Field lonization as a Plasma Source for the Plasma
Wakefield Accelerator
Patrick Muggli and Erdem Oz
University of Southern California, Los Angeles, CA 90089, USA
Kenneth A. Marsh
University of California, Los Angeles, CA 90095, USA
Abstract. Calculations of field-ionization of gas and vapors by short particle bunches are presented.
The results show that ionization can be achieved over more than two bunch lengths ar ahead of the
beam. Radially ionization extends over a distance much larger that the channel radius created by an
electron bunch in a plasma with a density optimum for acceleration in a uniform plasma [3]. Such selfionized plasmas could be appropriate for the plasma wakefield accelerator (PWFA), where the large
amplitude accelerating plasma wave is driven by the short bunch. Using self-ionized plasmas for the
PWFA would greatly simplify the experimental set up of a PWFA experiment and could lead to very
large energy gains in long, single self-ionized PWFA or SIPWFA module without staging.
INTRODUCTION
Acceleration of particles at rates much larger than those presently achieved in radiofrequency metallic structures has been demonstrated in plasmas. In the laser wakefield
accelerator (LWFA) scheme, gradients in excess of 100 GeV/m have been reached over
distances of the order of one millimeter [1]. The accelerating distance and thus the energy
gain was limited by the ability to maintain the laser intensity necessary to drive the plasma
wave over distance longer than approximately one Rayleigh length of focused beam.
In the plasma wakefield accelerator (PWFA) scheme a short electron or positron bunch
drives the relativistic plasma wave, or wake, to a large amplitude. The transverse
component of the wake provides a strong focusing force for the bunch particles. This
focusing allows for the bunch to be transported or "channeled" over distances much longer
than the beam beta function, the equivalent to the Rayleigh length for laser beams. The
PWFA thus offers the prospect for multi-GeV energy gains in meter-long modules with
GeV/m accelerating gradients, which makes the PWFA relevant to high-energy
electron/positron colliders. In addition plasmas do not suffer from surface breakdown,
pulse heating or radiation fatigue or damage as do metallic accelerating structures. The
accelerating structure is created by and for each bunch or train of bunches. The gas or
vapor to be ionized can be circulated and recycled on line. However, the creation of long
meters plasmas with densities in the 1016-1018 cm"3 appropriate for short bunch PWFA
experiments is still an open issue.
Recent PWFA experiments have shown acceleration of electrons by more than 200 MeV
with az~700um electron bunches in a 1.4m-long plasma with a density of
CP647, Advanced Accelerator Concepts: Tenth Workshop, edited by C. E. Clayton and P. Muggli
© 2002 American Institute of Physics 0-7354-0102-0/02/$19.00
620
. 1 4 cm"3 [2]. The linear theory for the PWFA and numerical simulations [3,4]
show that the accelerating gradient can be significantly increases when the bunch length az
is made shorter while keeping the number of particles per bunch N (or the charge) constant.
Experiments are planned to use bunches with az~100 um in a ~6xl015 cm"3 plasma to
demonstrate an energy gain >1 GeV over ~30 cm. In these experiments the plasma is
created by laser photo-ionization of a lithium vapor [5,6]. Photo-ionized plasmas are field
and current free, and can in principle be created with a very uniform plasma density. The
plasma density is varied by changing the laser pulse energy. They are thus very suitable for
the PWFA. However, photo-ionized plasmas are limited in length by the absorption of the
laser pulse photons and the resulting ionizing beam energy depletion. The plasma density is
proportional to the laser beam fluence (for 1-photon ionization process) incident on the
vapor [7] and is limited by damage to reflective optics. It will thus be difficult to reach the
plasma densities required for future experiments using electron bunches as short as
(Jz~30 um [4] or even 12 um [8], using photo-ionized plasmas.
As the acceleration gradient is increased by shortening the electron bunch, while keeping
the bunch charge or number of particles constant, the plasma density is increased according
to: kj<3z~^2 (for kpcr«l), where kp=(^pelc=(ne^I^Qme)ll2lc is the wave number of the
relativistic plasma wave, [3] to maintain the condition for maximum wake amplitude
excitation. As bunches are made shorter at constant charge, their radial space charge electric
field amplitude becomes larger, and can eventually exceed the value for field ionization [9]
of the ambient gas/vapor. In this case the field ionization by the particle bunch itself could
be used as the source of plasma for the PWFA. Using the plasma created by the particle
bunch to generate the plasma wake would greatly simplify the PWFA experimental set up,
and allow for very large energy gains through propagation over long self-ionized PWFA or
SIPWFA modules. In this paper, the ionization of various gases and vapors by the field of
short, ultra-relativistic particle bunches is investigated. Calculations provide an estimate for
the fractional ionization produced by the relativistic bunch, as well as for its spatial
distribution. For very short bunches (az=12 um) in low ionization potential vapors as
cesium and lithium, full ionization of the first electron of the gas/vapor over a length of
more than 2az ahead of the bunch is reached. In the radial direction full ionization extends
to a distance larger that that of the ion channel rc=(N/(2n)3/2ozne)1/2 [3] created by the bunch
in a preformed plasma with optimum plasma density for electron acceleration. In the
SIPWFA the plasma density is tuned by adjusting the density or pressure of the gas/vapor
that will be fully ionized. By choosing a vapor with a large ratio between the ionization
potential for the second electron to that of the first electron, the contribution to the plasma
density from ionization of the second electron can be minimized.
The dynamic formation of the plasma wake is a very nonlinear process. It must be
studied using numerical particle-in-cell simulations. These studies are in progress [10].
Preliminary numerical simulation results presented at the workshop by D. Bruhwiller et
a/., show that large amplitude accelerating plasma wakes can be driven by short particle
bunches traveling in a neutral lithium vapor. Preliminary experimental results [11], show
that an ultra-relativistic electron or positron bunch with az~700 um travelling in a gas can
create enough plasma density by impact ionization to "self-focus". The focusing of the
same beams in a millimeter long, high density gas jet ionized by impact ionization was also
observed [12].
e
FIELD IONIZATION
The radial, space charge electric field of a relativistic, symmetric bi-Gaussian bunch
propagating in the z direction is given by:
621
(
(
)
)
−r 2σ r
2
2
1
e N 1−e 2 2
E(r,z) = 1 3 2 e N 1 − e −r 2σ r e −z2 2σ2 z
r σr
E(r,z) = (2π )3 2 ε 0 σ rσ z
e −z 2σ z
((270*600,0,
2π ) ε 0 σ rσ z
r σr
1.2
1.2
Bunch Charge
Bunch
Bunch Charge
Charge
1
1
2
2
(1)
(1)
(1)
\X\
0.8
0.8
1/R
1/R
0.6
0.6
0.4
0.4
0.2
0.2
0.2
E ≈(1-exp(-R2/2))/R
Err≈(1-exp(-R2/2))/R
BunchField.graph
0 BunchField.graph
00
1
0
1
0
1
2
22
3
3
3
R=r/σ
R=r/σ
R=r/arrr
4
44
5
55
6
66
FIGURE 1: Radial profile of the Gaussian particle bunch, and of the radial electric field Er plotted as a
FIGURE
1: Radial
Radial profile
profile of
FIGURE
1:
of the
the Gaussian
Gaussian particle
particle bunch,
bunch, and
and of
of the
theradial
radialelectric
electricfield
field EEr rplotted
plottedasasaa
function
of r/σ
r. The maximum electric field is reached at r≈1.58σr (and z=0, Eq. 1). and decreases as 1/r at
function of
of r/a
r/σrr.. The
The maximum electric
field
at
(and z=0,
Eq. 1).
and decreases as 1/r at
function
electricelectric
field is
is reached
reached
at r≈1.58σ
p=4.58a
z=0,
r (and by
long
distances
(r>>σrmaximum
). The maximum
field
amplitude
is rgiven
Eq.Eq.
2. 1). and decreases as llr at
long distances
distances (r»a
(r>>σrr).
). The
The maximum
maximum electric
long
electric field
field amplitude
amplitude is
is given
given by
by Eq.
Eq. 2.
2.
where N is the number of particles in the bunch. Radially the field peaks at r≈1.59σr, and
where
N is
isdecreases
the number
number
of
in
the
peaks
atat r≈1.59σ
r ,r,and
where
N
the
of particles
particles
in the
the bunch.
bunch.
Radially
the field
field
peaks
r^l.59o
and
for
r>>σ
as
1/r
and
is proportional
proportional
toRadially
the enclosed
enclosed
bunch
charge
(Fig.1).
1).
The
for r»o
r>>σrrr decreases
decreases as
as llr
1/r and
and is
is proportional
to
the
bunch
charge
(Fig.
The
for
to
the
enclosed
bunch
charge
(Fig.
1).
The
maximum
field
is
reached
in
the
middle
of
the
bunch
(z=0),
and
is
given
by:
maximum field
field is
is reached
reached in
and
maximum
in the
the middle
middle of
of the
the bunch
bunch (z=0),
(z=0),
and isis given
given by:
by:
N
−19
−19 i N
E
[
GV
/
m
]
≈
5.2
×
10
(2)
N
r,max
Er,max [GV / m] ≈ 5.2 × 10 σ σ
(2)
r z
σ
σ
r
z
th
The
the
electron of
of an
an atomic
atomic gas/vapor
gas/vaporexperiencing
experiencingthe
theelectric
electricfield
field
The ionization
ionization rate
rate for
for
the iifhth electron
ionization
rate
for the
electron
gas/vapor
EThe
is
calculated
in
a
quasi-static
model of
andanis
isatomic
given by
by
[13]: experiencing the electric field
E
is
calculated
in
a
quasi-static
model
and
given
[13]:

E is calculated in a quasi-static model and is 5given
by
 φ 3 :32 2 E  
 2[13
2
−2 φ i   E a 
 φφ i 5 2 EEa − 33φφHi H EaE 
a e

ω i == 4ω
4ω 0  i 
(3)
ω
e  
(3)
i
0
(3)
 φφ H  EE
H
17
1.5
17
1.510
1017
1.5 10
101717
5510
He
He
a)
a)
b)
b)
b)
17
LiLi
1017
4410
Ar
Ar
17
10 17
11 10
1 10
IE
c
o
He
He
101717
3310
Xe
Xe
17
1017
2210
16
16
5510
10
510
Cs
Cs
Li
Li
00
00
IoniationRate1.graph
IoniationRate1.graph
100
100
Cs
101717
1110
Cs
Cs
200
300
400
400
100
200
300
400
Electric Field (GV/m)
Electric
500
500
500
Ar
Ar
Xe
Xe
00
00
0
IoniationRate2.graph
IoniationRate2.graph
500
500
1000
1000
1500
1500
2000
2000
500
1000
1500 2000
Electric
ElectricField
Field(GV/m)
(GV/m)
2500
2500
2500
Electric Field (GV/m)
Electric Field (GV/m)
FIGURE 2:
2: Ionization
Ionization rates (Eq. 3) as a function
FIGURE
function of
of the
the applied
appliedelectric
electricfield
fieldfor
forvarious
variousgases
gasesand
andvapors
vapors
FIGURE 2: Ionization rates (Eq. 3) as a function of the applied electric field for various gases and vapors
fora)
a)ionization
ionization of
of the
the first (i=1) and b) the second
for
second (i=2)
(i=2)atomic
atomicelectron.
electron.Lithium
Lithiumhas
hasthe
thelargest
largestφφ2/φ
ratio
1 ratio
2/φ
for
ionization
of the first (i=l) and
b) the second (i=2)rates.
atomic electron.
Lithium has the largest §J§i 1rati°
(seea)Table
Table
1), which
which
(see
1),
results in very different
different ionization
ionization rates. Note
Notethe
thedifferent
differentscales.
scales.
(see Table 1), which results in very different ionization rates. Note the different scales.
622
where <]). is the ionization potential for the fh electron, (00 =2e0<%A/e2«2.5 xlO"17 s is the
atomic frequency unit, Ea=^/(4m0a02)~5.l GV/m is the atomic electric field unit, and
(^13.6 eV is the ionization potential of hydrogen. In addition h is Planck's constant, and
a0 is the Bohr radius. Equation 3 describes the ionization rate when the electric field is
applied to the atom for a time interval much longer than that corresponding photo-ionization
frequency hv»etyi:
T>>
"T
<$i
(4)
which is of the order of 0.2 fs for (|)~3.9 eV (/=!, cesium case). For time intervals
comparable to this value multi-photon photo-ionization processes need to be taken into
account. The rate (Eq. 3) is a very nonlinear function of <|) and E, and is thus very material
dependent and is plotted on Fig. 2 for some alkali metals with low ionization potential:
cesium (Cs) and lithium (Li), and some noble gases: helium (He), argon (Ar), and xenon
(Xe). The ionization potential, as well as the ionization threshold for different gases and
vapors, are given in Table 1. The ionization threshold is defined here as electric field for
which the ionization rate equals the bunch period: v=c/(2n)1/2oz. Significant ionization can
be expected for fields exceeding these threshold values.
TABLE 1: Ionization potential (^ (from [14]), and field ionization threshold for various gas/vapors for
q7=100and 12 urn.
Gas/
Ionization
Ionization
Ionization
Ionization
Vapor
Level /
Potential
Threshold
Threshold
(GV/m)
(GV/m)
<|>,(eV)
a7=100 urn
G7=12 jim
Cs
3.894
4.4
1
3.9
2
23.1
55.1
60.5
Li
5.392
6.1
6.8
1
2
75.638
268.5
293.4
He
24.587
53.5
58.8
1
2
54.416
167.2
182.6
Ar
28.3
31.1
1
15.759
2
63.3
69.6
27.629
Xe
12.13
19.6
21.4
1
2
21.21
43.4
47.8
The total plasma density is the sum of the contributions from each level of ionization. The
ionization of the different level is assumed to be sequential. This model is appropriate since
the ionization threshold for the different levels are very different, and a large ionization
fraction for the fh level is reached before the ionization threshold of the (i+l)th level is
exceeded. The variation in time of the population of the fh level of ionization is given by the
rate equation:
where n0 is the population density of neutrals. In the model presented here, the bunch
approaches from z=-«>, and the electric field at a time t and a location z0 is obtained by
replacing z by z0-ctin Eq. 1. Equation 5 is then integrated in time from -«> to t using the
time varying field in Eq. 3. The plasma density is the sum of the ionization of all the levels:
ne=n1+2n2+3n3+... since the th level of ionization contributes / electrons to the plasma
density. The fractional ionization nf is the ratio of the plasma density to the neutral density.
623
FIELD
THE PWFA
PWFA
FIELD IONIZATION
IONIZATION FOR
FOR THE
For
for the
PWFA two
two
Forfield
field ionization
ionization to
to be
be an
an appropriate
appropriate way
way of
of creating
creating plasma
plasma for
the PWFA
condition
created by
by the
the early
early part
part of
of the
the
condition should
should be
be satisfied.
satisfied. First,
First, the
the plasma
plasma needs
needs to
to be
be created
bunch
drive the
the large
large amplitude
amplitude wake.
wake. In
In
bunch(>2a
(≥2σz)z) so
so that
that the
the bunch
bunch charge
charge can
can efficiently
efficiently drive
addition,
the bunch
bunch the
the focusing
focusing force
force of
of the
the
addition, ifif the
the ionization
ionization happens
happens early
early enough
enough in
in the
plasma
beta functions.
functions. On
On the
the contrary,
contrary, ifif
plasmawake
wake can
can propagate
propagate the
the bunch
bunch over
over many
many beam
beam beta
the
in the
the head
head of
of the
the beam
beam would
would be
be
theionization
ionization happened
happened late
late in
in the
the bunch,
bunch, the
the particles
particles in
lost
of the
the particle
particle beam.
beam. Second,
Second, the
the
lostover
overlong
long distances
distances through
through the
the natural
natural divergence
divergence of
plasma
distance to
to which
which the
the plasma
plasma
plasmamust
must extend
extend radially
radially to
to aa distance
distance larger
larger than
than the
the distance
electrons
are
by
the
bunch
electric
field.
The blow-out radius is of the order
order of
of
electrons3/2
are expelled
expelled
by
the
bunch
electric
field.
1/2 1/2
3/2
rrc=(N/(2n)
o
n
)
[2].
Numerical
simulations
pre-ionized
plasma
the
wake
z
e
=(N/(2π)
σ
n
)
[2].
Numerical
show
that
in
a
pre-ionized
plasma
the
wake
c
z e
amplitude
than rrcc [3].
[3]. In
In this
this
amplitudedrops
drops significantly
significantly when
when the
the plasma radius is made smaller than
case
restoring force
force
casethe
theplasma
plasmaelectrons
electrons are
are expelled
expelled outside the plasma and feel a smaller restoring
than
thanininthe
theinfinite
infiniteplasma
plasma case.
case.
1 •a)
a)
.........
0.8
0.8
:
0.6
0.6
:
0.4
0.4
'l'l
Ijl
σz=12 µm
I;/
'![
/
• /
s
.1
8 •
0.8
°'
|
0.6
0-6 •
o
0.4
0.4 -
\
LL
0.2
0.2
=110 pm
µm ".
aσz=110
*•• •••••--, σ<3_^\£
fjm •
=12 µm
z
:
.
'i
-10
-10
00
10
10
σozz=70
=70 µm
|jm I
^ ii A
;"
0
Ioniz(r)Cs.graph
-20
-20
r.......
σz=70 µm
0.2
0.2
00
1 •D)
b)
M
'?*
20
20
Ioniz(r)Li.graph
-20
0
-15
i^X _ . _ _ _ _ . _ _»_._.__ _ _ _
-10
-10
-5
-5
00
55
10
10
15
15
20
2C
Radius/σr
Radius/σ
Radius/a
Radius/ar
FIGURE 3:
3:Fractional
Fractional ionization
ionization as
as a function
function of radius created by the
FIGURE
the passage
passage of
of aasingle
singlebunch
bunchwith
with
70, and
and 12
12 um,
µm, and
and aσr=25
µm, N=2xl0
N=2×101010 particles
particles in a) Cs and b) Li. In the case of Li, the peak
aσz=100,
z=100,70,
r=25 um,
fractionalionization
ionizationfor
foraσz=100
µm is
is «y<10~
nf <10 -44 and not visible on Fig. b. With
fractional
With long
long (low
(low density)
density) bunches
bunches
z=100 um
thefractional
fractionalionization
ionizationisis significant
significant only
only around r^L58c
r≈1.58σrr where the electric field peaks
the
peaks (see
(see Fig.
Fig. 1).
1).With
With
short(high
(highdensity)
density)bunches,
bunches, the
the fractional
fractional ionization
ionization reaches nrij=l
short
extends from
from r≈0
r^O to
to many
many σor.r The
The
f =1 and extends
maximumfields
fieldsreached
reachedwith
with the
the three
three bunch
bunch lengths
lengths are: £=4.1,
E=4.1, 5.9, and 34.6 GV/m respectively
maximum
respectively (Eq.
(Eq. 2),
2),
Theyare
arelower
lower than
than the
the thresholds
thresholds for
for ionization
ionization of
of the
They
the i=2
i=2 electron,
electron, especially
especially for
for in
in the
the Li
Licase.
case.The
The
contributiontotothe
theplasma
plasmafrom
from second
second electron
electron ionization
ionization is
is thus
thus extremely
contribution
extremely small.
small.
Figure 33 shows
shows the
the radial
radial profile
profile of
of the
the fractional
fractional ionization
by
Figure
ionization created
created
by the
the passage
passage (-(10
∞<t<∞)of
of bunches
bunches with
with various
various aσzz and
and with
with σ
N=2×1010
particles
OO<K°O)
(Jzz=25
=25 µm,
um, Af=2xl0
particles or
or 3.2
3.2 nC,
nC,
Cs and
andLi
Li vapors.
vapors. In
In the
the long
long bunch
bunch case
case (a
(σzz=l
=110
ininCs
10 µm),
um), the
the maximum
maximum radial
radial field
field isis
GV/m (Eq.
(Eq. 2)
2) and
and isis between
between the
the ionization
ionization thresholds
thresholds for
max≈4GV/m
EEr,r,max~4
for the
the first
first electron
electron of
of Cs
Cs
andLi
Li (see
(see Table
Table 1).
1). The
The ionization
ionization occurs
occurs only
only around
arid
around the
the radial
radial location
location where
where the
the
electricfield
fieldpeaks,
peaks, r=4.59a
r≈1.59σrr,, and
and only
only around
around the
the middle
middle of
electric
of the
the bunch.
bunch. The
The bunch
bunch thus
thus
creates aahollow
hollow cylinder
cylinder of
of plasma.
plasma. A
A hollow
hollow plasma
plasma cylinder
creates
cylinder configuration
configuration could
could be
be
appropriatefor
foracceleration
acceleration of
of positrons.
positrons. Numerical
Numerical simulations
simulations show
appropriate
show that
that the
the accelerating
accelerating
gradientisis larger
larger in
in aa hollow
hollow plasma
plasma channel
channel than
than in
gradient
in aa homogeneous
homogeneous plasma
plasma [15].
[15]. As
As the
the
bunch
is
made
shorter,
the
fractional
ionization
can
reach
one,
and
extend
radially
towards
bunch is made shorter, the fractional ionization can reach one, and extend radially towards
r=0,and
and out
out to
to many
many times
times a
σr,, up
up to
to >200
>200 µm
µm (Fig. 3). The
/^O,
um in
in the
the case
case of
of σ(Jz=12
r
z=12 um (Fig. 3). The
channel
radius
calculated
using
the
optimum
plasma
density
obtained
from
channel radius calculated using the optimum plasma density obtained from the
the linear
linear theory
theory
forthe
theaσ=12
µmisis r=ll
rc=17um
µm and
and is
is much
much smaller
smaller than
than the
the self-ionized
plasma
radius.
z =12 um
for
self-ionized
plasma
radius. As
As
z
thebunch
bunchisismade
made shorter,
shorter, the
the fractional
fractional ionization
ionization also
also reaches
reaches one
the
one earlier
earlier in
in the
the bunch.
bunch.
Figure44shows
shows aa snapshot
snapshot of
of the
the fractional
fractional ionization
ionization created
Figure
created by
by σ(Jzz=12
=12 µm
um bunches
bunches in
in Cs
Cs
624
and Li, and shows that the full ionization (nf=l) extends to about 2az towards the beam
and Li,
andpre-ionized
shows that the
full ionization
to to
about
2σz towards
theelectron
beam
f=1) extends
head.
In the
plasma,
the plasma(nelectrons
begin
be displaced
by the
head.
In
the
pre-ionized
plasma,
the
plasma
electrons
begin
to
be
displaced
by
the
electron
bunch ~2-3az in front of the bunch peak and return on axis in the back of the bunch in the
bunch ≈2-3σ
of the
bunch
and4return
axisthe
in the
backum
of the
buncha in
the
z in front
optimum
plasma
density
case
[3]. peak
Figure
showsonthat
(Jz=12
creates
spatial
optimum
plasma
density
case
[3].
Figure
4
shows
that
the
σ
=12
µm
creates
a
spatial
z
fractional ionization pattern that extends both radially and longitudinally far enough so that
fractional
ionization
that extends
both radially
and longitudinally
far enough so
that
for
the PWFA
processpattern
the self-ionized
plasma
may appear
identical to a pre-ionized
plasma.
for
the
PWFA
process
the
self-ionized
plasma
may
appear
identical
to
a
pre-ionized
plasma.
In the linear theory for the PWFA largest wake amplitude is achieved in a plasma with a
In the linear
theory
the PWFA largest wake amplitude is achieved in a plasma with a
density
such that
kpofor
z~^l2 where kp=(toplc is the plasma wave number [3]. The gas/vapor
density
such
that
k
σ
≈√2
where
k p =ωpthat
/c isplasma
the plasma
wave
number
[3].maximum
The gas/vapor
p
z
pressure
can thus be adjusted
to reach
density.
Note
that the
radial
pressure can thus be adjusted to reach that plasma density. Note that the maximum radial
field
in
the
shortest
bunch
case
is
E
~34
GV/m
(Eq.
2),
below
the
threshold
values
r> majc
field in the shortest bunch case is Er, max≈34 GV/m (Eq. 2), below the threshold values ofof
Table
the higher
higher levels
levels of
ofionization
ionization(i>1)
(i>l)isisexpected,
expected,
Table1,1,and
andvery
verylittle
little contribution
contribution from
from the
especially
in
the
Li
vapor
case.
The
plasma
density
can
thus
be
adjusted
by
setting
the
especially in the Li vapor case. The plasma density can thus be adjusted by setting the
vapor
density (n
(n0=n
=ne,, nn^l).
Figure55shows
showsthe
thevapor
vapor
vapordensity
densityequal
equalto
tothe
the desired
desired plasma
plasma density
=1).
Figure
0
e
f
curves
neutral densities
densities
areeasier
easiertotoachieve
achieveininCs
Csthan
than
curvesofofCs
Csand
andLi,
Li, and
and shows
shows that
that high
high neutral
are
ininLi.
The
optimum
plasma
density
for
a
(J
=10
um
bunch
in
apre-ionized
pre-ionizedplasma
plasmaasasgiven
given
Li. The optimum plasma density for a σz=10 µm
bunch
in
a
by
~4xW 17 cm
cm"-33.. However,
However, with
withsuch
suchshort
shortbunches
bunchesthe
the
bythe
thelinear
lineartheory
theoryrelation
relation is
is nnee≈4×10
expelled
plasma
electrons
reach
relativistic
velocities, the
the wake
wakeexcitation
excitationbecomes
becomeshighly
highly
expelled plasma electrons reach relativistic velocities,
non-linear,
be required.
required.
non-linear,and
andaahigher
higherplasma
plasma density
density may
may be
a)
2
2
nf,i=1=1
1
1
0
0
-1
-2
b)
nf,i=1=1
-1
nf=0
-20
-2
-10
-10
00
10
20
nf=0
-20
-20
r/σr r
r/a
-10
-10
00
r/σ
r/ar r
10
20
FIGURE4:4:Fractional
Fractionalionization
ionization in
in the
the (r,z)
(r,z) plane
µm, σr=25 µm,
FIGURE
plane created
created by
by aabunch
bunchwith
withσaz=12
um,and
and
z=12 um, o'=25
1010 particles in a) Cs and b) Li. The ionization fraction n is 0 far from the beam and reaches 1 around
N=2×10
Af=2xl0
particles in a) Cs and b) Li. The ionization fraction nf f is 0 far from the beam and reaches 1 around
andbehind
behindthe
thebeam.
beam.The
The arrow
arrow shows
shows the
the beam
beam propagation
and
propagation direction,
direction,and
andcontours
contoursare
areplotted
plottedbybynfnsteps
f steps
0.1.The
Ther=G
r=σr rcontour
contourof
of the
the beam
beam is
is also
also shown.
µm
bunch
inina aplasma
ofof0.1.
shown. The
The channel
channelradius
radiusfor
foraaσaz=12
=12
um
bunch
plasma
z
densitygiven
givenby
bykkpG=^2
is rrcc~17
≈17 um.
µm.
pσz=√2is
density
Threeimportant
important additional
additional effects
effects need
need to
Three
to be
be taken
taken into
into account.
account.The
The first
first one
oneisisthe
the
contribution
of
impact
ionization
to
the
plasma
density
and
to
the
dynamics
of
contribution of impact ionization to the plasma density and to the dynamics ofthe
thewake
wake
excitation.The
Thecross
cross section
section for
for impact
impact ionization
ionization of
excitation.
of different
different materials
materialstypically
typicallypeaks
peaksatat
about
100
eV
[16].
The
cross-section
for
relativistic
particles
is
typically
two
to
about 100 eV [16]. The cross-section for relativistic particles is typically two tothree
three
ordersofofmagnitude
magnitude lower
lower than
than the
the peak
peak value.
value. Direct
impact
ionization
ofofthe
neutrals
oror
orders
Direct
impact
ionization
the
neutrals
ions by the relativistic beam particles is thus less important that that from the plasma
ions
by the
relativistic
beam
particles
thus
lessisimportant
that that
fromnumerically
the plasma
electron
when
blown out
by the
bunch.isThis
effect
presently being
studied
electron
when
blown
out
by
the
bunch.
This
effect
is
presently
being
studied
numerically
[17]. The second is the effect of the scattering of the bunch particles on the gas/vapor
[17].
The and
second
the effect
of the
scattering
the bunch
gas/vapor
neutrals
ions iswhen
travelling
over
a long, of
dense
plasma.particles
In orderontothe
achieve
the
neutrals
ionsdensities
when travelling
overatomic
a long,
densematerials
plasma.such
In order
to Ar
achieve
the
requiredand
plasma
low φ, large
number
as Cs or
may be
required
plasma
densities
low
<|
)
,
large
atomic
number
materials
such
as
Cs
or
Ar
may
used, therefore increasing the particles scattering and beam emittance growth. Emittancebe
used,
therefore
increasing
the particles
scattering
and beam
emittance
growth.
Emittance
growth
scales as
the material
atomic number
squared,
and could
be very
significant
in
growth
scales as the
material
atomic number
squared,
could ofbethevery
significant
multi-meter-long
PWFAs
[4 afterburner].
The third
is the and
ionization
gas/vapor
by thein
multi-meter-long
[4 wake
afterburner].
The
third
is the bunches.
ionization
of the gas/vapor
by the
fields of the largePWFAs
amplitude
driven by
short
particle
Anticipated
accelerating
fields
large
amplitude
wake driven
by short to
particle
accelerating
fieldsofinthe
short
bunch
experiments
are expected
be in bunches.
the 5-10Anticipated
GV/m, exceeding
the
fields
in short
experiments
arefor
expected
be Table
in the 1).
5-10 GV/m, exceeding the
ionization
rate bunch
threshold
for Li and Cs
exampleto(see
ionization rate threshold for Li and Cs for example (see Table 1).
625
The choice
choice of
of gas
The
gas or
or vapor
vapor requires
requires aa careful
careful analysis
analysis related
related to
to aa SIPWFA
SIPWFA module.
module. For
For
example lithium
lithium has
has aa low
eV), which
example
low ionization
ionization potential
potential for
for its
its first
first electron
electron (φ
((|)1 =5.4
7=5.4 eV), which
corresponds to
to aa low
low field
corresponds
field ionization
ionization threshold,
threshold, and
and aa relatively
relatively large
large ionization
ionization potential
potential for
for
its second
1 =75.7 eV),
its
second electron
electron (φ
((^=75.7
eV), which
which corresponds
corresponds to
to aa high
high field
field ionization
ionization threshold
threshold
(Table 1).
Lithium also
also has
has aa low
low atomic
atomic number
number (Z=3)
(Table
1). Lithium
(Z=3) and
and thus
thus aa low
low impact
impact ionization
ionization
cross section,
section, which
which minimizes
minimizes contributions
cross
contributions from
from primary
primary or
or secondary
secondary impact
impact ionization
ionization
processes to
to the
the wake
wake excitation,
excitation, and
processes
and also
also minimizes
minimizes emittance
emittance growth
growth from
from scattering.
scattering. In
In aa
heat-pipe oven
oven the
the vapor
vapor is
is contained
heat-pipe
contained by
by aa buffer
buffer gas.
gas. In
In this
this configuration
configuration the
the beam
beam can
can be
be
focused to
to aa small
small size
size at
at the
the He/Li
He/Li vapor
vapor boundary,
boundary, and
be
focused
and aa low
low ionization
ionization level
level may
may be
created in
in the
the He
He buffer
buffer section
section where
where the
the beam
beam size
size and
and the
the ionization
ionization potential
potential are
are large.
large.
created
Lithium may
may thus
thus be
be appropriate
appropriate for
for meters
meters long
long SIPWFA
SIPWFA aiming
aiming at
at achieving
achieving large
large energy
energy
Lithium
gain. Cesium,
Cesium, has
has aa lower
lower ionization
ionization potential
potential for
the first
first electron,
gain.
for the
electron, and
and may
may be
be appropriate
appropriate
for proof-of-principle
proof-of-principle experiments
experiments in
in short
for
short aa SIPWFA.
SIPWFA. However,
However, contribution
contribution to
to the
the
plasma
density
and
wake
excitation
from
the
second
ionization
level
must
be
evaluated.
plasma density and wake excitation from the second ionization level must be evaluated.
Gasses like
like Ar
Ar or
or Xe
Xe have
have similar
the successive
Gasses
similar ionization
ionization potentials
potentials for
for the
successive electrons
electrons and
and
could
be
used
to
achieve
large
plasma
densities
with
comparatively
low
pressures
could be used to achieve large plasma densities with comparatively low pressures through
through
ionization of
of multiple
multiple electrons
electrons per
per atom.
ionization
atom. Ionization
Ionization of
of multiple
multiple electrons
electrons may
may lead
lead to
to
plasma
density
gradients
with
negative
effects
on
the
PWFA
process.
plasma density gradients with negative effects on the PWFA process.
The beam
beam size
The
size σ
arr and
and charge
charge used
used here
here are
are typical
typical of
of those
those that
that have
have been
been used
used in
in recent
recent
long
bunch
(σ
≈700
µm)
PWFA
experiments
at
the
Stanford
Linear
Accelerator
long bunch (azz~700 pm) PWFA experiments at the Stanford Linear Accelerator Center
Center
(SLAC) [6].
Future experiments
experiments with
ultra-short bunches
µm at
(SLAC)
[6]. Future
with ultra-short
bunches (σ
(azz≈12
~12um
at SLAC),
SLAC), will
will
require aa smaller
beam (σ
<17µm) if
<1 is
p σrr<\
require
smaller size
size beam
(arr<17pm)
if the
the kkjp
is to
to be
be satisfied.
satisfied. Reaching
Reaching such
such small
small
σr may
may in
in turn
turn require
require lowering
N to
a
lowering N
to minimize
minimize the
the beam
beam energy
energy spread.
spread. In
In these
these
r
experiments
the beam
beam experiences
experiences multiple
multiple betatron
experiments
the
betatron oscillations
oscillations [18]
[18] and
and the
the beam
beam size
size
along the
the plasma
plasma is
is smaller
than at
plasma. The
along
smaller than
at the
the plasma.
The field
field ionization
ionization rate
rate varies
varies accordingly
accordingly
along the
the plasma,
plasma, and
and higher
higher ionization
ionization states
may be
be achieved.
achieved.
along
states may
1017
17
10
1016
16
10
1015
15
10
1014
14
10
1013
13
10
Cesium
Cesium
1012
12
Lithium
Lithium
I
10
11
10,11
10
10
10,10
10 0
200
200
1
VaporPressureLiCs.graph
VaporPressureLiCs.graph
400
400
600
600
Temperature
Temperature (K)
(K)
800
800
1000
1000
FIGURE 5: Vapor curves for cesium and lithium (after [14]). The buffer gas in a heat-pipe oven would be
FIGURE 5: Vapor curves for cesium and lithium (after [14]). The buffer gas in a heat-pipe oven would be
He(Z=2), for Li(Z=3), and Xe(Z=54) or Ar(Z=18) for Cs(Z=55). Neutral densities appropriate for short
He(Z=2), for Li(Z=3), and Xe(Z=54) or Ar(Z=18) for Cs(Z=55). Neutral densities appropriate for short
bunches are achieved at lower temperatures with Cs than Li.
bunches are achieved at lower temperatures with Cs than Li.
SUMMARY
SUMMARY
In the
the near
near future
future plasma
plasma wakefield
In
wakefield acceleration
acceleration will
will be
be performed
performed with
with shorter
shorter particle
particle
bunches
to
demonstrate
larger
acceleration
gradients
and
larger
energy
gains
bunches to demonstrate larger acceleration gradients and larger energy gains that
that previously
previously
measured. Long,
Long, high-density
high-density plasmas
plasmas are
these experiments.
measured.
are required
required for
for these
experiments. Calculations
Calculations
626
presented here show that plasma densities equal or larger than those required for short
bunch PWFA experiments in pre-ionized plasmas can be reached by field ionization of the
ambient gas/vapor by the driving bunch. The calculation results apply to electron and
positron bunches. In the examples presented here, the ionization process starts early in the
bunch (~2az ahead of the bunch center), and extends radially to a distance larger than the
excursion of the expelled plasma electrons in the electron-driven PWFA. The extent of the
field-ionized plasma could be large enough so that it would appear as a homogeneous preionized plasma for the wake excitation process. Field ionization by the driving bunch itself
could thus be an appropriate source of plasma for the PWFA, and lead to a self-ionized
PWFA or SIPWFA module. The transverse component of the plasma wake allows for the
channeling of the beam over many of its beta functions, and could lead to very large energy
gains in meters-long self-ionized plasmas. Self-ionization would suppress the need for an
external means of ionization of the gas/vapor and would greatly simplify the set up of a
PWFA. Staging of PWFA modules could be avoided using the SIPWFA. The wake
excitation by short bunches in a high density plasma is a highly nonlinear dynamic process
and is not described here. Numerical simulations will be used to determine whether large
energy gain, and good be quality can be achieved is a SIPWFA scheme. Some of the main
issues to be addressed numerically are the wake excitation, the effect of other ionization
processes such as impact ionization and field ionization by the wake fields, the propagation
of the beam and the erosion of the beam head, and the preservation of the beam quality.
ACKNOWLEDGMENTS
Work supported by US DoE Grants No DE-FG03-92ER40745 and DE-FG0392ER40727, and NSF Grant No PHY-0078715.
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627