# An improved electron beam dynamics design for laboratory plasma-
astrophysical studies: a technical note111The note is supplementary to the
framework of laboratory plasma-astrophysical studies previously carried out at
Deutsches Elektronen-Synchrotron in Refs. [3-5].

Ye Chen and Chun-Sung Jao<EMAIL_ADDRESS>
###### Abstract

A technical note is given regarding our previous laboratory plasma-
astrophysical studies [C.-S. Jao et al., High Energy Density Physics 32, 31-43
(2019) and Y. Chen et al., Nucl. Instrum. Methods Phys. Res., Sect. A 903, 119
(2018)]. In this note, an upgraded accelerator beamline design is proposed
based on a feasible experimental setup in a realistic laboratory environment.
The improved design aims to provide milliampere (mA) mega-electron-volt (MeV)
quasi-continuous (cw) electron beams for plasma-astrophysical applications.
Such a design utilizes a so-called mixed-guiding-field magnetic system right
after the cut disk structure (CDS) booster cavity to provide a periodic
longitudinal focusing field. The transportation of the produced cw beam with
large energy spread to the plasma cell location is improved. The magnetic
field serves as well as a seeding field in the plasma environment for the
growth of electromagnetic instabilities. In conjunction with the appliance of
a circular collimator at the exit of the CDS, the new design allows production
of quasi-cw beams with a three orders higher number density at the entrance of
the plasma cell compared to the previous design for a seeding magnetic field
of about 50 mT while the locally enhanced electric field at the cathode is up
to 8 GV/m. The associated beam dynamics simulation results are presented. As
proof of principle studies, the produced electron beams are applied in
nonlinear plasma-astrophysical simulations for exploring the growth of the
instabilities. The extracted parameters and/or distributions from the
generated electron beams in the laboratory environment are used in these
particle-in-cell simulations. The obtained results are presented and
discussed.

##  1 Introduction

Laboratory plasma-astrophysics has became one of the most significant branches
engrossing in astrophysics over the last decades. As an alternative method
apart from the classical observation and numerical simulation [1, 2], it
provides a novel methodology for exploring the mechanisms of astrophysical
phenomena in a realistic laboratory environment based on advanced technologies
and modern instrumentation in high energy physics. A research program was
launched at Deutsches Elektronen-Synchrotron for establishing a so-called
plasma-astrophysical laboratory for experimental studies of beam-plasma
instabilities [3, 4, 5].

With feasible experimental setups in such a laboratory environment, particle
beams need to be produced in a controlled manner to fulfill a set of
compromised requirements between several crucial electron beam parameters for
developing the instabilities. This does include not only the duration
(preferably continuous), the average current ($\geq$mA), the average energy
(several MeV at least), the number density, but also other instability-
mechanism-oriented quality properties of the required particle beam. In
principle, a long beam (continuous) of high average current (mA) is required
as the energy source to excite the plasma wave. A higher beam energy is
expected to induce stronger perturbation of the instability due to the fact
that plasma wave excitation is led by the drift energy of streaming particles.
Higher beam energy ($\geq$MeV) results also in a higher growth rate of the
instability. From this point of view the appliance of RF injectors for our
purpose can be much more beneficial due to higher beam energy and principally
better beam qualities compared to DC guns [6, 7]. A matched number density of
the particle beam with the plasma density is of another crucial importance to
grow the instability. This requires specially designed beam transport system
for properly focusing the beam with large energy spread at the plasma cell
location while maintaining other beam quality parameters as required.

A preliminary design was reported in [3]. It demonstrated electron bunch
extraction based on field emission from a designed metallic needle cathode and
quasi-cw beam formation using velocity bunching over subsequent radio-
frequency (RF) cycles of a cut disk structure (CDS) downstream of an L-band RF
electron gun. In this note, an improved beam dynamics design is proposed and
will be further discussed based on 3D particle tracking simulations (Sec. 2).
Based on the improved design, the quasi-cw beams with fulfilled requirements
produced in a realistic laboratory environment are applied in nonlinear
plasma-astrophysical simulations for studying the beam-plasma instabilities.
This involves the electrostatic streaming instability [8, 9, 10], the
filamentation instability [11, 12] and a non-resonant streaming instability
(i.e. Bell’s instability[13]). Among those instabilities, the electrostatic
instability is commonly known as the unstable mode with the shortest
wavelength and the shortest development time [14]. As a competition mode, the
prior existence of the electrostatic instability is critical to the laboratory
observation of other aforestated instabilities, in terms of its parametric
dependencies on the realistic particle beams produced in the laboratory and
thereby plausible suppression schemes for it. The obtained results consisting
of case studies for the so-called cold beam, warm beam and quasi-cw beam will
be shown in this note (Sec. 3). A summary and an outlook will be given in Sec.
4.

##  2 Design of electron beam dynamics

Figure 1 shows a proposed experimental setup. It consists of a specially
designed needle cathode, an L-band RF injector [6], a pair of focusing
magnets, a cut disk structure (CDS), a circular collimator at the exit of the
CDS, a periodic solenoidal focusing system, a plasma cell and a beam dump. The
field-emitted (FE) electron bunch is generated from the needle cathode sitting
on the backplane of the 1.3 GHz copper resonator through highly enhanced
electric field gradient up to 8 GV/m. The electron bunches produced over
subsequent RF cycles are accelerated by the RF gun and velocity debunched
through the 1.3 GHz CDS forming a quasi-cw beam by jointing existing gaps in
the time domain profile of the produced FE beam between neighbouring RF
periods. A set of required beam parameters are produced in [3], however, the
scheme of beam focusing is not efficient for transporting the produced
particle beam to the plasma cell location due to large beam energy spread.
Consequently, the resulting electron number density when transported to the
entrance of the plasma cell location are limited to approx.
$10^{7}-10^{8}$/cm3 for an enhanced electric field gradient of 5.7 GV/m at the
cathode.

Figure 1: A proposed laboratory experimental setup for the enhancement of
electron number density in the plasma cell.

As shown in Fig. 1, a periodic magnetic focusing system is applied. Such a
system provides a longitudinal mixed magnetic field formed by overlapping
fields of an array of solenoidal lenses (e.g. four solenoids exemplarily shown
in Fig. 1). A mathematical description of the periodical field on the
longitudinal axis can be represented using a Fourier series [15]

$Bz(z)=B_{0}[1+\Sigma_{n}a_{n}cos(k_{0}z)],$ (1)

where $Bz$ represents the mixed guiding magnetic field in the longitudinal (z)
direction and $B_{0}$ stands for the magnetic field strength of the solenoid.
The term $k_{0}$=2$\pi$/$T$ with $T$ denoting the period of the solenoidal
array. The symbol $a_{n}$ is defined as the Fourier coefficient with
$n=1...\infty$. The mixed guiding field $Bz(z)$ is numerically reconstructed
by using four solenoidal lenses periodically arranged in the beam propagation
direction and used in the following beam dynamics simulations.

Figure 2 shows the rms size of the particle beam in the longitudinal direction
with the gray diagram indicating the location of the plasma cell. The
evolution of the beam size is compared among multiple cases with the seeding
magnetic field $B_{seed}$ (maximum field amplitude of $B_{z}$) as a variable.
The gray dashed curve shows the pattern of the longitudinal magnetic field
used for the simulations. The black curve shows in the seeding-field-free case
a significant beam size growth within the plasma cell while the ones in cyan,
blue and red show a suppressed behavior of the beam divergence and a much
reduced beam size within the plasma cell for an applied seeding field of 10,
25 and 50 mT, respectively. A significant role plays the applied seeding
magnetic field in maintaining the beam size and therefore in increasing the
electron number density.

Figure 2: Significant improvement of beam focusing within the plasma cell
using a periodic magnetic focusing system. $\sigma_{rms}$: rms beam size.

In Fig. 3, the right inset shows the transverse beam profile right after the
CDS where the majority of electrons is concentrated in the central part of the
bunching area while for a large beam-halo area much less electrons are
populated. Therefore, properly collimating the beam may increase the electron
number density while maintaining a sufficiently high beam current. For such a
purpose, a circular collimator is applied at the exit of the CDS. Figure 3
shows the electron number density as a function of the seeding magnetic field
(i.e. 1-60 mT) with the radius of the collimator as a variable (i.e. 0.5, 1.0
and 2.0 mm). The left inset of Fig. 3 shows the average beam current plotted
versus the collimator radius. As shown, with a collimator radius of 0.5 mm and
a seeding magnetic field of 50 mT, the average beam current can be maintained
at about 10 mA for a reached number density on the order of $10^{10}$/cm3.

Figure 3: Simulated electron number density within the plasma cell as a
function of the seeding magnetic field and the collimator radius.

Furthermore, Fig. 4 shows an exemplary temporal profile of the produced quasi-
cw beam based on the improved beam dynamics design. The base current is
increased to 1 mA and the electron number density within the plasma cell is on
the order of $10^{10}$/cm3. In the following, such electron beams produced in
the realistic laboratory environment are used for the nonlinear plasma-
astrophysical simulations.

Figure 4: Temporal profile of a produced quasi-cw electron beam with a base
current of about 1 mA over RF periods at $\lambda=231$ mm.

##  3 Plasma-astrophysical studies using realistic laboratory electron beam
distributions

To understand the physical conditions for beam-plasma instabilities to occur,
the extracted parameters and/or distributions of the realistic laboratory
electron beams (Sec. 2) are used for plasma-astrophysical simulations. A fully
relativistic particle-in-cell code (OSIRIS) [16, 17, 18] is employed.
Regarding a basic setup for simulations [4], a uniform stationary plasma is
filled in a two-dimensional open-boundary system ($43.14$ cm $\times~{}5.9$
cm) and an electron beam with an initial width of 0.17 cm is applied in the
simulation. The number density of stationary plasma is set as $n_{0}=10^{13}$
cm-3. The width of the electron beam is roughly equal to the skin length of
the background plasma ($c/\omega_{pe}$ with $\omega_{pe}$ standing for the
electron plasma frequency). In exemplary simulations, the injection rate is
set as 1.78 GHz instead of the RF frequency of 1.3 GHz based on the assumption
that the growth rate of the involved instability can only be slightly varied
by changing the injection rate. The settings indicate that the electron
bunches are injected into the simulation system every 100 $\omega_{pe}^{-1}$.

Referred to the results from 3D particle tracking simulation, as indicated in
Fig. 5, a typical momentum distribution of the produced quasi-cw electron beam
is shown for a single RF cycle. Due to the time-of-flight effect, a beam
momentum shift with a positive slope ($p_{z}$=10 to 4 $m_{e}c$ with $m_{e}$
and $c$ denoting the electron mass and the speed of light, respectively) is
clearly observed in the longitudinal direction at the entrance of the plasma
cell. A fundamental question arises when a realistic particle distribution is
applied in our simulations, in terms of the differences in exciting the
streaming instability compared to the conventional assumption of using a cold
beam and/or a warm beam. These assumptions essentially refer to the beam
temperature which strongly influences the growth rate of the electrostatic
streaming instability with the shortest development time among the
instabilities of our interest (Sec. 1). The linear growth rate of the
electrostatic instability, in particular, decreases with increasing the beam
temperature[8, 9, 10, 4, 5]. Therefore, the impacts of different beam
distributions including the produced quasi-cw beam onto the electrostatic
instability are worth of first investigations. Nonlinear PIC simulations are
thus carried out for three cases, namely, the cold beam case, the warm beam
case and the quasi-cw beam case. A homogeneous Maxwell-Boltzmann momentum
distribution centering at $7m_{e}c$ is set with a thermal velocity
($v_{th,e}$) of 0.01$c$ and 0.2$c$ for the cold and warm beam cases,
respectively. For the quasi-cw beam case, a realistic beam momentum
distribution is used, as shown in Fig. 5.

Figure 5: The longitudinal momentum distribution of a quasi-cw electron beam
imported in PIC simulations. Figure 6: Spatial profiles of the longitudinal
electric field components (left panels, in units $m_{e}c\omega_{p}/e$) and the
corresponding Fourier plots (right panels).

Figure 6 shows the spatial profile of longitudinal electric field components
(panel a) and the corresponding Fourier plots (panel b) for the three cases at
T=5.783 ns (1024 $\omega_{pe}^{-1}$), respectively. As shown, the electric
field perturbation is present as the electron beam is propagating through the
plasma environment. The major perturbations are along X=0, at which position
the electron beam is traveling along a central path with an initial width of
0.17 cm. The wavelength (wave number) of the major perturbations is about 1.05
cm (5.88 cm-1) indicating a reciprocal relation of the wave number to the skin
length of the background plasma. This is well resolved in the PIC simulations.
It can therefore be confirmed from the Fourier plots in Fig. 6 (b), that the
electrostatic instability occurs in all three cases. Furthermore, the cold
beam case shows the strongest electric field perturbation while in the other
two cases the instability strengths are on the same order of magnitude and
both are much reduced. It should be noted, that the warm beam case turns out
to be a good approximation of the quasi-cw beam. Note, in addition, that the
appliance of a quasi-cw beam produced in a realistic laboratory environment
may be even more beneficial, in terms of the suppression of the electrostatic
instability for relieving its competition with the target instability, for
studying the beam-plasma instabilities compared to the ideal cold beam case
which is typically used in conventional simulations.

##  4 Conclusion

An improved experimental setup is proposed for studies of beam-plasma
instabilities in the laboratory environment. The associated beam dynamics
design is carried out to provide quasi-cw electron beams with required
parameters. With a circular collimator placed at the exit of the debunching
structure, the appliance of a periodic longitudinal magnetic focusing system
renders significant improvements in production of mA and MeV quasi-cw electron
beams with a three orders higher number density within the plasma cell for a
seeding magnetic field of approx. 50 mT, in comparison to the previous design
at a conservative working point of the field emitter.

The produced electron beams are employed in the plasma-astrophysical
simulations for investigating the parametric dependencies and thereby the
suppression scheme for the competition of the fast-growing electrostatic
streaming instability with other instabilities of our interest. Numerical
results for multiple cases (i.e. cold beam, warm beam and realistic beam) show
the well-resolved growing electrostatic instability in the beam-plasma system.
The obtained wave numbers well agrees with theoretical predictions. A further
comparison of electric field perturbations between the multiple cases
demonstrates the strongest instability growth in the cold beam case while
comparable growths of the instability for the warm beam and the quasi-cw beam.
The results show the warm beam model can be a good approximation of the
realistic quasi-cw beam, and that the appliance of the quasi-cw beam may be
even more beneficial for studying the beam-plasma instabilities in comparison
to the ideal cold beam model conventionally used in many theoretical and/or
numerical analysis.

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