# Ion dynamics driven by a strongly nonlinear plasma wake

V K Khudiakov1,2, K V Lotov1,2, M C Downer3 1Budker Institute of Nuclear
Physics, Novosibirsk, Russia  2Novosibirsk State University, Novosibirsk,
Russia  3University of Texas at Austin, Austin, TX, USA

###### Abstract

In plasma wakefield accelerators, the wave excited in the plasma eventually
breaks and leaves behind slowly changing fields and currents that perturb the
ion density background. We study this process numerically using the example of
a FACET experiment where the wave is excited by an electron bunch in the
bubble regime in a radially bounded plasma. Four physical effects underlie the
dynamics of ions: (1) attraction of ions toward the axis by the fields of the
driver and the wave, resulting in formation of a density peak, (2) generation
of ion-acoustic solitons following the decay of the density peak, (3) positive
plasma charging after wave breaking, leading to acceleration of some ions in
the radial direction, and (4) plasma pinching by the current generated during
the wave-breaking. Interplay of these effects result in formation of various
radial density profiles, which are difficult to produce in any other way.

  *

Keywords: plasma wakefield acceleration, strongly nonlinear wake, ion-acoustic
solitons

##  1 Introduction

Acceleration of electrons and positrons in near-light-speed plasma waves is a
rapidly developing field of research [1, 2, 3, 4, 5]. The highest performing
plasma accelerators are based on strongly nonlinear wakefield by a
relativistic electron bunch or a high-intensity laser pulse in the blowout
regime [6, 7]. In this case, the field of the driver is strong enough to
completely expel electrons from its immediate wake, forming a so-called
‘bubble’.

Usually, in plasma-based accelerators, the accelerated beam (called a witness)
follows closely behind the driver. The time between the two beams is too short
for plasma ions to respond, except in the case of very dense electron drivers
[8, 9, 10, 11, 12] or special regimes in which a small perturbation of the ion
density stabilizes the electron driver [13]. For this reason, the ion dynamics
has not attracted as much attention as other wakefield features, and has been
studied mainly in the context of the wakefield lifetime [14, 15, 16, 18, 17]
or the overall energy balance in the system [19]. However, the ion motion can
result in formation of exotic, slowly changing plasma density profiles, which
may be useful for diagnostics [20, 19], driver guiding [1], beam stabilization
[21, 22], wakefield enhancement [24, 23], radiation generation [25], or making
field structure favorable for positron acceleration [26, 29, 27, 28]. Ion
density valleys or peaks can appear both in strongly nonlinear [30, 31] and
almost linear [14, 15, 20, 17, 32, 16, 18, 33] regimes and for all kinds of
drivers. As more dense drivers become available [34], the importance of ion
dynamics will increase. Despite the visual similarity of the observed density
structures, the mechanisms of their formation are different in each case. In
this paper, we study the ion dynamics after the wakefield excitation by a
dense electron beam in a radially bounded plasma in the bubble regime. In this
regime, the driver can act on the ions not only by means of the excited wave,
but also directly by its own field and by the field of the current arising in
the plasma.

The paper is organized as follows. In section 2, we describe the problem under
study and the methods. Then we focus our attention on four effects, some of
which are specific to electron drivers in radially bounded plasmas. The first
effect is that ions from some paraxial region receive a negative radial
momentum and form a density peak on the axis (section 3). In our case, both
the electric field of the beam and the ponderomotive force of the plasma wave
are responsible for the peak formation. The second effect is the appearance of
ion-acoustic solitons (section 4). The on-axis density peak eventually breaks
up into several smaller, diverging peaks, which propagate at the ion-sound
velocity. The third effect is related to fast electrons that appear when the
bubble collapses (section 5). These electrons escape from the plasma column
and leave an uncompensated ion charge behind, which, in turn, pushes outer ion
layers radially. The escape of fast electrons also creates a compensating
current in the plasma, which ultimately leads to the fourth effect (section
6): the bulk of the ions contracts to the axis, forming a high-amplitude
compression wave. In section 7, we summarize the main findings.

Figure 1:  Wakefield excitation by the equilibrium beam: (a) radial dependence
of the plasma ion density $n_{i}$ shortly after ionization and (b) the
simulated density of the electron bunch $n_{b}$ (orange) and plasma electrons
$n_{e}$ (blue) during the first three wakefield periods. The bunch propagates
to the left.

##  2 Statement of the problem

We consider the case that corresponds to the recent E224 experiment [19] at
the SLAC Facility for Advanced aCcelerator Experimental Tests (FACET) [35]. In
the E224 experiment, the temporal evolution of the plasma density profile was
measured and compared to numerical simulations. The achieved quantitative
agreement proved the validity of the simulation method, so we use the same
approach and the same set of parameters in our study.

Figure 2: (a) The ion density map $n_{i}(r,t)$. The numbers in circles
indicate the features discussed in the text and detailed in Supplementary
Materials, movie 1. (b) The average ion density $\langle{n}_{i}\rangle(t)$ in
the region $r<0.05k_{p}^{-1}$. The averaged density does not depend on the
simulation grid step, unlike the density on the axis.

An electron bunch of energy 20 GeV, charge 2 nC, root-mean-squared (rms)
radius $30\,\mu$m and rms length $55\,\mu$m passes through a chamber filled
with lithium vapor of atomic density $n_{0}=8\times 10^{16}\,\rm{cm}^{-3}$
[36, 37]. The bunch head field-ionizes the lithium and creates a plasma column
of radius about $40\,\mu$m (figure 1). The plasma focuses the rest of the
bunch down to the equilibrium radius of about $4\,\mu$m. After a short period
of initial equilibration, most of the bunch propagates almost without changing
its shape and excites a strongly nonlinear plasma wave in a plasma of constant
radius. The equilibration stage (first 32 cm of bunch propagation in the
plasma) is simulated with a 2d3v (axisymmetric) version of the particle-in-
cell code OSIRIS [19, 38]. Then, the equilibrium profiles of plasma and beam
are imported into the quasistatic axisymmetric 2d3v code LCODE [39, 40] as
initial conditions, and the simulations continue up to thousands of wakefield
periods at a constant longitudinal position $z$. In the equilibrium,
approximately 60% of the beam (1.2 nC) propagates in the plasma and
participates in driving the plasma wave.

To present the results in a more general form, we measure times in units of
$\omega_{p}^{-1}$, distances in $k_{p}^{-1}=c/\omega_{p}$, fields in
$E_{0}=m_{e}c\omega_{p}/e$, and densities in $n_{0}$, where
$\omega_{p}=\sqrt{4\pi n_{0}e^{2}/m_{e}}$ is the plasma frequency, $m_{e}$ is
the electron mass, $e$ is the elementary charge, and $c$ is the speed of
light. For $n_{0}=8\times 10^{16}\,\rm{cm}^{-3}$, $\omega_{p}^{-1}\approx 60$
fs, $k_{p}^{-1}\approx 19\,\mu$m, and $E_{0}\approx 27\,\rm{GV/m}\approx 0.9$
MGs. For lithium ions of the mass $m_{i}\approx 12850\,m_{e}$, the
characteristic timescale of ion response is
$\tau_{i}=\omega_{p}^{-1}\sqrt{m_{i}/m_{e}}\approx 113\,\omega_{p}^{-1}\approx
7$ ps.

The radius of the simulation window is $500\,k_{p}^{-1}$, the radial grid step
is $0.005\,k_{p}^{-1}$, and the time step is $0.005\,\omega_{p}^{-1}$. The
plasma consists of two species (electrons and ions) with $2.5\times 10^{5}$
equally weighted macro-particles in each. The number of plasma macro-particles
slightly increases due to impact ionization of the surrounding neutral vapor
[19], but this effect is not significant at the times considered here.

##  3 On-axis density peak

Formation of the ion density peak near the axis (figure 2, feature 1) is
routinely observed in simulations of various plasma-based accelerator schemes
[14, 32, 15, 16, 41, 31, 17, 20]. In the bubble regime, the peak appears
because of the strong radial electric field of an electron beam located inside
the bubble [30, 8]. In turn, in moderately nonlinear wakes, the inward radial
force on the ions arises from specific properties of the plasma wave. The
ponderomotive force of the wave is such that it pushes the plasma towards the
axis [14, 15]. Another explanation of the same effect is that plasma electrons
have different oscillation amplitudes at different radii, which leads to an
average charge separation and, consequently, to a nonzero radial field
experienced by the ions [16, 18]. The ions gain the radial momentum gradually
over many wave periods rather than during a short time of driver passage.

Figure 3: The total charge density of plasma and bunch,
$\rho=e(n_{i}-n_{e}-n_{b})$. Figure 4: The radial momentum $p_{r}(t)$ of ions
initially located at different radial positions $r_{0}$ (colored lines) and
the radial electric field $E_{r}(t)$ at $r=0.5\,k_{p}^{-1}$ (black dashed
line). Figure 5: The radial ion momentum $p_{r}(r)$ at different times. The
line $\omega_{p}t=7$ shows the momentum gained mainly because of the beam
field; the line $\omega_{p}t=140$ corresponds to the arrival of near-axis ions
to the axis and shows the combined effect of the beam and the wave.

In the considered case of a dense electron driver, the contributions of both
the driver and the wave are important. The total charge density in the axial
region is such that there is a large negative charge of the beam in the first
bubble, and then narrow negatively charged and extended positively charged
regions alternate (figure 3). The ions first receive an inward push from the
driver and then experience an oscillating force with a non-zero average
(figure 4). Relative contributions of the initial push and wave force are
comparable in value. In the considered case, the wave contribution is roughly
5 times stronger: the line $\omega_{p}t=7$ in figure 5 characterizes the
initial push, while the difference between this line and the line
$\omega_{p}t=140$ is due to the wave force. As a consequence, the time when
ions from the near-axis region reach the axis is shorter than it would be if
they received only the initial push, $140\,\omega_{p}^{-1}$ instead of
$10^{3}\omega_{p}^{-1}$. The latter number follows from the slope of the blue
line in figure 5.

Figure 6: Equidistantly selected ion trajectories $r(t)$ plotted over the map
of the radial electric field $E_{r}(r,t)$. The field is time-averaged over
$20\omega_{p}^{-1}$ to reduce the effect of plasma oscillations.

##  4 Ion-acoustic solitons

The movement of the ions toward the axis, followed by the formation of a
density peak there, gives rise to ion-acoustic solitons, which are seen as
diverging density ridges in figure 2 (feature 2). When the first ions cross
the axis, they not only create the density peak, but also a positive ambipolar
potential, which attracts additional electrons to maintain average plasma
quasi-neutrality. The resulting radial electric field reverses the direction
of the next portion of ions (figure 6). The counterstreaming ions form an off-
axis density peak accompanied by an ambipolar potential and an electric field
directed away from the peak. The peak moves radially outward against the
background of inward-moving ions (figure 7). As the soliton moves away from
the axis, there is nothing to prevent the next portion of ions from passing
through the soliton, reaching the axis and forming the next density peak
there, which gives rise to the next soliton, and so on. Soliton generation
continues as long at there is ion motion towards the axis. The dynamics of ion
and electron density, ion velocity and radial electric field during soliton
formation is detailed in the Supplementary materials, movie 1. Similar
solitons, but not the mechanism of their formation, were also observed in
simulations [31].

Figure 7: (a) The phase portrait of plasma ions (orange points) and the radial
electric field $E_{r}$ (blue dashed line) at $\omega_{p}t=1200$. The electric
field is time-averaged over $20\omega_{p}^{-1}$ to reduce the effect of plasma
oscillations. (b) The corresponding radial dependencies of electron ($n_{e}$)
and ion ($n_{i}$) densities. The electron density is smoother because of the
high electron temperature, and this leads to an ambipolar potential. The
dashed lines show the soliton parameters discussed in the text.

To make sure that the observed feature really behaves as a soliton, we compare
its velocity with the velocity expected for a soliton. We follow the approach
outlined in Ref. [42] and start from the hydrodynamic equations for one-
dimensional ion motion:

$\displaystyle\frac{\partial n}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{\partial(nv)}{\partial x},$ (1)
$\displaystyle\frac{\partial v}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{\partial}{\partial
x}\left(\frac{v^{2}}{2}+\frac{\phi}{\tilde{m}_{i}}\right),$ (2)
$\displaystyle\frac{\partial^{2}\phi}{\partial x^{2}}$ $\displaystyle=$
$\displaystyle n_{\infty}e^{\phi/T}-n,$ (3)

where we use dimensionless quantities: ion density $n=n_{i}/n_{0}$,
electrostatic potential $\phi$ (normalized to $m_{e}c^{2}/e$), ion velocity
$v=v_{i}/c$, unperturbed ion density $n_{\infty}$, electron temperature
$T=T_{e}/(m_{e}c^{2})$, and ion mass $\tilde{m}_{i}=m_{i}/m_{e}$. The ions are
cold.

We are looking for a stationary solution, vanishing at infinity, propagating
with velocity $u$ and depending on $\zeta=x-ut$. Then the equations can be
integrated:

$\displaystyle
n=\frac{n_{\infty}}{\sqrt{1-\phi/\phi_{0}}}=\frac{n_{\infty}u}{u-v},$ (4)
$\displaystyle
v=u\left(1-\sqrt{1-\phi/\phi_{0}}\right)=\frac{n-n_{\infty}}{n}u,$ (5)
$\displaystyle\left(\frac{\partial\phi}{\partial\zeta}\right)^{2}=2n_{\infty}\left(T\bigl{(}e^{\phi/T}-1\bigr{)}+2\phi_{0}\bigl{(}\sqrt{1-\phi/\phi_{0}}-1\bigr{)}\right),$
(6)

where $\phi_{0}=\tilde{m}_{i}u^{2}/2$.

Let $n_{m}$, $v_{m}$ and $\phi_{m}$ be the maximum values of the solution
reached where $\partial\phi/\partial\zeta=0$. They are related as follows:

$\displaystyle\phi_{m}=\phi_{0}\frac{n_{m}^{2}-n_{\infty}^{2}}{n_{m}^{2}},$
(7) $\displaystyle
T\left(e^{\phi_{m}/T}-1\right)+2\phi_{0}\left(\sqrt{1-\phi_{m}/\phi_{0}}-1\right)=0.$
(8)

We analyze the outermost soliton, as its properties are least affected by the
cylindricity. Its numerical parameters can be taken from figure 7: background
ion velocity $v_{-}=-1.6\times 10^{-4}$, maximum ion velocity $v_{+}=2.6\times
10^{-4}$, background density $n_{\infty}=0.7$, and peak ion density
$n_{m}=2.45$. From the equation (5), we find the soliton propagation velocity
with respect to the ion background

$u=\frac{n_{m}}{n_{m}-n_{\infty}}(v_{+}-v_{-})=6.0\times 10^{-4}$ (9)

and in the laboratory frame

$u_{l}=u+v_{-}=4.2\times 10^{-4}.$ (10)

On the other hand, the propagation velocity of the density peak can be
directly measured in figure 2(a):

$u_{l,\>sim}=4.4\times 10^{-4},$ (11)

which agrees with the theoretical value (10).

Figure 8: The electron distribution function $f_{sim}(v_{r})$ observed in
simulations near the outermost soliton at $\omega_{p}t=1200$ and the Maxwell
distribution $f_{th}$ for the estimated temperature $T_{e}=2.4$ keV.

The velocity and peak density of the soliton determine the maximum potential
and electron temperature from the equation (8): $\phi_{m}=3.9\times 10^{-3}$,
$T=4.7\times 10^{-3}$, $T_{e}=2.4$ keV. The ion sound velocity for this
temperature is close to the soliton velocity (9):

$c_{s}=\sqrt{\frac{T_{e}}{m_{i}}}=6.1\times 10^{-4}c.$ (12)

The Maxwell distribution for this temperature is close to the electron
velocity distribution observed in simulations (figure 8). Taken together, this
proves that the observed features are ion-acoustic solitons.

Figure 9: (a) Selected trajectories of plasma electrons in the co-moving
window: blue (orange) lines correspond to electrons escaping after the first
(second) bubble, and green lines to other electrons. (b) The position of these
electrons at $\omega_{p}t=18$ on the momentum plane ($p_{r},p_{z}$). The thin
line shows the parabola $p_{z}=p_{r}^{2}/(2m_{e}c)$.

##  5 Wave breaking

The wavebreaking in the considered strongly nonlinear regime leads to ejection
of some electrons from the plasma with high (relativistic) velocities (figure
9). These electrons acquire a large momentum when the bubbles collapse and
escape from the plasma column in the form of diverging tail waves. Other types
of wavebreaking discussed in Ref. [43] do not result in such high energies.
Most of the high-energy electrons appear in the tail of the first bubble. They
carry away about 40% of the wakefield energy and form a density ridge visible
on the electron density map (figure 1). Subsequent bubbles also generate tail
waves, but weaker ones. The electrons ejected from different bubbles are
initially located at approximately the same radii (figure 9(a)).

After the high-energy electrons leave the plasma, the excess positive charge
concentrates in its outer layer (figure 3) and generates a radial electric
field around the plasma and inside this layer (figure 10(a)). The layer
thickness (of the order of $k_{p}^{-1}$) is much greater than the Debye length
corresponding to the electron temperature [44], because the near-boundary
electrons oscillate (figures 1 and 3) and respond to the average field as if
their temperature were high. The ions located in this layer are then
accelerated in the radial direction, forming the feature 3 in figure 2. As the
plasma expands radially, the region of the strong electric field also moves to
larger radii (figure 10(b)). Similar processes occur when a moderately
nonlinear plasma wave breaks [45, 46] or the plasma is heated by a strong
laser pulse [47].

The escaping high-energy electrons also have a large positive longitudinal
momentum, that is, they move in the direction of driver propagation. This is a
common feature of wavebreaking, which follows from the basic wakefield
properties. If the driver evolves slowly and propagates in an unperturbed
plasma where the particles are initially in rest, then all plasma properties
depend on the longitudinal coordinate $z$ and time $t$ only in their
combination $\xi=z-ct$, and there is a relation between the relativistic
factor $\gamma$ and the longitudinal momentum $p_{z}$ of plasma electrons [48]

$(\gamma-1)m_{e}c^{2}-e\Phi-cp_{z}=0,$ (13)

where $\Phi$ is the so-called wakefield potential or pseudo-potential related
to the longitudinal electric field $E_{z}$ as

$\Phi(r,\xi)=\int_{\xi}^{\infty}E_{z}(r,\xi^{\prime})\,d\xi^{\prime}.$ (14)

By introducing

$W=\frac{e\Phi}{m_{e}c^{2}}+1$ (15)

we obtain from equation (13)

$p_{z}=\frac{p_{r}^{2}}{2m_{e}cW}+\frac{1-W^{2}}{2W}.$ (16)

Outside the region of strong wakefields, $e\Phi\ll m_{e}c^{2}$, and $W\approx
1$. Therefore, any electron escaping from the plasma with a large radial
momentum $p_{r}$ must also have a large positive longitudinal momentum

$p_{z}\approx\frac{p_{r}^{2}}{2m_{e}c},$ (17)

and simulations confirm this (figure 9(b) and movie 2 in Supplementary
materials).

Figure 10: Evolution of (a),(b) radial electric field $E_{r}$ and (c),(d)
azimuthal magnetic field $B_{\phi}$ at different timescales. At the longer
timescale, the fields are time-averaged over $20\omega_{p}^{-1}$.

##  6 Plasma pinching

As forward-moving fast electrons leave the plasma column, an average current
of the opposite direction appears inside the plasma. This current creates a
strong azimuthal magnetic field (figure 10(c),(d)), which in turn exerts an
inward radial force on the return current of plasma electrons. A charge
separation electric field (blue area at $\omega_{p}t\lesssim 500$ in figure
10(b)) transmits this force to ions and accelerates them toward the axis.
Thus, most of the ions move inward under the magnetic force, while a smaller
(outer) part moves outward, entrained by the escaping fast electrons (figure
2(a)). The inward-directed force is nonlinear in radius and is strongest at
the periphery, so the far ions overtake the near ones and form a compression
wave (feature 4 in figure 2(a)), which behaves like a soliton and is similarly
accompanied by charge-separation electric fields (figure 10(b)). When high
density spikes appear in the near-axis region, they generate ion-acoustic
solitons by the mechanism discussed in section 4. As a result, the dense
plasma pinch formed near the axis is highly inhomogeneous (feature 5 in figure
2(a)).

At even longer times, ionization of the surrounding neutrals occurs, and we
observe an increase of ion density beyond the initial plasma radius (feature 6
in figure 2(a)). We will not discuss this feature, as it is described in
detail in Ref. [19].

Figure 11: Radial profiles of the ion density at different times after the
drive beam passage.

##  7 Discussion

The technique in which an additional beam or discharge passes through the
plasma before the main pulse to form the desired density profile is widely
used in plasma-based wakefield accelerators [1]. The long-term evolution of a
strongly nonlinear plasma wave can provide additional opportunities for this.
We considered the case typical of a high-amplitude plasma wave in a finite-
radius plasma and observed a variety of ion density profiles formed at
different times after beam passage (figure 11). These are a narrow density
spike at the center of a nearly uniform plasma at $\omega_{p}t\sim 260$, a
density channel with steep high-density walls at $\omega_{p}t\sim 1500$, and a
narrow filament with an order of magnitude increased density at
$\omega_{p}t\sim 3000$. A fine-scale density structure is superimposed on the
latter two. At the heart of this variety are the four effects discussed. Two
of them, the on-axis density peaking and the formation of ion-acoustic
solitons, do not require a boundary and will also work in an unbounded plasma.
Two others, the radial acceleration of sheath ions and the plasma pinching,
are characteristic only of radially bounded plasmas.

The authors thank T. Silva and J. Vieira for providing the OSIRIS output data
from [19], which were used as input data for the simulations presented, and
thank M. Hogan, A. Sosedkin, V. Yakimenko, and R. Zgadzaj for helpful
discussions. This work was supported by the Russian Science Foundation,
project 20-12-00062.

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