Exploring ultra-high-intensity wakeelds in carbon nanotube arrays an eective plasma-density approach A. Bonatto

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Exploring ultra-high-intensity wakeﬁelds in carbon nanotube arrays:

an eﬀective plasma-density approach

A. Bonatto∗

Graduate Program in Information Technology and Healthcare Management, and the Beam Physics Group,

Federal University of Health Sciences of Porto Alegre, Porto Alegre, RS, 90050-170, Brazil

G. Xia and O. Apsimon

Department of Physics and Astronomy, The University of Manchester, Manchester, M13 9PL, United Kingdom

and The Cockcroft Institute, Sci-Tech Daresbury, Warrington, WA4 4AD, United Kingdom

C. Bontoiu, E. Kukstas, V. Rodin, M. Yadav, and C. P. Welsch

Department of Physics, The University of Liverpool, Liverpool, L69 3BX, United Kingdom

and The Cockcroft Institute, Sci-Tech Daresbury, Warrington, WA4 4AD, United Kingdom

J. Resta-L´opez†

ICMUV, Instituto de Ciencia de los Materiales, Universidad de Valencia, 46071 Valencia, Spain

(Dated: March 10, 2023)

Charged particle acceleration using solid-state nanostructures has attracted attention in recent

years as a method of achieving ultra-high-gradient acceleration in the TV/m domain. More con-

cretely, metallic hollow nanostructures could be suitable for particle acceleration through the exci-

tation of wakeﬁelds by a laser or a high-intensity charged particle beam in a high-density solid-state

plasma. For instance, due to their special channelling properties as well as optoelectronic and

thermo-mechanical properties, carbon nanotubes could be an excellent medium for this purpose.

This article investigates the feasibility of generating ultra-high gradient acceleration using carbon

nanotube arrays, modelled as solid-state plasmas in conventional particle-in-cell simulations per-

formed in a two-dimensional axisymmetric (quasi-3D) geometry. The generation of beam-driven

plasma wakeﬁelds depending on diﬀerent parameters of the solid structure is discussed in detail.

Furthermore, by adopting an eﬀective plasma-density approach, existing analytical expressions,

originally derived for homogeneous plasmas, can be used to describe wakeﬁelds driven in periodic

non-uniform plasmas.

I. INTRODUCTION

High-energy particle accelerators are predominantly

based on radiofrequency (RF) technology. However,

standard RF technology is limited to gradients of the

order of 100 MV/m due to surface breakdown [1]. Thus,

larger and more expensive accelerator facilities are nec-

essary in order to obtain higher energy particle beams.

Therefore, R&D into novel accelerator techniques is im-

portant to overcome the present acceleration limitations

towards more compact and cost-eﬀective solutions. Sev-

eral alternative paths towards high-gradient acceleration

are currently being investigated, e.g. techniques using

dielectric microstructures [2–4] or plasmas as accelerat-

ing media. For instance, plasma wakeﬁeld acceleration

(PWFA) methods based on gaseous plasma have been

shown to produce gradients of up to approximately 100

GV/m [5–8]. For typical gaseous plasmas used as ac-

celeration media, the maximum achievable accelerating

gradient is limited by the so-called plasma wave-breaking

limit, which depends on the plasma density. In the lin-

ear regime, this limit, known as the cold non-relativistic

∗abonatto@ufcspa.edu.br

†javier2.resta@uv.es

wave breaking ﬁeld [9], is given by

E0[V/m] = mec ωp/e '96pn0[cm−3],(1)

where meand eare the electron mass and charge, re-

spectively, cis the speed of light in vacuum, ωp=

[n0e2/(0me)]1/2is the plasma frequency, n0is the

plasma density and 0is the vacuum permittivity. To

surpass present PWFA limits, solid-based acceleration

media, such as crystals or nanostructures could oﬀer a

solution. The density of charge carriers (conduction elec-

trons) in solids is four or ﬁve orders of magnitude higher

than those in a gaseous plasma, thus oﬀering the pos-

sibility to obtain ultra-high gradients on the order of

E0∼1–10 TV/m, if the same linear theory is assumed.

Solid-state wakeﬁeld acceleration using crystals was

proposed in the 1980s and 1990s by T. Tajima and oth-

ers [10, 11] as an alternative particle acceleration tech-

nique to sustain TV/m acceleration gradients. In the

original Tajima’s conceptual scheme, a metallic crystal

is excited by a laser (laser driven), generating a longi-

tudinal electric wakeﬁeld which can be used as an ac-

celerating structure. Then, if a witness beam of charged

particles is injected into the crystal with an optimal injec-

tion angle for channelling and with the right phase with

respect to the wakeﬁeld, the channelled particles can ex-

arXiv:2210.12830v3 [physics.plasm-ph] 8 Mar 2023

perience acceleration. To reach accelerating gradients on

the order of 1–10 TV/m, crystals must be excited by ul-

trashort X-ray laser pulses within a power range of TW–

PW, which makes the practical realisation of the concept

very challenging. It has only recently become a realistic

possibility since the invention of the single-cycled optical

laser compression technique by G. Mourou et al. [12]. Al-

though simulation studies of X-ray wakeﬁeld acceleration

have been performed [13], it has not been experimentally

demonstrated yet. Alternatively, channelled particles in

crystals can also be accelerated by means of electric wake-

ﬁelds excited by ultrashort, relativistic electron bunches

(beam driven). In this case the energy losses of a driving

bunch can be transformed into the acceleration energy of

a witness bunch.

If natural crystals (e.g. silicon) are used for the solid-

state wakeﬁeld acceleration, the beam intensity accep-

tance is signiﬁcantly limited by the angstrom-size chan-

nels. In addition, such small size channels increase the

dechannelling rate and make channels physically vulner-

able to high energy interactions, thus increasing the dam-

age probability by high power beams.

Over the past decade there have been great advances

in nanofabrication techniques [14–16] that could oﬀer an

excellent way to overcome many of the limitations of nat-

ural crystals. Metallic nanostructures and metamaterials

[17, 18] may oﬀer suitable ultra-dense plasma media for

wakeﬁeld acceleration or charged particle beam manip-

ulation, i.e. channelling, bending, wiggling, etc. This

also includes the possibility of investigating new paths

towards ultra-compact X-ray sources [17].

In this context, the use of nanotube structures for gen-

erating ultra-high gradients is attracting attention [19–

21]. For instance, carbon nanotube (CNT) based struc-

tures can help to relax the constraints to more realistic

regimes with respect to natural crystals. CNTs present

the following advantages with respect to natural crystals:

(i) larger degree of dimensional ﬂexibility and thermo-

mechanical strength; (ii) transverse acceptances of the

order of up to 100 nm, i.e. three orders of magnitude

higher than a typical silicon channel; (iii) lower dechan-

nelling rate; (iv) less disruptive eﬀects such as ﬁlamenta-

tion and collisions. Therefore, CNTs are a robust candi-

date for solid-state wakeﬁeld acceleration.

Wakeﬁelds in crystals or nanostructures can be induced

by means of the excitation of high-frequency collective

motion of conduction electrons through the crystalline

ionic lattice. This collective oscillation of conduction

electrons in metals, excited by external electromagnetic

ﬁelds is commonly referred to as plasmon [22]. For in-

stance, the excitation of surface plasmonic modes [23–

26], driven either by charged particle beam [27–30] or

by laser [13, 31], could be used as a collective mecha-

nism to generate high acceleration gradients in metallic

nanotubes. To be eﬀective, the driver dimensions should

match the spatial (∼nm) and time (sub-femtoseconds)

scales of the excited plasmonic oscillations. Wakeﬁeld-

driving sources working on these scales can be experimen-

tally accessible nowadays or in the near future. For in-

stance, attosecond X-ray lasers are possible thanks to the

pulse compression technique invented by Donna Strick-

land and Gerard Mourou [32]. In the case of beam-driven

wakeﬁelds, future upgrades of the experimental facility

FACET-II at SLAC [33, 34] might allow the access to

”quasi-solid” and ultra-short electron beams, with densi-

ties up to ∼1024 cm−3and sub-micrometer bunch length

scale. Recent studies have reported that ultra-short and

high-density electron beams could lead to a nonlinear

plasmonic regime, generating acceleration gradients be-

yond TV/m in micro- and nano-tubes. This is also known

as the crunch-in regime [35–37] and could be a potential

step towards the realisation of compact PeV colliders [38].

In this article we study the feasibility of generating

ultra-high acceleration gradients in nanostructures based

on CNTs. In addition, we show that, under proper condi-

tions, by adopting an eﬀective density, existing analytical

estimates, originally derived for wakeﬁelds driven in ho-

mogeneous plasmas in the linear regime, can be used to

describe the wakeﬁelds excited in such nanostructures. In

particular, for multiple target conﬁgurations (single hol-

low plasma channel/CNT, CNT arrays) the amplitude of

the longitudinal beam-driven wakeﬁeld is evaluated and

compared to – and shown to be in agreement with – ana-

lytical estimates for the amplitude of longitudinal beam-

driven wakeﬁelds, obtained by using an eﬀective-density

approach, as described along this work. Moreover, exist-

ing results [21] for CNT beam-driven wakeﬁelds, previ-

ously simulated using 2D Cartesian geometry have been

revisited with the code FBPIC (Fourier–Bessel Particle-

In-Cell) [39], using a 2D axisymmetric geometry. Regard-

ing the structure of this work, in Section II the simulation

model is described. Section III investigates the case of

beam-driven wakeﬁelds in single tubes, and the role of

key parameters, such as tube wall thickness and aper-

ture, is systematically studied. Section IV focuses on the

case of an aligned multichannel structure, representing a

CNT array. Highly-ordered nanotube bundles would al-

low to fabric macroscopic samples with transverse width

on the order of centimetres, thus being able to cover all

the transverse cross sections for beam waist-sizes on the

order of tens and hundreds of micrometres. In this case,

where we deal with an inhomogeneous plasma structure,

the feasibility of using an eﬀective-density approach is

investigated. Finally, some conclusions are drawn in Sec-

tion V.

II. SIMULATION MODEL

Hollow plasma channels (HPC), consisting of cylin-

drical shells populated by a uniform, pre-ionized cold

plasma of two species (ions and electrons), are adopted

here as a ﬁrst-order approximation to describe a CNT, or

a larger structure made of CNT bundles, as shown in Fig.

1. The carbon ions are simulated as cold ions with mass

mi= 12 mp, where mpis the proton mass, and charge

qi=Z e =e, where Zis the atomic number and ethe

fundamental charge. Because of the single-level ioniza-

tion, for a given ion density ni, the electron density ne,

initially cold as well, will have the same value (ne=ni).

The choice of Z= 1 was made aiming to obtain conserva-

tive, lower bound estimates for the wakeﬁeld amplitudes

to be driven in carbon-based solid state plasmas.

Regarding the target geometry, two distinct conﬁgura-

tions are investigated. First, “large” hollow plasma chan-

nels (HPC), with µm-wide apertures are used as targets.

Such structures could be built with CNT bundles, as

shown in Fig. 1, with much larger dimensions than those

of a nanostructured CNT, and thus capable of channeling

∼µm electron beams. In the second conﬁguration, mul-

tiple concentric HPCs, with thicknesses (and gaps) of a

few nm, are adopted to describe CNT array targets. For

both cases, the walls are modelled as uniform plasmas,

with an average, eﬀective density, which is presented and

discussed along this document.

Although this collisionless ﬂuid model does not take

into account the solid state properties emerging from the

ionic lattice, such as, for example, the presence of polari-

tons, previous studies have shown that the wakeﬁeld for-

mation and electron acceleration processes in crystalline

structures are only slightly aﬀected by the ionic lattice

force [40]. Therefore, neglecting the ionic eﬀects at a ﬁrst

approximation might be justiﬁed, and – if this is the case

– conventional particle-in-cell (PIC) codes might be an

useful tool to investigate ultra high-gradient acceleration,

as well as plasmon modelling in solids [13, 36, 41]. As it

has been already shown [41], the PIC method can be very

suitable to model solid-sate based plasmons, since it self-

consistently solves the ﬁelds and the motion of a large

assembly of charged particles for the required time (∼

sub-fs) and spatial (∼nm) scales.

Due to the high computational cost of 3D PIC simu-

lations, the 2D Cartesian geometry is often adopted. In

such geometry, CNT walls are modelled as ﬂat plasma

sheets, with ﬁnite thickness and length, and inﬁnite

width. However, this geometry is known to aﬀect the

spatial derivatives of the ﬁelds [42] if applied to describe a

non-slab-like system. Given the close-to-cylindrical sym-

metry of the physical system under consideration, a PIC

code with a spectral solver can provide an accurate 3D

description of the system, at a computational cost similar

to the cost of performing 2D Cartesian PIC simulations

[43].

In this work, the Fourier-Bessel Particle-in-Cell

(FBPIC) code [39] is adopted to perform the simulations

using the cylindrical CNT hollow plasma channel model.

Although particles in FBPIC have 3D Cartesian coor-

dinates, its solver uses a set of 2D radial grids, each of

them representing an azimuthal mode m(m= 0,1, . . . ).

While the ﬁrst mode (m= 0) describes axisymmetric

ﬁelds, higher-order modes can be added to model

departures from the cylindrical symmetry. For example,

a linearly polarised laser can be computed by adding

the mode m= 1. An interesting feature of the spectral

solver implementation in FBPIC is the mitigation of

spurious numerical dispersion, including the zero-order

numerical Cherenkov eﬀect [44].

Compact high-energy electron beams are often re-

ported in literature with dimensions ranging from a frac-

tion to a few micrometers [34, 45]. Therefore, in this

work, beams with near-µm RMS sizes are used as drivers

to excite the intense wakeﬁelds in hollow plasma chan-

nels, which are under investigation in this section. The

beam driver is assumed to have a bi-Gaussian density

proﬁle,

nb(ξ, r)/n0= (nb/n0)e−ξ2/(2σ2

ξ)e−r2/(2σ2

r),(2)

where ξ≡z−ct is the beam co-moving coordinate, cis

the speed of light in vacuum, nb≡(Q/e)/[(2π)3/2σξσ2

is the peak beam-density, n0is the initial plasma electron

density, Qis the beam charge and σξ,σrare the beam

longitudinal and radial RMS sizes, respectively. The

beam has initial kinetic energy Ek0, and energy spread

δEk0/Ek0= 1%. In addition, Ek0is chosen to ensure

that the corresponding relativistic factor γsatisﬁes the

condition γ1, in order to increase the beam stiﬀness.

III. SINGLE TUBE IN 2D AXISYMMETRIC

GEOMETRY

As a ﬁrst approach, a single HPC is adopted as the

medium for the beam-driven wakeﬁeld excitation. Fig-

ure 1 depicts a schematic of the system, in which an

electron beam (driving source) is injected into a hollow

plasma channel. The plasma is conﬁned in the channel

wall, assumed to be made up of CNT bundles. In princi-

ple, modern techniques allow for the fabrication of macro-

scopic materials based on aligned single and multi-wall

CNT bundles or CNT forest ﬁlms [46–48]. In this nanos-

tructured materials, the density proﬁle of the plasma

can be controlled by the packaging conﬁguration of the

CNTs. By varying parameters from this structure, such

as internal radius, wall thickness, and plasma density,

it is possible to verify how the wakeﬁeld intensity is af-

fected. In order to accommodate a near-µm beam, the

hollow plasma channel also has a micrometer-scale.

Typical electron densities (ne) in solid-state plasmas lie

within the range of 1019 cm−3≤ne≤1024 cm−3[49, 50].

Aiming to maintain conservative estimates for the am-

plitude of the wakeﬁelds to be excited in these materials,

the lower limit of this range is chosen as the initial den-

sity, n0= 1019 cm−3, for both electrons and ions. In

other words, ne=ni=n0, where niis the ion den-

sity. Although this density is much lower than that of

a CNT wall (∼1023 cm−3), it could represent electrons

in gaps and hollow spaces of (partially ionized) targets

made with CNT arrays or bundles. Moreover, for the

chosen beam and plasma parameters, the wakeﬁelds are

excited approximately in the linear regime. Hence, the

obtained results can be scaled up to higher densities. If

FIG. 1: Top: Schematic model for beam-driven

wakeﬁeld simulation using a hollow cylinder of

solid-state plasma conﬁned in a wall of thickness

w=rout −rin and length Lp. Bottom: the cylinder wall

could be made of CNT bundles (not to scale).

the plasma electron and peak beam densities (neand

nb, respectively) are increased accordingly, then the ra-

tio Ez/E0, where Ezis the longitudinal wakeﬁeld, should

remain constant. In this case, the wakeﬁeld amplitude for

a higher density can be estimated by multiplying the ra-

tio Ez/E0obtained from the lower density simulation by

the new value of E0, calculated for the higher density.

For a density of ne= 1019 cm−3, a plasma wave-

length of λp= 10.6µm is obtained. From now on,

this quantity (λp) is adopted as the characteristic length

scale to deﬁne the HPC and beam dimensions as fol-

lows. The HPC has a length Lp= 10 λp, internal ra-

dius rin = 0.1λp, external radius rout = 0.5λp, and

wall thickness w=rout −rin = 0.4λp. Regarding the

beam, both the longitudinal and radial RMS sizes are

σξ=σr= 0.1λp. For such dimensions, a charge of

Q= 33 pC is chosen, providing a normalized peak beam-

density of nb/n0= 1.1, i.e., right after the transition

from an overdense to a underdense propagation in the

plasma, in order to ensure that the beam will experience

linear focusing forces [51, 52]. The initial beam-energy is

Ek0= 1 GeV, with an energy spread of δEk0/Ek0= 1%,

and the transverse normalized beam emittance is null.

Such parameters were chosen to produce a stiﬀ beam,

able to drive a stable wakeﬁeld along its propagation.

Since the amplitude of the wakeﬁeld is evaluated and

compared in multiple situations along the investigation,

this is a relevant matter.

Figure 2 shows PIC simulation results for the afore-

mentioned parameters, taken at a propagation distance

of z= 53 µm, corresponding to Lp/2=5λp. Despite

the betatron motion of individual beam particles, due

to its high initial energy, the beam density proﬁle re-

mains mostly unchanged during the propagation. Fig-

ure 2(a) depicts the beam (transparent-blue-green-yellow

color scale) and plasma-electron (purple color scale) den-

sities, in units of the initial plasma density n0, both be-

ing saturated (capped) in order to enhance the visualiza-

tion of how these densities are mutually aﬀected. At this

propagation distance, the beam density has been slightly

modulated, being lower within the CNT walls due to the

beam-wall interference. Regarding the plasma electrons,

after being radially expelled by the beam, they experi-

ence a strong restoring force due to the carbon ions (not

shown in this ﬁgure), which remain mostly undisturbed.

As a consequence, these electrons are tightly focused, cre-

ating on-axis density spikes 20 to 35 times higher than

the initial plasma density n0, This behaviour has been

described by Sahai et al. as the crunch-in regime [35–

37]. The periodic transverse motion of plasma electrons,

caused by the competition between the Coulomb repul-

sion and the restoring force due to the ions, creates an

intense wakeﬁeld, which could be used as an accelerating

structure for a witness beam. Figure 2(b) shows the ac-

celerating (negative) phase of the longitudinal wakeﬁeld

Ez(ξ, r) peaking at |Emax

z| ' 105 GV/m. This value

is approximately one third of the cold non-relativistic

wave breaking ﬁeld E0calculated for the chosen den-

sity. The transverse wakeﬁeld, W⊥(ξ, r) = Er−cBθ,

with Erthe radial component of the electric ﬁeld and Bθ

the azimuthal component of the magnetic ﬁeld, is shown

in Fig. 2(c). While appreciable amplitudes can be seen

within the CNT wall for both focusing and defocusing

phases of the transverse wakeﬁeld, inside the CNT (i.e.,

for r < rin) there are regions in which the transverse

wakeﬁeld is approximately null. Due to this interesting

feature, the use of hollow plasma channels to mitigate

beam quality degradation caused by transverse eﬀects is

an active ﬁeld of research [53–58].

A. Tube aperture and wall thickness

Parameter scans might be helpful to determine the

optimal system dimensions or aspect ratios to achieve

as high amplitude wakeﬁeld as possible. In this sense,

while maintaining the beam parameters and plasma den-

sity ﬁxed, we have investigated the dependence of the

longitudinal wakeﬁeld on both, tube aperture and wall

thickness. In the ﬁrst case, the inner tube radius rin has

been varied. For instance, Fig. 3, plotted for a ﬁxed wall

thickness w= 0.2λp, illustrates the tube electron density

and beam density for the extreme (smallest and largest)

investigated values of inner radius rin, for a propagation

distance of approximately z= 70 µm. While in Fig. 3(a),

plotted for rin = 0.05 λp< σr(= 0.1λp), the beam trans-

versely overlaps the tube wall, in Fig. 3(b), plotted for

rin = 0.50 λp> σr, the tube inner radius is larger than

the transverse beam size. The beam density color scale

was saturated (capped) at 75% of its maximum value,

in order to improve the visualisation of how this quan-

tity is aﬀected by the interference with the tube wall.

This saturation has been adopted for all beam density

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Exploringultra-high-intensitywakeeldsincarbonnanotubearrays:aneectiveplasma-densityapproachA.BonattoGraduatePrograminInformationTechnologyandHealthcareManagement,andtheBeamPhysicsGroup,FederalUniversityofHealthSciencesofPortoAlegre,PortoAlegre,RS,90050-170,BrazilG.XiaandO.ApsimonDepartmentofPhysi...

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Exploring ultra-high-intensity wakeelds in carbon nanotube arrays an eective plasma-density approach A. Bonatto

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