Nonlinear Dynamics of High

Brightness Electron Beams and Beam

Plasma Interactions: Theories,
Simulations, and Experiments
Courtlandt L. Bohn
–
Northern Illinois University
Summary:
Funding for this project commenced on 1 July 2004
, and it augments suppo
rt that Northern Illinois
University’s
Beam Physics and Astrophysics Group
(founded in August 2002)
has from other sponsors
.
According to the respective Statement of Work, the overarching objective is: “To enhance substantially
the understanding of the fu
ndamental dynamics of nonequilibrium high

brightness beams with space
charge.” Associated problems fall within two categories, (1) the physics of phase mixing, and (2) the
physics of beam

halo formation. Both are related to the general phenomenon of rapi
d ‘collisionless’
relaxation in Coulomb systems, i.e., those in which the interparticle force respects an inverse

square
dependence on the interparticle separation. In addition to charged

particle beams, self

gravitating stellar
systems, e.g
., galaxies, a
re another class of Coulomb systems, in which case rapid collisionless evolution
is called “violent relaxation”. Thus, a deeper understanding of the evolution of nonequilibrium beams has
application to galactic dynamics, as well.
Related work involves
probing the microscopic, i.e., orbital, dynamics in space

charge potentials.
We
seek
to answer question
s of nonlinear dynamics, such as the following
: What conditions lead to a
significant population of globally chaotic orbit
s in a space

charge potential?
How does the presence of
globally chaotic orbits influence the time scales that characterize the macr
oscopic evolution of the beam?
To what extent does "noise", both white noise (granularity arising from discrete particles) and colored
noise (such as wi
ll arise from machine imperfections and space

charge fluctuations), influence the
population of chaotic orbits?
More generally, u
nder what conditions can the six

dimensional phase space
of a single particle (the conventional Vlasov

Poisson formalism) adeq
uately account for the dynamics
associated with the full 6
N

dimensional phase space of the beam? If it cannot adequately do so, might
augmenting the Vlasov

Poisson formalism with a judiciously constructed Fokker

Planck model make up
the difference?
In ke
eping with recent findings that localized details generally do matter as concerns beam evolution,
NIU
is
dev
ising
theoretical tools that center on multiresolution analysis, an example of which is the
application of wavelet techniques to space

charge algori
thms, with the ultimate goal of developing
improved simulation codes.
An auxiliary effort is underway to develop methods for efficiently
quantifying orbital chaos.
The overarching goal of the NIU work
is
a fundamentally correct and
comprehensive understa
nding of the dynamics of nonequilibrium beams, especially as applies to high

brightness electron sources and accelerators.
Accomplishments
(since 1 July 2004)
:
We investigated the validity of the Vlasov

Poisson equations for calculating properties of sys
tems of
N
c
harged particles governed by time

independent Hamiltonians. Through numerical experiments we
v
erified that there is a smooth convergence toward a continuum limit as
N
and the particle charge
q
0 such that the system charge
Q
qN
remains fixed.
However, in real systems
N
and
q
are always
finite, and the assumption of the continuum limit must be questioned. We demonstrated that Langevin
simulations can be used to assess the importance of discreteness effects, i.e., granularity, in systems for
w
hich the physical particle number
N
is too large to enable orbit integrations based on direct summation
of interparticle forces.
By applying Langevin techniques to
a beam bunch in thermal equilibrium
, we
assess
ed
the conditions under which
the continuum l
imit can be safely
assumed for
this system.
In the
process we show
ed
, especially for systems supporting a sizable population of chaotic orbits that roam
Coarse

grained potential

density pair (in dimensionless
form):
Φ
(
x
)
= (
Ω
2
/2)[(
a
/
b
)
2
x
2
+
y
2
+(
c
/
b
)
2
z
2
] +
Φ
sc
(
x
),
n
(
x
) = exp[

Φ
(
x
)];
∇
2
Φ
sc
(x
)
=

n
(
x
).
Chao
tic

Mixi
ng
Rate
(unit
=
avera
ge
orbita
l
frequ
ency)
Theory
Simulation
Total Particle Energy
Coarse

grained potential

density pair (in dimensionless
form):
Φ
(
x
) = (
Ω
2
/2)[(
a
/
b
)
2
x
2
+
y
2
+(
c
/
b
)
2
z
2
] +
Φ
sc
(
x
),
n
(
x
) = exp[

Φ
(
x
)];
∇
2
Φ
sc
(x
)
=

n
(
x
).
Chao
tic

Mixi
ng
Rate
(unit
=
avera
ge
orbita
l
frequ
ency)
Theory
Simulation
Total Particle Energy
globally through phase space, that for the continuum limit to be valid,
N
must sometimes be
surprising
ly
la
rge (~10
10
).
Otherwise the influence of granularity on particle orbits cannot be ignored.
We investigated how collective space

charge modes and colored noise conspire to produce a beam halo
with much larger amplitude than could be generated by eithe
r phenomenon separately. For the collective
modes, we used the lowest

order radial eigenmodes calculated self

consistently for a configuration
corresponding to a direct

current, cylindrically symmetric, warm

fluid Kapchinskij

Vladimirskij
equilibrium. Th
e ultimate source of colored noise is unavoidable machine errors that act on the beam and
self

consistently
perturb its internal space

charge force. We demonstrated that its presence quickly
launches statistically rare particles to ever

growing amplitude
s by continually kicking them back into
phase with the collective

mode oscillations. The halo amplitude is essentially the same for purely radial
orbits as for orbits that are initially purely azimuthal; orbital angular momentum has no statistically
signi
ficant impact. Factors that do have an impact include the amplitudes of the collective modes and the
strength and autocorrelation time of the colored noise. The underlying dynamics ensues because the noise
breaks the Kolmogorov

Arnol’d

Moser tori that ot
herwise would confine the beam. These tori are
fragile; even very weak noise will eventually break them, though the time scale for their disintegration
depends on the noise strength. Both collective modes and noise are thus demonstrated to be centrally
i
mportant to the dynamics of halo formation in real beams.
This phenomenology may have implications for beam loss in, for example, Fermilab’s Booster
synchrotron, and in fact our work attracte
d the attention of the Booster G
roup. We showed not only that
c
olored noise acting for a short time on a beam with strong space charge, as in a proton linac for, e.g.,
driving an intense source of spallation neutrons, can cause serious emittance degradation and halo growth,
but also this is true if the noise acts on a
beam with weak space charge for a long time, as in a synchrotron
or storage ring. Consequently, a University of Rochester Ph.D. student at Fermilab, Mr. Phil Yoon, has
been modifying the code ORBIT to simulate the Booster with noise, this being done unde
r the guidance
of C. Bohn and W. Chou. His preliminary findings are that the presence of noise would indeed lead to
continuous emittance growth and beam loss. The ultimate question is whether it is possible to identify
sources of such noise in the Booste
r.
To pinpoint the essential dynamical ingredients inherent to time

dependent space

charge potentials, we
constructed a family of 1D models corresponding to breathing spherically symmetric waterbags. They
offer several attractive advantages: (1) the dist
ribution function of particles in the phase space of a single
particle, the charge

density profile, and the space

charge potential are all analytic; (2) the space

charge

depressed tune parameterizes the family of models, so one can readily explore the rang
e from zero space
charge to the space

charge limit; (3) the equilibrium configurations are all stable, and (4) whereas all of
the equilibrium configurations are integrable in keeping with spherical symmetry, a range of models
admits chaotic orbits once the
y are made to breathe. Hence, when it is present, this deterministic chaos is
fully attributable to the time dependence alone. We have used these models for several purposes: (1)
relating the population of globally chaotic orbits to the amplitude of the
breathing mode and to
the space

charge

depressed tune;
(2) exploring how orbits mix through phase space in response to the time
dependence,
and to the presence of stochastic noise
;
(3) determining to what extent halo formation in
these models can be
attrib
uted to chaotic behavior;
(4)
discover
ing
a fast
indicator of
orbital
transitions
betwe
en regular and chaotic dynamics, one that devolves from constructing an integral of motion in time

dependent potentials.
We also have been exploring another possible me
asure of chaos, one that involves pattern recognition in
orbit segments. One cannot possibly use standard techniques like Lyapunov exponents (which are
calculated from long

time

averaged orbital behavior) to decipher transient chaos. What people do inste
ad
is freeze the time

dependent potential at each time step and then compute, e.g., Lyapunov exponents in
each frozen potential. But this is also completely unsatisfactory because it side

steps the essence of the
problem: it ignores the actual dynamics of
both the potential and the orbit(s). What really matters is how
fast the frequencies that characterize the evolution of the orbit in the time

dependent potential change.
This is why we turned to developing a new method. By applying the measure to orbit
s generated from
different codes, but with statistically th
e same initial conditions, one
has a precision method for
detailed
code verification. Differences originating from different computational algorithms
will
show up. This is
our principal motivatio
n for this work.
We
successfully built and implemented a
n
N

body, particle

in

cell
(PIC)
Poisson solver based on wavelet
mathematics
.
Presently we are working on
wavelet denoising of small

scale structure/granularity that has
unphysical or minor conseque
nces to the beam properties. Once finished, we should have a code that is
not only more accurate than its predecessors, but also
considerably faster.
Once the wavelet code has
been brought to maturity, we can then
study complicated, but
generic
, nonequil
ibrium
beams in which
multiscale physics is important.
A key advantage is ‘compression’: with wavelets one can represent a
density profile or potential, even if complicated, by a small number of wavelet coefficients, where ‘small’
is with respect to the n
umber of macroparticles or grid points in a
N

body PIC simulation. This means
one can store compactly the entire time

dependent density

potential pair computed in the course of a real
machine simulation. This offers two i
mmediate new applications.
First
, one can now populate the real
time

dependent potential with a large number, say 10
6

10
7
, of test particles. These particles feel the
space

charge potential but contribute negligibly to it. This affords a means to compute accurately the
formation and di
stribution of a diffuse halo. Actually, it may come to pass that the accuracy of such
a
computation would be limited primarily by uncertainties in the real boundary conditions (hardware
irregularities) rather than in detai
led space

charge dynamics.
Secon
d, one can accurately pursue
computations that require the inclusion of retardation due to the finite speed of the electromagnetic field.
This consideration is inherent to self

consistent interactions between wake fields and the beam
, an
important example
being
coherent synchrotron radiation (CSR). We envision eventually
doing
fully 3D
simulations of bunch dynamics in magnetic bending systems, like bunch compressors, that include self

consistently generated CSR.
2003

2005 Publications:
1.
C.L. Bohn and I.V.
Sideris, “
Chaotic Orbits in Thermal

Equilibrium Beams: Existence and
Dynamical Implications
”, Phys. Rev. ST
–
Accel. Beams
6
,
034203 (2003).
2.
H.E. Kandrup and I.V. Sideris, “
Smooth Potential Simulations and N

Body Simulations
”, Astrophys.
J.
585
,
244 (2003
).
3.
H.E. Kandrup, I. Vass, and I.V. Sideris, “
Transient Chaos and Resonant Phase Mixing in Violent
Relaxation
”, Mon. Not. Roy. Astron. Soc.
341
,
927 (2003).
4.
H.E. Kandrup, I.V. Sideris, B. Terzić, and C.L. Bohn, “
Supermassive Black Hole Binaries as Galactic
Blenders
”, Astrophys. J.
597
,
111 (2003).
5.
H.E. Kandrup, I.V. Sideris, and C.L. Bohn, “
Chaos and the Continuum Limit in Charged

Particle
Beams
”, Phys. Rev. S
T
–
Accel. Beams
7
,
014202 (2004).
6.
N. Barov, J.B. Rosenzweig, M.C. Thompson, and R Yoder
, “
Energy Loss of a High Charge Bunched
Electron Beam in Plasmas
”, Phys. Rev. ST
–
Accel. Beams
7:
061301 (2004).
7.
R.A. Kishek, S Bernal, C.L. Bohn, D. Grote, I. Haber,
H. Li, P.G. O'Shea, M. Reiser, and M. Walter
,
“
Simulations and Experiments with Space

Charge

Dominated Beams
”, Phys. Plasmas
10
,
2016 (2003).
8.
C.L. Bohn and I.V. Sideris
, “
Fluctuations Do Matter: Large Noise

Enhanced Halos in Charged

Particle Beams
”, Phys.
Rev. Lett.
91
, 264801 (2003).
9.
I.V. Sideris and H.E. Kandrup, “
Noise

Enhanced Parametric Resonance in Perturbed Galaxies
”,
Astrophys. J.
602,
678 (2004).
10.
I.V. Sideris
, “
The Validity of the Continuum Limit in the Gravitational N

body Problem
”, Celestial
Mech
anics
90,
149 (2004).
11.
I.V. Sideris and C.L. Bohn, “
Production of Enhanced Beam Halos via Collective Modes and Colored
Noise
”, Phys. Rev. ST
–
Accel. Beams
7:
104202 (2004).
12.
C.L. Bohn
, “
Collective Modes and Colored Noise as Beam

Halo Amplifiers”,
AIP Conf.
Proc.
737
,
456 (2004).
13.
J.R. Cary and C.L. Bohn
, “
Computational Accelerator Physics Working Group Summary”,
AIP Conf.
Proc.
737
, 231 (2004).
14.
Y.

E. Sun, P. Piot, K.

J. Kim, N. Barov, S. Lidia, J. Santucci, R. Tikhoplav, and J. Wennerberg,
“
Generation of Angu
lar

Momentum

Generated Electron Beams from a Photoinjector”,
Phys. Rev. ST
–
Accel. Beams
7:
123501 (2004).
15.
H.E. Kandrup, C.L. Bohn, R.A. Kishek, P.G. O'Shea, M. Reiser, and I.V. Sideris, “Chaotic
Collisionless Evolution in Galaxies and Charged

Particle Be
ams” Ann. NY Acad. Sci.
1045
, 12 (2005).
16.
C.L. Bohn, “Chaotic Dynamics in Charged

Particle Beams: Possible Analogs of Galactic Evolution”,
Ann. NY Acad. Sci.
1045
, 34 (2005).
17.
B. Terzić and I.V. Pogorelov, “Wavelet

Based Poisson Solver for Use in Particle

in

Cell
Simulations”, Ann. NY Acad. Sci.
1045
, 55 (2005).
18.
I.V. Sideris, “Characterization of Chaos: A New, Fast, and Effective Measure”, Ann. NY Acad. Sci.
1045
, 79 (2005).
19.
P.S. Yoon, W. Chou, and C.L. Bohn
, “
Simulations of Err
or

Induced Beam Degradation in
F
ermilab's
Booster Synchrotron
”,
Proceedings of the 2005 P
article Accelerator Conference
(
in press
).
20.
I.V. Sideris
, “
Characterization of the Chaotic or Regular Nature of Dynamical Orbits: a New, Fast
Method
”,
Proceedings of the 2005
Particle Accelerator Co
nference
(in press).
21.
B. Terzić, I.V. Pogorelov, D. Mihalcea, and C.L. Bohn, “
Wavelet

Based Poisson Solver for Use in
Particle

in

Cell Simulations
”,
Proceedings of the 2005 Particle Accelerator Conference (
in press
).
22.
D. Mihalcea, C.L. Bohn, U. Happek, and
P. Piot
, “
Longitudinal Electron Bunch Diagnostics Using
Coherent Transition Radiation
”,
Proceedings of the 2005 Particle Accelerator Conference (
in press
).
23.
G.
T. Betzel, I.V. Sideris, and C.L. Bohn, “Chaos in Time

Dependent Space

Charge Potentials”,
Proceed
ings of the 2005 Particle Accelerator Conference (in press).
Current Research Staff:
Courtlandt L. Bohn
Professor, Accelerator Physics,
Principal Investigator
Bela Erdelyi
Assoc. Professor, Accelerator Physics (working on RIA technology)
Philippe Piot
Assoc. Professor, Accelerator Physics (effective 15 August 2005)
Nick Barov
Postdoctoral Research Associate (now at Far

Tech, Inc.)
Daniel Mihalcea
Postdoctoral Research Associate
Ioannis V. Sideris
Postdoctoral Research Associate
Balša Terzić
Postdoc
toral Research Associate
Gregory T. Betzel
Current Graduate Student (MS, potential PhD)
Marwan Rihaoui
Current Graduate Student (MS, potential PhD)
Ileana Vass
Current (Univ. of Florida astrophysics) Graduate Student (PhD)
Dan Bollinger
Graduate Stude
nt (MS, 2005, presently at Fermilab)
Contact Information:
Courtlandt L. Bohn (PI)
Department of Physics
Northern Illinois University
DeKalb, IL 60115
PHONE: 815

753

6473
FAX: 815

753

8565
EMAIL:
clbohn@niu.edu
WEB:
http://nicadd.niu.edu
Stroboscopic snapsho
ts of
mixing in a breathing spherical
waterbag. The first
5
plots illustrate evolution of
3
clumps
of 1000 test particles
initially tightly
localized in phase
space:
(blue) wildly chaotic orbits, (green)
regular orbits,
(red) sticky chaotic orbits. The
6
th
plot shows
the complete
surface of section
computed with 500 test particles initially
distributed according to the waterbag density profile.
Representative orbit:
chaotic
epoch,
regular
epoch
Measuring
transient chaos
in
an orbit (right panel, top
right) from
an
N

body simulation of the University of Ma
ryland 5

beamlet
exper
iment (left panel). Right panel:
(top
left)
orbital irregularity
vs. time;
everything
under the red
lin
e is more regular than
chaotic; (bottom
left)
x

coo
rdinate vs. time;
(bottom
right
)
surface of section.
Th i s f i g u r e s y mb o l i z e s c l
o s e c o l l a b o r a t i o n
b e t we e n NI U a n d t h e Un i v. o f Ma r y l a n d UME R Gr o u p.
E
x p e r i me n t s
v s.
s i mu l a t i o n s o f t h e F e r mi l a b/NI CADD
p h o t o i n j e c t o r: ( t o p l e f t ) l o n g i t u d i n a l b u n c h s h a p e a t
c a t h o d e ( s i mu l a t i o n ); ( t o p r i g h t ) t r a n s v e r s e p a r t i c l e
d i s t r i b u t i o n a t c a t h o d e ( s i
mu l a t i o n, o b t a i n e d b y d i g i t i z i n g
t h e l a s e r s p o t
o n t h e p h o t o c a t h o d e
); ( b o t t o m) n o r ma l i z e d
t r a n s v e r s e e mi t t a n c e ( s i mu l a t i o n v s. e x p e r i me n t ).
Co n v e n t i o n a l v s. wa v e l e t

b a s e d ( CP CG) I mp a c t 3 D v s.
P
a r me l a
i n s i mu l a t i n g
a 1 3 3 p C b u n c h

c h a r g e
( AES/J La b )
p h o t o i n j e
c t o r.
Co
nventional (upper) vs. wavelet

based (CPCG)
Impact3D
(
lower):
transverse
beam spots
at different
z

locations along
a 133 pC bunch

charge (
AES/JLab
)
photoinjector.
Measured longitudinal density profile of compressed 3 nC
bunches
in the Fermilab/
NICADD photoinjector
(red
curve)
vs.
that resulting from a P
armela
simulation
using
20,000 macroparticles (blue curve).
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