn(E0) times
(x,y)
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Attachment
“different Monte Carlo scheme”
In the probabilit y of Pnewborn ,
MacroSize , E0
(nx,ny)
0 
 MacroSize * 

add N newborn of 
, Ei 
 Pnewborn N newborn

(x,y)
We are having different Monte Carlo scheme to control the number of macroparticles and
their macrosize (weight of macroparticle) without changing physics feature
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ORBIT EC Module Benchmarks: Outline
Proton Bunch with 3D SC pot. grid
Electron Cloud Region
with few (maybe only one) longitudinal slices
Pipe
• Benchmark of electron motion in EM field
• Benchmark of secondary emission surface model
Implementation of Furman and Pivi’s Model
with simplifications and flexible Monte Carlo scheme to save calculation time
Secondary energy spectrum
Electron cloud development in a cold proton bunch
• Benchmark of two stream model
Analytically solvable model of electron-proton instability (effect of eCloud on pBunch)
Setup in ORBIT
Instability and growth rate
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Y. Sato
37
Indiana University; SNS, ORNL
x
Tracker Benchmark
mv 2


R
2    0  R
H
z
Test for 2D solver
and the tracker.
y
E
e 
mx  H  y
c
e 
my  eE y  H  x
c
mz  eE z
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E-Cloud Development in ORBIT
PSR b eam line dens it y (sc aled)
c omp lete SE mod el (0 )=0. 5 (P ivi and Fu rman )
O R BIT E-Clo ud m odul e = 
No kick on the proton bunch
to compare the results with
Pivi and Furman’s
PRST-AB 6 034201 (2003)
ini
O R BIT E-Clo ud m odul e =  * 0.95
electron's density (nC/m)
ini
10
EC peak height is sensitive to SEY
1
The same parameterization
with Furman and Pivi’s
but the different Monte Carlo
scheme from theirs.
Thus, this EC development shows
sufficient match.
0 .1
0.0 1
0
2 00
40 0
t, ns ec
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Field inside coasting proton beam
uniform
p-bunch
radius a

c
e r 2
Er  2r 
0 a2
r2
B  2r   0 ce 2
a
e r
Er 
20 a 2
0
r

B 
ce 2   0  0 c  Er  Er
2
a
c
Force on a proton
2
e 2 r

e
r
2

Fr  eEr  vs B  
1    2
2
20 a
2 0 a 2
40
Analytically Solvable Electron Cloud Model
uniform
p-bunch
ap,bp
uniform
e-cloud
ae~ap, be~bp
Ref: D. Neuffer et. al. NIM A321 p1 (1992)
The equations of motion inside the two streams (no momentum spread) :






2
..

e
p y p  y p
e 2 e y p  ye
2

 m p y p   m p Q  0  y p  2
 0 b p a p  b p  0 be ae  be 


p ,e
e 2  p ye  y p
e 2 e ye  ye
 ..
a



p ,e
x x
 me y e    b a  b    b a  b 
0
p
p
p
0
e
e
e

b p ,e   y yp ,e
same for x direction except the switch a p ,e  b p ,e
With takin g averages, the equations of mothon become :



2
2




y

Q

y


Q

 0
p
p ,V
0
 p
 ..
 y   Q  2 y  y
e ,V
0
e
p
 e
..




e 2 e
2
Q p ,V  0  
y p  ye
0m p be ae  be 

where 
e 2 p
Q  2 
 e,V 0
0 mebp a p  bp 


41
Analytically Solvable Electron Cloud
Model, cont
  ep frequancy 
Assuming harmonic oscillation in both centroid motion
y p ,c  Ap Expi (n  t ) ,
ye ,c  Ae Expi (n  t ) ,
the equation of motion become



2
2
2
2





Q

Q

n



A

Q

p
0
p
p Ae

 2
2
2


Q



A

Q

0
e
e Ap
 e

d
dt
 longitudin al harmonic
n  
of ep mode


p


 
t p

,
p
d


dt e t



e
and lead the amplitude ratio and the dispersion relation (no momentum spread):
Ae
Qe2
 2
,
2
Ap Qe    0 
Q
2
e
   0 
2
Q
2


 Q p2  n    0   Qe2Q p2
2
The relation is valid under linear force inside the streams
The dispersion relation has complex solutions (instability: growth and damping) for


the ep frequency  ~ Qe 0 and  ~ n  Q  0 , slow wave,
and satisfies the threshold condition:
Qp 
~
Q Qe n  Qe  Q
; Qp  1e/ 2 , Qe  1p/ 2
The larger beam current, the fewer electrons cause instability
42
Two Stream Model in ORBIT
To study the two stream model in ORBIT, we use SNS parameters
ae  be  a p  b p  30 mm, 1 GeV protonbeam , betatron tune Qx  Qy  6.2
12
1
 
 0  2 T  6.646 [ s 1 ],  p  0.165.510
248m  2.5 Bunch factor  2.326  10 [m ]
14
Qe   e  0  172.171
Q p   p  0  2.79616 
;  e  p  neutraliza tion factor
which are most unstable at the longitudinal harmonic number n = 178. n  Qe  Q
For sufficient electron cloud, exceeding the threshold, the dispersion relation
for n = 178 has a growth mode as one of 4 roots of 
:
 2 0  171.961  0.716 i ,
Ae Ap
2
 116.1
for   0.01
So, if we initialize the electron cloud
and proton beam as slow waves with
n=178 modulation and proper phase
relationship, we can expect EC
centroid oscillation to grow.
43
Two Stream Model in ORBIT:
Initial EP-streams for Growth Mode
To reduce the calculation time, we adopt the periodic structure of
12
L=248m/178=1.393m having 20 longitudinal nodes. N p   p L  3.24110
Initial proton bunch
KV distribution (Rp=30mm) –needs very (32 points) symmetric structure
0.01mm centroid modulation (slow wave) in vertical direction
more than 400,000 macroprotons to satisfy at least 10 particles/grid-cell
Initial electron cloud
KV distribution (Re=26mm) –needs to receive
linear force inside p-bunch
Re 2
400,000 macroelectrons with e   R p  p
Ae Ap  , growth mode  0.01mm centroid modulation in vertical direction

0.007
0.006
p-bunch
e-cloud
The change in the transverse
momentum of protons is in
perfect agreement with analytic
calculations except for the round
shoulder
P/P0, mrad

 
0.005
0.004
0.003
0.002
ORBIT
Analytical
0.001
0.000
0
5
10
15
20
25
r, mm
30
35
40
45
50
44
Two Stream Model in ORBIT:
Centroid Motion in Growth Mode
10 turns in the periodic structure requires about 10 min in SNS 16 CPUs
The growth of both electron and proton centroids matches for
first several turns.
They also shows good match with theoretical amplitude ratio.
45
Two Stream Model in ORBIT: Growth Rate
Turn by turn, the 1st FFT Harmonic amplitude of
the proton bunch centroid oscillation increases.
Larger neutralization factor makes e-cloud
exceeds p-bunch radius rapidly. Thus, We can
apply the analytic two stream model just for the
first several turns.
1


Q p 0
2
Q p 0
Qe

n  Qe
2
Qe
 
Q
The ORBIT growth rate is ~20% lager than the theory.
   for low 
However, we need to remind that initial centroid
modulations are set for analytical model of
[Re=Rp=30mm], but we adopt Re=26mm to ensure
linear force. Each proton spends outside of the ecloud in some part of its trajectory.
Good match under the regulations.
46
Attachment for “32 points”
symmetry
y
py
px
x
47
Electron Cloud Properties to
Cold Proton Bunch ----- Outline
• ECloud growth to Proton pulse profile (cold proton)
The same head p-pulse (tail slope dep.)
The same tail p-pulse (trapped electron dep.)
Effect of carry-over electrons before peak of ECloud
• Reproduce PSR results using cold proton pulse in ORBIT:
Recovery after “Clearing Gaps”
Estimation of proton loss late in ED area
Prompt and Swept Electrons
Parameter shifting (SEY, proton loss late)
Another estimation of proton loss late in EC area
as a function of beam intensity
48
PBunch Triangular Profiles: Same Head
Energy band
250eV-300eV
corresponds
to SEY peak
120ns
PB tail
110ns
PB tail
100ns
PB tail
90ns
PB tail
The energy distribution of
surface hitting electrons
of off-peak-band reduces
ECloud growth
49
PBunch Triangular Profile: Same Tail
Effect of primary electrons
stored in the head of proton
bunch
Effect of carry over
electrons surviving
beam gap
50
PBunch Triangular Profile:
ORBIT Results
• The longer tail of pBunch causes the larger eCloud and its
higher growth rate.
• The steeper tail pBunch gives the higher energy of hitting
electrons (E_0 > 250eV) and the lower SEY that may lead
the lower growth rate.
• The carry-over and primary electrons before the peak of
pBunch have the (weak) effect of lowering eCloud growth
rate. The pBunch effect to drive electrons may be
mitigated by the existence of inside electrons during the
beginning of pBunch tail slope.
• To have long head in pBunch profile is a possible way to
accumulate more protons in a bunch.
51
Recovery of ECloud: PSR and ORBIT
7.5 uC proton pulse
Measured in PSR
4~5 turn recovery
of EC peak
after swept
In ORBIT simulation,
4~5 turn recovery to 7.5uC
SEY_max=2.0
Proton loss rate ~1E-8
Both surface current and line
density of EC show the same
number of recovery turns
52
Recovery of ECloud: Carry-over Electrons
and New Primary Electrons
Energy range of electrons hitting surface
keeps almost same in recovery turns.
The amount of carry-over electrons has
little effect on the energy range.
Same carry-over electrons
in PB head of [2nd turn, 2E-8]
and [3rd turn, 1E-8]
Difference of EC peaks comes
from new primary electrons
(Trailing Edge Multipaction)
53
Recovery of EClouds: ORBIT Results
• To the number of turns of recovery after electron sweeper,
we compare our ORBIT simulation with the PSR result.
• From the comparison, we estimate the proton loss rate in
the area of electron cloud detector is around 1E-8 per turn
for maximum SEY 2.0.
• The energy band of electrons hitting vacuum chamber
keeps almost same in recovery turns.
• In eCloud development, the effect of carry over electrons
before pBunch peak (electrons inside pBunch) is much
weaker than the effect of primary electrons after pBunch
peak.
54
Prompt-Swept Electrons: Profile Dependence
Actual proton pulse profiles
measured in PSR.
Now, they seek to have square one.
All my simulations for prompt-swept and recovery
adopt nonotched profile except for below one.
~6uC/pulse, prompt e
and swept e slopes
are flipped in different
pulse profiles.
ORBIT suggests that
a desirable proton
profile depends on
beam intensity.
55
Prompt-Swept Electrons: ORBIT Results
• The two experimental features: (1) swept electron slope is
flatter than prompt electron slope, (2) swept electron slope
has saturation in high intensity beam.
• In ORBIT simulations for constant proton loss rate and
constant maximum SEY to beam intensity, the features (1)
and (2) are reproduced qualitatively, but not quantitatively.
The feature (2) is reproduced only for proton loss rate ~
4E-6 per turn.
• To fit experimental slopes of prompt-swept electrons, we
assume that proton loss rate is an increase function of
beam intensity. The fitting proton loss rate function is
exponential.
56
PSR Instability Characteristics
Rapid beam loss follows after high
frequency centroid motion on BPM.
345 s
~1000 turns
BPM V signal
230 s
CM42 (4.2 C)
(Circulating Beam
Current)
115 s
Vertical difference signals (blue) from a
short stripline BPM and beam pulses from a
wall current monitor (red) in turn-by-turn.
0
V oscillation shift from after-peak to
center of peak of pBunch in 1000 turns
Bk86, p98
57
Frequency of unstable motion in PSR
Frequency spectra of the BPM vertical difference signal in unstable
motion agrees with two-stream electron-proton model as function of
beam intensity. This is a good evidence of e-p instability.
70
60
Factor 2 difference of beam
intensity gives frequency shift
by a factor of ~Sqrt[2] as model
expects.
Amplitude
50
6.1 C
40
30
3 C
20
10
0
0
50
100
150
200
250
300
Frequency (MHz)
2 Nre c 2 (1  e )
e  Qe0  2f 
,
b(a  b) R
f  230 MHz (6.1 C )
Bk85, p140-4
58
PSR Layout with EC & e-p
Diagnostics
Skew Section
Stripper Foil
Merging Dipole
H- B
ea
Matching Section
m
ROED/ES1
Final Bend
Line
ction
a
r
t
Ex
H-/H0 Dump Line
Skew Quad
C Magnets
Bump Magnets
Circumference = 90m
Beam energy = 798 MeV
Revolution frequency =2.8 MHz
Bunch length ~ 250 ns (~63 m)
Accumulation time ~ 750 ms
~2000 turns
ED92
ED02
ED22
Pinger
plates
WM41 and WC41
For single turn, just use ES1
For multiple turns, use ES41
072304 – SNS
rf buncher
ED51
ES41
ED42
59
2002 PSR Parameter List
PSR kinetic energy (800 MeV; cp = 1463. MeV; γ = 1.853; β=0.842)
Beam Kinetic Energy
T
798 MeV
Betatron tunes
x, y
3.19, 2.19
Inc. Tune Shifts @ 100 A
x, y
-0.22, -0.18 calc
Chromaticities
x
-1.22.07 meas, -0.8 calc
(normalized)
y
-1.14.06 meas, -1.3 calc
Transition 
T
3.1
Phase Slip factor

-0.19
Max rf voltage
Vrf
18 kV
Synchrotron Tune (10kV)
0
0.00042
Buncher Harmonic, freq
h, f
1, 2.795 MHz
Mean Pipe Radius
b
0.05 m
Circumference
C=2R
90.2 m
60
Frequency spectra of unstable
motion agrees with model
2 Nrec2 (1  fe )
e  Qe0  2f 
,
b(a  b) R
f  230 MHz (6.1C)
70
60
Amplitude
50
Np~4E13 proton/pulse
6.1 C
40
Np~2E13 proton/pulse
30
3 C
20
10
0
0
50
100
150
200
250
300
Frequency (MHz)
Bk85, p140-4
61
Well Established PSR Instability
Characteristics
p
1

p
0
Instability Signals
eV
1  cos(BW / 2(indeg ))
 0  0mpc2
BPM V signal
CM42 (4.2 C)
(Circulating Beam
Current)
5E13 protons/pulse
 Growth time ~ 75 s or
~200 turns
 High frequency ~ 70 – 200
MHz
 Controlled primarily by rf
buncher voltage See Next
Threshold Intensity (  C/pulse)
8
6
4
2
0
0
5
10
15
20
rf Buncher Voltage (kV)
Bk86, p98
62
“Explanation” of the linear threshold vs
rf voltage curves
 Threshold condition when frequency spreads
overlap (Landau damping applicable)
Qp2
2eNrpc 2
1
64 Q Q e




2
2 2
2
Q
 Q b(a  b)R 9
Q Qe
Qp2
Q
2
N
Q  (n  Q)  Q 
p
 N.L.  Q e  V  N V
p
 R.H.S and L.H.S. for Qe/Qe constant
p
1

p
0
eV
1  cos(BW / 2)
2
 0  0mp c
 Finally, the threshold condition becomes
N  const N V
or
NV
Also
 p 
N  

 p 
2
63
Turn-by-turn vertical oscillations compared with
beam profile during evolution of unstable
motion
~1000 turns
345 s
230 s
115 s
0
 Vertical difference signals
(blue) from a short
stripline BPM and beam
pulses from a wall current
monitor (red).
 WM41VD.4B
 WC41.4B
 Data taken Apr. 14,
1997
 Data at t, t+115 s,
t+230 s, t+345 s
You can see the shift of oscillation
from after-peak to center of peak. We
might reproduce the shift and the
beam profile in our ORBIT calculation
(072304 discussion)
Bk70, p16
64
Cross-section of electron-sweeping
detector and its collection region
Acceptance of New Detector-=75
(Particles inside blue lines hit detector region-V=-100 volts)
0.05
Collector
0.04
0.03
Repeller Grid
Pipe
(V=0)
Detector
(V=0)
Slots & Screen
Y (meters)
0.02
0.01
0.00
-0.01
Plate
(V=-100)
-0.02
Pulsed Electrode
-0.03
Accepted fractional area=0.296
-0.04
-0.05
-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06
X (meters)
65
Sample Electron Data from
Electron Sweeper

Signals have been timed
correctly to the beam pulse
(wall current monitor)

“Prompt” electrons strike the
wall peak at the end of the
beam pulse; basically acts a
large area RFA until HV pulse
applied


Beam Pulse
HV pulse
10ns
Note ~10 ns transit time delay
between HV pulse and swept
electron signal is expected
Swept electron signal is
narrow (~10 ns) with a tail that
is not completely understood
 May be due to
secondaries created at
ground screen, walls of
slots and repeller screen
 Reduced by higher
repeller voltage
Maybe, the overlap of
swept time and next pB
gives the longer slope
Check slope between no
centroid oscillation and
exit c. oscill.
Swept electron signal
The same ES (see HV)
Electron Signal (prompt electron)
Bk 98, p 51
7.7 C/pulse, bunch length = 280 ns, 30 ns injection notch, signals averaged for 32 macropulses,
repeller = - 25V, HV pulse = 500V
Bk 98, p 51
66
Recovery after “Clearing Gap” of
electrons
Beam Pulse
7.5 micro C
HV pulse
E-sweeper signal
Bk xx, p yy
67