# Excitation of wakefields in carbon nanotubes: a hydrodynamic model approach

P. Martín-Luna<EMAIL_ADDRESS>Instituto de Física Corpuscular (IFIC),
Universitat de València - Consejo Superior de Investigaciones Científicas,
46980 Paterna, Spain  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  C.
Bontoiu  Department of Physics, The University of Liverpool, Liverpool L69
3BX, United Kingdom,
The Cockcroft Institute, Sci-Tech Daresbury, Warrington WA4 4AD, United
Kingdom  G. Xia  Department of Physics and Astronomy, The University of
Manchester, Manchester M13 9PL, United Kingdom
The Cockcroft Institute, Sci-Tech Daresbury, Warrington WA4 4AD, United
Kingdom  J. Resta-López<EMAIL_ADDRESS>Instituto de Ciencia de los
Materiales (ICMUV), Universidad de Valencia, 46071 Valencia, Spain

###### Abstract

The interactions of charged particles with carbon nanotubes may excite
electromagnetic modes in the electron gas produced in the cylindrical graphene
shell constituting the nanotube wall. This wake effect has recently been
proposed as a potential novel method of short-wavelength high-gradient
particle acceleration. In this work, the excitation of these wakefields is
studied by means of the linearized hydrodynamic model. In this model, the
electronic excitations on the nanotube surface are described treating the
electron gas as a 2D plasma with additional contributions to the fluid
momentum equation from specific solid-state properties of the gas. General
expressions are derived for the excited longitudinal and transverse
wakefields. Numerical results are obtained for a charged particle moving
within a carbon nanotube, paraxially to its axis, showing how the wakefield is
affected by parameters such as the particle velocity and its radial position,
the nanotube radius, and a friction factor, which can be used as a
phenomenological parameter to describe effects from the ionic lattice.
Assuming a particle driver propagating on axis at a given velocity, optimal
parameters were obtained to maximize the longitudinal wakefield amplitude.

##  I Introduction

Carbon nanotubes (CNTs) were discovered by S. Iijima in 1991 [1] and they can
be thought of as a sheet of graphene (a hexagonal lattice of carbon) rolled
into a cylinder. CNTs can exhibit metallic or semiconductor properties
depending on their rolling pattern (i.e. on their radius and geometrical
angle). Thus, as a consequence of their unique thermo-mechanical and
electronic properties and dimensional flexibility, CNTs have been widely
studied in both theoretical and experimental aspects. As a hollow structure,
CNTs may be used for transporting and focusing charged particles similar to
crystal channeling. In particular, experimental results on 2 MeV
$\text{He}^{+}$ ions [2] and 300 keV electrons [3] channeling in CNTs have
been reported.

On the other hand, solid-state wakefield acceleration using crystals was
proposed in the 1980s and 1990s by T. Tajima and others [4, 5, 6] as an
alternative particle acceleration technique to sustain TV/m acceleration
gradients. In the original Tajima’s conceptual scheme [4], a longitudinal
electric wakefield is excited by a laser (laser driven) in the crystal so that
a properly injected witness beam may be accelerated. Similarly, the ultrashort
charged particle bunches (beam driven) can excite electric wakefields so that
the energy loss of the driving bunch can be transformed into an increment of
energy of a witness bunch. However, the angstrom-size channels of natural
crystals pose a limitation for the beam intensity acceptance and the
dechanneling rate. In this context, CNTs can obtain wider channels in two
dimensions and longer dechanneling lengths [7, 8], which together with their
remarkable electronic properties, larger degree of dimensional flexibility and
thermo-mechanical strength, make them a robust candidate for TeV/m
acceleration. Consequently, carbon nanostructures (CNTs or even graphene
layers) are currently being widely studied for wakefield acceleration [9, 10].

Wakefields in CNTs are excited through the collective oscillation of electrons
on the nanotube’s surface, often referred to as plasmons. This excitation
arises from the interaction between the driving bunch and the CNT, leading to
the generation of these wakefields. The electronic excitations on the single-
wall CNT surface produced by the interaction with charged particles have been
theoretically studied using a dielectric theory [11, 12, 13], a hydrodynamic
model [14, 15, 16], a two-fluid model [17, 18], a quantum hydrodynamic model
[19] and a kinetic model [20, 21, 22, 23]. However, while these articles
mostly explore properties such as the energy loss and stopping power or, at
most, evaluate the induced surface electron density and/or induced potential
[16, 22, 23], they do not address the induced longitudinal and transverse
wakefields, which could provide acceleration and focusing, respectively, for a
witness charge.

Thus, this article is motivated by the need to study the wakefields excited by
charged particles moving paraxially inside CNTs. Additionally, the article
aims to optimize the parameters of the CNT in order to obtain the highest
longitudinal wakefield for particle acceleration purposes.

This work is organized as follows. In Sec. II the general expressions are
derived for the longitudinal and transverse wakefields excited by the
interaction of a charged particle with a CNT in the realm of the hydrodynamic
model. This model has been chosen because of its simplicity and its good
agreement with the dielectric formalism in random-phase approximation [15]. In
Sec. III, after investigating the influence of different model parameters in
terms of dispersion relations, the CNT parameters are optimized to achieve the
highest longitudinal wakefield for a given driving velocity. Finally, the main
conclusions of this study are presented in Sec. IV.

##  II Linearized hydrodynamic theory

In this work, a linearized hydrodynamic theory [14, 15, 18] is adopted, in
particular a model which includes single-electron based excitations on
nanotube surfaces and was described by Wang and Miškovic in [15], although we
will use the SI units instead of atomic units. In this theory, a single-wall
CNT is modelled as an infinitesimally thin and infinitely long cylindrical
shell with a radius $a$. The delocalized electrons of the carbon ions are
considered as a two-dimensional free-electron gas that is confined over the
cylindrical surface of the CNT with a uniform surface density $n_{0}$. It is
considered a driving point-like charge $Q$ travelling parallel to the $z$-axis
inside the tube with a constant velocity $v$ (see Fig. 1). Consequently, its
position as a function of time $t$ is
$\mathbf{r}_{0}(t)=\left(r_{0},\varphi_{0},vt\right)$ in cylindrical
coordinates. As a consequence of the presence of the driving charge $Q$, the
homogeneous electron gas will be perturbed and can be modelled as a charged
fluid with a velocity field $\mathbf{u}(\mathbf{r}_{a},t)$ and a perturbed
density per unit area $n_{1}(\mathbf{r}_{a},t)$, where
$\mathbf{r}_{a}=(a,\varphi,z)$ are the coordinates of a point at the
cylindrical surface of the tube. As the electron gas is confined to the
cylindrical surface, the normal component to the surface of the tube of the
velocity field $\mathbf{u}$ is zero.

Figure 1: Scheme of the considered charge $Q$ travelling parallel to the
$z$-axis inside a tube.

In the linearized hydrodynamic model, the electronic excitations on the tube
wall can be described by three differential equations: (i) the continuity
equation

$\frac{\partial n_{1}\left(\mathbf{r}_{a},t\right)}{\partial
t}+n_{0}\nabla_{\|}\cdot\mathbf{u}\left(\mathbf{r}_{a},t\right)=0,$ (1)

(ii) the Poisson’s equation

$\nabla^{2}\Phi(\mathbf{r},t)=\frac{1}{\varepsilon_{0}}\left[en_{1}\left(\mathbf{r}_{a},t\right)\delta(r-a)-Q\delta\left(\mathbf{r}-\mathbf{r}_{0}\right)\right],$
(2)

and (iii) the momentum-balance equation

$\begin{split}\frac{\partial\mathbf{u}\left(\mathbf{r}_{a},t\right)}{\partial
t}=&\frac{e}{m_{e}}\nabla_{\|}\Phi\left(\mathbf{r}_{a},t\right)-\frac{\alpha}{n_{0}}\nabla_{\|}n_{1}\left(\mathbf{r}_{a},t\right)+\\\
&\frac{\beta}{n_{0}}\nabla_{\|}\left[\nabla_{\|}^{2}n_{1}\left(\mathbf{r}_{a},t\right)\right]-\gamma\mathbf{u}\left(\mathbf{r}_{a},t\right).\end{split}$
(3)

In these equations, $\mathbf{r}=(r,\varphi,z)$ is the position vector,
$\nabla=\hat{\mathbf{r}}\frac{\partial}{\partial
r}+\hat{\bm{\varphi}}\frac{1}{r}\frac{\partial}{\partial\varphi}+\hat{\mathbf{z}}\frac{\partial}{\partial
z}$,
$\nabla_{\|}=\hat{\bm{\varphi}}\frac{1}{a}\frac{\partial}{\partial\varphi}+\hat{\mathbf{z}}\frac{\partial}{\partial
z}$ differentiates only tangentially to the tube surface, $\Phi$ the electric
scalar potential, $e$ the elementary charge, $m_{e}$ the rest mass of the
electron, $\varepsilon_{0}$ the vacuum electric permittivity and $\delta$ the
Dirac delta. Equation (3) shows the sum of four different contributions. The
first term in the right-hand side is the force on electrons on the nanotube
surface due to the tangential component of the electric field generated by the
driving charge $Q$ and the consequent perturbed density $n_{1}$. The second
and third terms are related to the parts of the internal interaction force in
the electron gas. In particular, the second term takes into account the
possible coupling with acoustic modes defining the parameter
$\alpha=v^{2}_{F}/2$ (in which $v_{F}=\hbar(2\pi n_{0})^{1/2}/m_{e}$ is the
Fermi velocity of the two-dimensional electron gas; $\hbar$ is the reduced
Planck constant), and the third term is a quantum correction that arises from
the functional derivative of the Von Weizsacker gradient correction in the
equilibrium kinetic energy of the electron fluid [18] and describes single-
electron excitations in the electron gas, where the parameter
$\beta=\frac{1}{4}(\frac{\hbar}{m_{e}})^{2}$ has been defined. The last term
is introduced to satisfy the non-conservation of the system and represents a
frictional force on electrons due to scattering with the ionic-lattice
charges, where $\gamma$ is the friction parameter. The friction parameter may
be also used as a phenomenological parameter to take into account the
broadening of the plasmon resonance in the excitation spectra of different
materials [11].

Taking into account that the electric potential vanishes at
$r\rightarrow\infty$ and is finite at the origin $r=0$, the potential can be
expanded in terms of the modified Bessel functions $I_{m}(x)$ and $K_{m}(x)$
of integer order $m$ (i.e. a Fourier-Bessel expansion). The total potential
inside the nanotube ($r<a$) can be calculated as
$\Phi_{in}=\Phi_{0}+\Phi_{ind}$ with the Coulomb potential $\Phi_{0}$ due to
the driving charge and the induced potential $\Phi_{ind}$ due to the
perturbation of the electron fluid on the CNT surface. The total potential
outside the nanotube ($r>a$) will be denoted as $\Phi_{out}$. Thus, the
Fourier-Bessel expansion of these three components is

$\begin{split}\Phi_{0}(r,\varphi,\zeta)=&\frac{1}{4\pi\varepsilon_{0}}\frac{Q}{\left\|\mathbf{r}-\mathbf{r}_{\mathbf{0}}\right\|}=\frac{Q}{4\pi^{2}\varepsilon_{0}}\sum_{m=-\infty}^{+\infty}\int_{-\infty}^{+\infty}\mathrm{d}k\\\
&e^{ik\zeta+im\left(\varphi-\varphi_{0}\right)}I_{m}\left(|k|r_{\text{min}}\right)K_{m}\left(|k|r_{\text{max}}\right),\end{split}$
(4)

$\begin{split}\Phi_{ind}(r,\varphi,\zeta)=&\frac{Q}{4\pi^{2}\varepsilon_{0}}\sum_{m=-\infty}^{+\infty}\int_{-\infty}^{+\infty}\mathrm{d}k\\\
&e^{ik\zeta+im\left(\varphi-\varphi_{0}\right)}I_{m}\left(|k|r_{0}\right)I_{m}(|k|r)A_{m}(k),\end{split}$
(5)

$\begin{split}\Phi_{out}(r,\varphi,\zeta)=&\frac{Q}{4\pi^{2}\varepsilon_{0}}\sum_{m=-\infty}^{+\infty}\int_{-\infty}^{+\infty}\mathrm{d}k\\\
&e^{ik\zeta+im\left(\varphi-\varphi_{0}\right)}I_{m}\left(|k|r_{0}\right)K_{m}(|k|r)B_{m}(k),\end{split}$
(6)

where $k$ is the wavenumber, a comoving coordinate $\zeta=z-vt$ has been
defined and $r_{\text{min}}=\text{min}(r,r_{0})$,
$r_{\text{max}}=\text{max}(r,r_{0})$. The unknown coefficients $A_{m}(k)$ and
$B_{m}(k)$ can be calculated if the following boundary conditions are imposed:
(i) the continuity of the electric potential at the nanotube surface

$\left.\Phi_{in}(r,\varphi,\zeta)\right|_{r=a}=\left.\Phi_{out}(r,\varphi,\zeta)\right|_{r=a},$
(7)

and (ii) the discontinuity of the radial component of the electric field due
to perturbed density $n_{1}$ of the electron fluid

$\left.\frac{\partial\Phi_{out}(r,\varphi,\zeta)}{\partial
r}\right|_{r=a}-\left.\frac{\partial\Phi_{in}(r,\varphi,\zeta)}{\partial
r}\right|_{r=a}=\frac{en_{1}(a,\varphi,\zeta)}{\varepsilon_{0}}.$ (8)

Thus, the coefficients $A_{m}(k)$ and $B_{m}(k)$ are given by the following
non-dimensional functions:

$A_{m}(k)=\frac{\Omega_{p}^{2}a^{2}\left(k^{2}+m^{2}/a^{2}\right)K_{m}^{2}(|k|a)}{D_{m}(k)},$
(9)
$B_{m}(k)=\frac{D_{m}(k)+\Omega_{p}^{2}a^{2}\left(k^{2}+\frac{m^{2}}{a^{2}}\right)K_{m}(|k|a)I_{m}(|k|a)}{D_{m}(k)},$
(10)

where $\Omega_{p}=\sqrt{\frac{e^{2}n_{0}}{\varepsilon_{0}m_{e}a}}$ is the
plasma frequency,

$D_{m}(k)=kv(kv+i\gamma)-\omega_{m}^{2}(k),$ (11)

and

$\begin{split}\omega_{m}^{2}(k)=&\alpha\left(k^{2}+\frac{m^{2}}{a^{2}}\right)+\beta\left(k^{2}+\frac{m^{2}}{a^{2}}\right)^{2}+\\\
&\Omega_{p}^{2}a^{2}\left(k^{2}+\frac{m^{2}}{a^{2}}\right)K_{m}(|k|a)I_{m}(|k|a).\end{split}$
(12)

Consequently, the resonant excitations occur when $kv=\omega_{m}(k)$ if the
damping or friction factor $\gamma$ vanishes.

On the other hand, the longitudinal and transverse electric wakefields inside
the tube are, respectively,

$\displaystyle W_{z}(r,\varphi,\zeta)$
$\displaystyle=-\frac{\partial\Phi_{in}}{\partial
z}=\frac{Q}{4\pi^{2}\varepsilon_{0}}\sum_{m=-\infty}^{+\infty}\int_{-\infty}^{+\infty}\mathrm{d}k\,ke^{im\left(\varphi-\varphi_{0}\right)}$
(13)
$\displaystyle\times\left(I_{m}\left(|k|r_{\text{min}}\right)K_{m}\left(|k|r_{\text{max}}\right)\sin(k\zeta)\right.$
$\displaystyle+I_{m}\left(|k|r_{0}\right)I_{m}(|k|r)\left[\text{Re}\left[A_{m}(k)\right]\sin(k\zeta)\right.$
$\displaystyle\left.\left.+\text{Im}\left[A_{m}(k)\right]\cos(k\zeta)\right]\right)=W_{z0}+W_{z1}+W_{z2},$

$\displaystyle W_{r}(r,\varphi,\zeta)$
$\displaystyle=-\frac{\partial\Phi_{in}}{\partial
r}=-\frac{\partial\Phi_{0}}{\partial
r}-\frac{Q}{4\pi^{2}\varepsilon_{0}}\sum_{m=-\infty}^{+\infty}\int_{-\infty}^{+\infty}\mathrm{d}k$
(14)
$\displaystyle|k|e^{im\left(\varphi-\varphi_{0}\right)}I_{m}\left(|k|r_{0}\right)I_{m}^{\prime}(|k|r)$
$\displaystyle\times\left[\text{Re}\left[A_{m}(k)\right]\cos(k\zeta)\right.-\text{Im}\left[A_{m}(k)\right]\sin(k\zeta)]$
$\displaystyle=W_{r0}+W_{r1}+W_{r2},$

where $I^{\prime}_{m}(x)=dI_{m}(x)/dx$ and the properties
$\text{Re}\left[A_{m}(k)\right]=\text{Re}\left[A_{m}(-k)\right]$ and
$\text{Im}\left[A_{m}(k)\right]=-\text{Im}\left[A_{m}(-k)\right]$ were
considered; Re and Im denote the real and imaginary part, respectively. The
previous integrals have been separated in three different terms: the first
terms $W_{z0}$ and $W_{r0}$ come from the Coulomb potential and the other
terms $W_{z1},W_{z2}$ and $W_{r1},W_{r2}$ from the induced potential. It is
important to note that the third terms $W_{z2}$ and $W_{r2}$ can be
analytically integrated if the damping factor vanishes ($\gamma\rightarrow
0^{+}$):

$\displaystyle
W_{z2}(r,\varphi,\zeta)=\frac{Q}{4\pi^{2}\varepsilon_{0}}\sum_{m=-\infty}^{+\infty}\int_{-\infty}^{+\infty}\mathrm{d}k\,ke^{im\left(\varphi-\varphi_{0}\right)}$
(15) $\displaystyle
I_{m}\left(|k|r_{0}\right)I_{m}(|k|r)\text{Im}\left[A_{m}(k)\right]\cos(k\zeta)$
$\displaystyle=\frac{-Q}{2\pi\varepsilon_{0}}\sum_{m=-\infty}^{+\infty}e^{im\left(\varphi-\varphi_{0}\right)}k_{m}I_{m}\left(k_{m}r_{0}\right)I_{m}\left(k_{m}r\right)$
$\displaystyle\Omega_{p}^{2}a^{2}\left(k_{m}^{2}+\frac{m^{2}}{a^{2}}\right)K_{m}^{2}\left(k_{m}a\right)\left|\frac{\partial
Z_{m}}{\partial k}\right|_{k=k_{m}}^{-1}\cos\left(k_{m}\zeta\right),$

$\displaystyle
W_{r2}(r,\varphi,\zeta)=\frac{Q}{4\pi^{2}\varepsilon_{0}}\sum_{m=-\infty}^{+\infty}\int_{-\infty}^{+\infty}\mathrm{d}k\,|k|e^{im\left(\varphi-\varphi_{0}\right)}$
(16) $\displaystyle
I_{m}\left(|k|r_{0}\right)I_{m}^{\prime}(|k|r)\text{Im}\left[A_{m}(k)\right]\sin(k\zeta)$
$\displaystyle=\frac{-Q}{2\pi\varepsilon_{0}}\sum_{m=-\infty}^{+\infty}e^{im\left(\varphi-\varphi_{0}\right)}k_{m}I_{m}\left(k_{m}r_{0}\right)I_{m}^{\prime}\left(k_{m}r\right)$
$\displaystyle\Omega_{p}^{2}a^{2}\left(k_{m}^{2}+\frac{m^{2}}{a^{2}}\right)K_{m}^{2}\left(k_{m}a\right)\left|\frac{\partial
Z_{m}}{\partial k}\right|_{k=k_{m}}^{-1}\sin\left(k_{m}\zeta\right),$

where it has been defined the quantity
$Z_{m}(k)=\text{Re}[D_{m}(k)]=(kv)^{2}-\omega_{m}^{2}(k)$ and $k_{m}$ are the
(positive) roots of $Z_{m}(k)$, i.e. the condition of the plasma resonance
$k_{m}v=\omega_{m}(k_{m})$.

##  III Results and discussion

As it can be deduced from Eqs. (15)-(16), the roots $k_{m}$ given by the
plasma resonance are essential to describe the behaviour of the wakefields.
For this reason, this section begins with a detailed analysis of the
dispersion relation.

###  III.1 Dispersion relation

In the following calculations, unless otherwise indicated, it is assumed that
the surface electron density of a single-wall carbon nanotube can be
approximated by the electron-gas density of a graphite sheet:
$n_{0}=1.53\times 10^{20}$ m-2 [24, 15]. Figure 2 shows the dispersion curves
$\omega_{m}(k)$ for the first modes at different CNT radii. If the radius is
too small, the first mode $m=0$ does not satisfy the resonance condition for
high velocities, while modes with $m>0$ have a solution $k_{m}$ for those
velocities, as seen in Fig. 2(a) at $a=1$ nm. If the CNT radius increases, the
resonance condition can be satisfied for a wider range of high velocities, as
it is depicted in Fig. 2(b) for $a=100$ nm. Furthermore, it can be seen that
the modes $\omega_{m}(k)$ converge for a sufficiently large value of the
wavenumber $k$. Therefore, if the resonance condition $k_{m}$ is sufficiently
large, then all modes will have a similar value of $k_{m}$. Note that a large
value of $k_{m}$ indicates that the associated wavelength of the wakefield
($\lambda_{m}=2\pi/k_{m}$) will be smaller. Moreover, if the surface density
$n_{0}$ increases, then the dispersion curves increase and the resonance
conditions are obtained for higher $k_{m}$.

Figure 2: Dispersion curves $\omega_{m}(k)$ for several angular-momentum modes
for (a) $a=1$ nm and (b) $a=100$ nm. The resonances $k_{m}$ are the
intersection of the $kv$ lines (plotted for $v=0.1c,v=0.3c$ and $v=0.5c$) with
the dispersion curves $\omega_{m}(k)$. Figure 3: Dispersion relation
$\omega_{0}^{2}(k)$ and contribution from the different three terms of Eq.
(12) for $a=1$ nm. Figure 4: Longitudinal wakefield contributions on axis
($r=0$) for a proton travelling on axis ($r_{0}=0$) at different velocity
values: (a) $v=0.05c$, (b) $v=0.10c$, (c) $v=0.15c$ and (d) $v=0.20c$. The CNT
radius is $a=1$ nm and the friction parameter is $\gamma=0.01\Omega_{p}$ in
cases (a) and (b), and $\gamma=0.0001\Omega_{p}$ in (c) and (d). Note that the
driving proton is at $\zeta=0$.

Furthermore, it is analysed the fundamental mode $\omega_{0}(k)$, since it is
the only mode which contributes for a particle travelling on axis (or if the
wakefield is calculated on axis). Figure 3 depicts the dependence of the
fundamental mode $\omega_{0}(k)$ of the resonant frequency on the wavenumber
$k$ for a CNT with a radius $a=1$ nm as well as the contribution of the three
addends in Eq. (12). It can be observed that the contribution from the
$\Omega_{p}$ term dominates for low values of $k$, whereas the contribution
from the $\beta$ term is dominant for larger values of $k$. Nevertheless,
there is an intermediate region, $10^{10}\,\text{m}^{-1}\lesssim k\lesssim
10^{11}\,\text{m}^{-1}$, where the three contributions exhibit considerable
similarity. The resonance condition $k_{m}$ for a given velocity $v$ is given
by the intercept of $(kv)^{2}$ (a parallel line to the $\alpha$ contribution)
with $\omega_{m}^{2}(k)$. Hence, the resonance condition cannot be satisfied
if $v<\alpha$, where $\alpha=v_{F}/\sqrt{2}$. In general, two resonances
$k_{m}$ can exist (as seen in Fig. 3 for $v=0.1c$) and then the contribution
from both resonances must be summed in Eqs. (15)-(16). However, the
contribution from the resonance with a larger value of $k_{m}$ is, in general,
totally negligible taking into account the exponential behaviour of the Bessel
function $K_{m}(x)=\sqrt{\frac{\pi}{2x}}e^{-x}$ for $x\rightarrow\infty$ and
the factor $\left|\frac{\partial Z_{m}}{\partial k}\right|_{k=k_{m}}^{-1}$
that decreases rapidly for high values of $k_{m}$. Therefore, by resonance
condition we practically mean the first root $k_{m}$ (the resonances shown in
Fig. 2). Consequently, for high velocities $v$, the resonance condition
$k_{m}$ is obtained (if it is satisfied) in the region where the $\Omega_{p}$
term dominates and contribution from the parameters $\alpha$ and $\beta$ is
negligible. Thus, the parameters $\alpha$ and $\beta$ only become important if
the resonance condition is obtained in the intermediate region.

###  III.2 Electric wakefields

In the following calculations, it is considered that the point-like charged
particle is a proton, i.e. $Q=e$. Figure 4 shows the three different
contributions to the longitudinal wakefield (cf. Eq. (13)) for different
driving velocities. It can be seen that the Coulomb term is only important
near the driving particle, whereas $W_{z1}$ and $W_{z2}$ are responsible for
the plasmonic excitations and are practically identical, except in the
proximity of the driving particle. Moreover, the wavelength of the wakefield
increases with the velocity $v$ as it was deduced from the dispersion
relation. Besides, the wakefield amplitude decreases as the proton speed
increases since we are approaching to the velocities that do not satisfy the
resonance condition. The friction parameter $\gamma$ produces an exponential
decay of the wakefield ($W_{z1}+W_{z2}$) with the distance behind the driving
charged particle. For this reason, the value of $\gamma$ has been diminished
in Fig. 4(c)-(d) in order to see the plasmonic excitations. Thus, when the
friction parameter $\gamma$ is very small, $W_{z2}$ follows a cosine pattern
and the longitudinal wakefield can be approximated by $W_{z}=W_{z0}+2W_{z2}$
($W_{z2}$ calculated using Eq. (15)), as shown in Fig. 5.

Figure 5: Comparison between Eq. (13) for small $\gamma$ and the approximation
using Eq. (15) to calculate $W_{z2}$ for $a=1$ nm and $v=0.1c$.

Figure 6 depicts an example of the longitudinal and transverse wakefield
inside a CNT. Here, to better appreciate the wakefield details, only the
contribution from the plasmonic excitation is shown, taking into account that
the Coulomb contribution is negligible except near the driving particle. In
Fig. 6 one remarkable thing is that plasmonic wakefields increase with the
radial distance $r$ because of the dependence on $I_{m}(|k|r)$ and
$I_{m}^{\prime}(|k|r)$ (both increasing functions with the argument) of
$W_{z}$ and $W_{r}$, respectively. A similar reasoning can be used to show
that the wakefields are higher if the particle travels off axis ($r_{0}\neq
0$), although in this case higher order modes ($|m|>0$) should be computed.
These effects become more important for lower velocities, since the value of
the resonance wavenumber increases.

Figure 6: (a) Longitudinal and (b) transverse components of the plasmonic
wakefields inside a CNT considering the following parameters: $a=1$ nm,
$v=0.05c$ and $\gamma=0.01\Omega_{p}$.

Furthermore, it is interesting to note that there is a phase offset of $\pi/2$
between the longitudinal and transverse wakefields. These results agree with
the Panofsky-Wenzel theorem [25] and are similar to what is observed for
wakefields excited in homogeneous plasmas in the linear regime [26]. As a
consequence, there are periodical regions where the witness charged particles
can simultaneously experience both acceleration and focusing (if they travel
off-axis), as it is depicted in Fig. 7.

Figure 7: Wakefields (including the Coulomb term) at $r=a/2$ for $a=1$ nm,
$v=0.05c$, $\gamma=0.01\Omega_{p}$ and $r_{0}=0$. The red (blue) arrows
indicate the regions where a positive (negative) witness charged particle
would experience both acceleration and focusing simultaneously.

###  III.3 Optimization of the CNT parameters

As pointed out in the previous section, the plasmonic excitations can be
approximated by $2W_{z2}$ when the friction parameter $\gamma$ converges to
zero in the considered system. Therefore, Eq. (15) can be used to efficiently
optimize key parameters, such as $n_{0}$, $a$ and $v$, to enhance the
longitudinal wakefield amplitude. Concretely, this section is focused on the
plasmonic excitation created on axis by a proton travelling on axis, i.e.
$r=r_{0}=0$. Figure 8(a) depicts the amplitude of $W_{z2}$ as a function of
the radius $a$. The maximum of the wakefield increases with the surface
density $n_{0}$ and moves to smaller radii $a$. It is also worth noting that
these plots do not depend on the surface density if both the radius and the
wakefield amplitude are normalized to the plasma wavelength $\lambda_{p}=2\pi
c/\Omega_{p}$ and the peak maximum $W_{z2}^{max}$, respectively (see Fig.
8(b)). Thus, there is an optimum radius (in units of $\lambda_{p}$) for a
given driving velocity regardless of the surface density $n_{0}$. This optimum
radius $(a/\lambda_{p})^{max}$ is proportional to the driving velocity $v$, as
it can be seen in Fig. 9(a). On the other hand, Fig. 9(b) shows the peak
maximum $W_{z2}^{max}$ as a function of $v$ for different surface densities.
The maximum wakefield is obtained for lower velocities (as long as they
satisfy the resonance condition) and increases with the surface density. Thus,
driving particles with low velocities can excite more efficiently plasmonic
modes in CNTs. However, at very low velocities, the resonance condition cannot
be satisfied. As a result, the curves in Fig. 9(b) do not originate at $v=0$.

Figure 8: (a) Amplitude of $W_{z2}$ as a function of the radius $a$ for
different values of $n_{0}$ and $v$ ($n_{g}=1.53\times 10^{20}$ m-2 is the
electron-gas density of a graphite sheet). (b) Amplitude of $W_{z2}$
(normalized to the maximum of Fig. 8(a)) as a function of $a/\lambda_{p}$.
Figure 9: (a) Optimum radius and (b) maximum longitudinal wakefield associated
as a function of $v$ for different surface densities ($n_{g}=1.53\times
10^{20}$ m-2 is the electron-gas density of a graphite sheet).

In summary, for a given velocity, the maximum longitudinal wakefield is
obtained for the optimum CNT radius given by Fig. 9(a) and a surface density
as high as possible. Finally, it is important to remark that for high
velocities we should take into account the relativistic effects which are not
considered in this manuscript. Nevertheless, it is worth mentioning that a
behavior qualitatively similar to that shown in Fig. 8(b) was also observed in
Fig. 4 of [9], where particle-in-cell (PIC) simulations were used with a
driving bunch composed of 1 GeV electrons (with an energy spread of 1$\%$). In
that case, the maximum wakefield was obtained for $a/\lambda_{p}\approx
0.10c$, a value quite similar to what was obtained in Fig. 9(a) as
$v\rightarrow c$. Therefore, the CNT radius optimization carried out in this
manuscript might provide a good approximation of its value.

##  IV Conclusions

The linearized hydrodynamic model in conjunction with the Poisson’s equation
has been used to study the electric wakefields generated by a point-like
charge travelling parallel to the axis in a CNT. General expressions have been
derived for the longitudinal and transverse wakefields and their dependencies
on the surface density, the CNT radius and the velocity of the driving charged
particle have been numerically studied and related to the dispersion relation.
It has been shown that the friction parameter produces an exponential decay of
the excited plasmonic modes. If the friction is negligible, the plasmonic
excitations can be approximated by twice the Eqs. (15)-(16). This
approximation for the longitudinal wakefield was used to perform an
optimization of the CNT radius (in units of the plasma wavelength) for a given
driving velocity. Interestingly, at least at a qualitative level, the results
agree with those obtained in [9] through PIC models. Hence, the linearized
hydrodynamic model might be used to obtain an approximation of the optimum
radius without requiring time-consuming PIC simulations. In future works, a
quantitative comparison will be performed.

It is worth mentioning that the wakefields generated by a point-like charge
could be used as a Green’s function to compute the wakefields excited by a
driving bunch with an arbitrary charge distribution. It has been shown that a
single proton may excite GV/m wakefields in CNTs with nanometric radii.
Consequently, a beam with $10^{6}$ protons would be able to excite PV/m
wakefields, provided that the beam size is small enough to simply consider the
sum of the contribution of each single particle. However, the obtention of
beams with such small sizes can prove to be extremely challenging.
Nevertheless, in the case of ultra-relativistic driving charged particles, the
optimal radius and wavelength associated with peak wakefields are estimated to
be $\gtrsim 100$ nm. Besides, space-charge effects are mitigated in beams with
highly relativistic velocities. As a consequence, it would be easier to obtain
beams which can be propagated inside the CNT and witness beams which fit in
the periodic regions where they can experience acceleration and focusing for
ultra-relativistic driving velocities. Even in this case, a driving beam with
$10^{6}$ protons may excite TV/m wakefields, which are more than four orders
of magnitude higher than those obtained with conventional RF cavities. Hence,
the use of nano-structures may open new possibilities to obtain ultra-high
particle acceleration gradients.

###### Acknowledgements.

This work has been supported by Ministerio de Universidades (Gobierno de
España) under grant agreement FPU20/04958, and the Generalitat Valenciana
under grant agreement CIDEGENT/2019/058.

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# A central degeneracy induced outer core in fermionic dark matter halos

Weikang Lin<EMAIL_ADDRESS>South-Western Institute For Astronomy
Research, Yunnan University, Kunming 650500, Yunnan, P. R. China

###### Abstract

Several galaxy-scale observations challenge the predictions of the cold dark
matter paradigm based on cosmological simulations. The cusp-core problem is
one of the most outstanding. However, current simulations are unable to
consider quantum statistical effects. For a fermionic dark matter halo, a
degenerate inner core induces an extended outer core that dominates in gravity
in the region relevant for rotation curve observation. We study the properties
of this outer core and show that its density and radius are correlated. The
prediction remarkably agrees with the measured scaling relation in both the
slope and the magnitude. Such consistency suggests that the observed cored
profiles do not contradict massive dark matter but indicate its fermionic
nature.

##  I Introduction

Accumulated observations suggest that the dark matter (DM) halos are cored for
dwarf galaxies and low-surface-brightness (LSB) galaxies [1, 2, 3, 4, 5]. This
differs from the results of DM-only N-body simulations that predict cuspy
profiles at the center [6]. Such a discrepancy is called the “cusp-core”
problem and is difficult to explain with baryon effects for those DM-dominated
systems [7, 8, 9].

New DM physics are motivated to resolve the problem [10], such as fuzzy DM
[11] and DM self-interactions [12]. But those proposals remain controversial.
For example, it is argued that light boson DM cannot correctly account for the
core properties [13, 14, 15] and the required self-interaction strength is
constrained by other considerations [16, 17]. Nonetheless, the cusp-core
problem is thought to disfavor collisionless massive DM candidates, if
complicated baryon effects are not involved. This work shall show the
opposite.

One fundamental limitation of current N-body simulations is their incapability
to capture the quantum statistical effects. This is rooted in the fact that
particles in simulations are distinguishable, not to mention the
unrealistically huge test particle mass. Recently, several works investigated
the role played by quantum statistics on DM halos at galaxy scales [18, 19,
20, 21, 22, 23]. In a regime where DM is degenerate at the center, the DM
density profiles are cored. Most efforts were devoted to this core, which is
caused by the Fermi degenerate pressure preventing the system from
gravitational collapse [24]. Unfortunately, to account for the observed core
size ranging from hundreds of pc to a few tens of kpc [3, 5], the required DM
mass is about one to two orders of magnitude smaller than the mass lower bound
obtained from the cosmic large-scale structure [25, 26, 27].

There is more to the degenerate regime and the DM profile has a low-density
but extended plateau enclosing the high-density core [21]. Due to its dominant
mass over the inner compact core, this plateau should be more relevant for the
rotational curve observation for dwarf-size galaxies [28]. We call this
plateau an “outer core”. The properties of this outer core have been
completely overlooked in the literature, which we study and show that there is
significant implication regarding the cusp-core problem.

##  II Method

We highlight here the key steps in obtaining the fermionic isothermal
profiles, adopting the natural units where $c=\hbar=k_{\textsc{b}}=1$. We
consider a spherical halo of fermionic DM in dynamical equilibrium,

$\frac{dP}{dr}=-\frac{G\rho}{r^{2}}M(r)\,,$ (1)

and DM follows a nonrelativistic Fermi-Dirac distribution presumably due to
violent relaxation [29],

$f(p)=\frac{g}{2\pi^{2}}\frac{p^{2}}{\exp\big{[}(\frac{p^{2}}{2m_{\rm
d}}-\mu_{\rm eff})/T_{\rm d}\big{]}+1}\,,$ (2)

where $g$ is taken to be $2$ but can be generalized. The effective chemical
potential is

$\mu_{\rm eff}=\mu-m\phi\,,$ (3)

where $\phi$ is the gravitational potential and $\mu$ is the real chemical
potential. Assuming there is no net particle flow at any radius, we take $\mu$
to be spatially independent. This treatment corresponds to the Thomas-Fermi
approach applied in [18]. The above settings are expected to break down at a
large radius $r$ but should be a good approximation for the region we are
interested in.

The mass density and pressure are then calculated in a standard statistical
mechanics way and it can be derived that [27],

$\displaystyle\rho$ $\displaystyle=\sqrt{2}Q_{*}\left(\frac{T_{\rm d}}{m_{\rm
d}}\right)^{\frac{3}{2}}J_{\nicefrac{{3}}{{2}}}^{\textsc{f}}(\Delta)\,,$ (4)
$\displaystyle P$ $\displaystyle=\frac{2^{3/2}Q_{*}}{3}\left(\frac{T_{\rm
d}}{m_{\rm
d}}\right)^{\frac{5}{2}}J_{\nicefrac{{5}}{{2}}}^{\textsc{f}}(\Delta)\,,$ (5)

where $\Delta\equiv\mu_{\rm eff}/T_{\rm d}$ is the degree of degeneracy,
$Q_{*}=\frac{gm_{\rm d}^{4}}{2\pi^{2}}$ and
$J_{s}^{\textsc{f}}(\Delta)=\int_{0}^{\infty}\frac{z^{s-1}{\rm
d}z}{\exp(z-\Delta)+1}$. Eq. (2) reduces to the classical Maxwell-Boltzmann
distribution when $\Delta<0$ and describes a degenerate fermionic system when
$\Delta>0$.

The last equation is Poisson’s equation for the gravitational potential,

$\frac{d\phi}{dr}=\frac{GM(r)}{r^{2}}\,.$ (6)

We have ignored the general relativistic correction (see the RAR model [21]).
The above conditions require the resultant halo to be isothermal, i.e.,
$T_{\rm d}$ is spatial independent. So, the solutions are fermionic isothermal
spheres.

With the above settings, the DM density profile is then fully determined by
the following 2nd-order differential equation about $\Delta$,

$\Delta^{\prime\prime}+\frac{2}{z}\Delta^{\prime}=-J_{\nicefrac{{3}}{{2}}}^{\textsc{f}}(\Delta)\,,$
(7)

where ′ denotes the derivative with respect to $z\equiv r/r_{*}$. The
characteristic radius $r_{*}$ is,

$\displaystyle r_{*}$ $\displaystyle=\frac{2\pi}{g^{1/2}}\left(\frac{m_{\rm
d}}{2T_{\rm d}}\right)^{1/4}\left(\frac{m_{\rm p}}{m_{\rm
d}}\right)^{2}\ell_{\rm p}\,,$
$\displaystyle=\left(\frac{2}{g}\right)^{\frac{1}{2}}\left(\frac{30\,\text{km/s}}{\sigma_{v}}\right)^{\frac{1}{2}}\left(\frac{5\,\text{keV}}{m_{\rm
d}}\right)^{2}\times 0.23\,{\rm pc}\,,$ (8)

where $\sigma_{v}=\sqrt{T_{\rm d}/m_{\rm d}}$ is the (1-D) DM velocity
dispersion, $m_{\rm p}=2.4\times 10^{24}$ keV the reduced Planck mass and
$\ell_{\rm p}=\frac{1}{m_{\rm p}}=8.2\times 10^{-35}$ m. Note that $r_{*}$ is
macroscopic for a wide range of massive DM candidates, although still small
compared to kpc scales.

Figure 1: Halo density profiles normalized by the central density. Curves are
labeled by the central degree of degeneracy $\Delta_{0}$. When an outer core
appears, the value of $\Delta_{\rm oc}$ is shown in parenthesis.

The solutions to Eq. (7) are characterized by the central degeneracy, denoted
as $\Delta_{0}$. Resultant DM profiles are shown in Figure 1 presented as the
normalized density versus $r/r_{*}$ with their values of $\Delta_{0}$ labeled
in curves. The profiles obtained are consistent with previous studies [18, 19,
20, 21, 22, 23]. Two classes of profile are readily identified, one with
$\Delta_{0}<0$ and the other with $\Delta_{0}>0$. Profiles with $\Delta_{0}<0$
are nothing but the classical cored isothermal sphere obtained by imposing a
finite central density [30] and suffer unstable gravitational collapse [31].
Profiles with $\Delta_{0}>0$ have a high-density but compact inner core
followed by a low-density but extended outer core. It has been shown that the
outer core can explain the individual rotation curve of LSB galaxies [28]. As
more cored profiles of dwarf-size galaxies are determined, a universal scaling
relation between the core density and radius is reported and supported by
increasing observations [32, 33, 8]. We therefore focus on the properties of
the outer core, in particular its relation between density and radius.

##  III Analysis and results

Intuitively, the outer core appears because of the sudden drops of the density
right outside the inner core, making the right-hand side of Eq. (1) very small
and allowing the pressure to vary slowly for a large range. The density
profile of the outer core and beyond is in a region where DM has a Maxwell-
Boltzmann distribution and
$J_{\nicefrac{{3}}{{2}}}^{\textsc{f}}(\Delta)\simeq\frac{\sqrt{\pi}}{2}e^{\Delta}$.
It implies that the outer core radius should behave like a King’s radius such
that it is correlated with the outer core density. This can be clearly seen
with the solution to Eq. (7) in the outer core region, which reads,

$\Delta\simeq\Delta_{\rm oc}+\frac{B}{z}-\frac{\sqrt{\pi}}{12}e^{\Delta_{\rm
oc}}z^{2}\,,$ (9)

where the constants $\Delta_{\rm oc}$ and $B$ are determined by patching the
outer solution with the inner solution. The second term decreases with $z$
quickly and can be ignored. The outer core radius $r_{\rm oc}$ can be
estimated by the location at which the third term becomes unity, that is,
$z_{\rm oc}\propto\exp(-\Delta_{\rm oc}/2)$ and thus $\frac{r_{\rm
oc}}{r_{*}}\propto\frac{1}{\sqrt{\rho_{\rm oc}}}$.

Figure 2: Comparison between the predicted outer core density-radius relation
to the best-fit parameters given in [5] (black crosses) and [4] (black
triangles). Red symbols are predictions with an empirically motivated scaling
relation between the DM velocity dispersion $\sigma_{v}$ and the galactic
baryon mass $M_{\rm b}$. Green dotted lines are predictions with some fixed DM
velocity dispersion labeled in lines.

Numerically, we define $r_{\rm oc}$ to be the location where the logarithmic
slope of the density profile is $-1.5$ to match the core radius in the Burkert
profile [34], that is $\frac{d\ln\rho}{d\ln r}\big{\rvert}_{r_{\rm{c}}}=-1.5$.
Whereas, the outer core density $\rho_{\rm oc}$ is defined to be the density
at the peak of the logarithmic slope of the density profile in the outer core
region. From these definitions along with Eq. (II), we find that $\rho_{\rm
oc}$ and $r_{\rm oc}$ are correlated by,

$\rho_{\rm oc}=1.4\times
10^{8}\left(\frac{\sigma_{v}}{30\,{\text{km/s}}}\right)^{2}\left(\frac{1\,{\text{kpc}}}{r_{\rm
oc}}\right)^{2}\,\frac{M_{\odot}}{{\rm kpc}^{3}}\,.$ (10)

This relation contains no other free parameter, such as the DM mass, as long
as the outer core is induced. We compare the above prediction with the best-
fit core densities and radii of a Burkert profile fitted to the rotation
curves of LSB galaxies [5] and dwarf galaxies [4].111Note that, Ref. [4]
performed the analysis with a pseudo-isothermal profile instead of a Burkert
profile. So, we have scaled the measured cored radii provided in Ref. [4] by a
factor of $\sqrt{3}$ to match the definition of the core radius here. We also
have removed the data point of galaxy Haro 36 as the uncertainty of the core
radius is above 100%. The comparison is shown in Figure 2. Remarkably, with
$\sigma_{v}\sim 40$ km/s motivated by the stellar velocity dispersion of the
dwarf galaxies with a similar baryon mass range, the predicted $\rho_{\rm
oc}$-$r_{\rm c}$ relation (green lines) matches the measurements (black
symbols) very well.

The predicted $\rho_{\rm oc}$-$r_{\rm c}$ relation with a constant
$\sigma_{v}$ is somewhat steeper than the measurements, i.e., the absolute
slope is larger. This is because the variation of $\sigma_{v}$ with the galaxy
size has not been taken into account. To do that, we assume the following
scaling relation between $\sigma_{v}$ and $M_{\rm b}$ motivated by the
empirical Faber-Jackson relation for dispersion-supported galaxies,

$\sigma_{v}=\left(\frac{M_{\rm
b}}{10^{9}\,M_{\odot}}\right)^{1/4}\sigma_{v}^{\rm ref}\,,$ (11)

where $M_{\rm b}$ is the baryon mass of the galaxy and $\sigma_{v}^{\rm ref}$
is some reference DM velocity dispersion at $M_{\rm b}=10^{9}M_{\odot}$. We
take $\sigma_{v}^{\rm ref}=40$ km/s motivated by the stellar velocity
dispersion for a dwarf galaxy of similar size [35]. Note that the Faber-
Jackson relation is between the luminosity (or the baryon amount) and the
stellar velocity dispersion for elliptical galaxies. However, it is reasonable
to assume that the invisible DM velocity dispersion is correlated with the
baryon amount similarly.

For each galaxy, we use the galactic baryon mass (the sum of the masses of
disk, bulge, and gas) given in [36, 4] to infer the DM velocity dispersion
according to Eq. (11). Then, we take the measured core radius and the inferred
$\sigma_{v}$ as the inputs of Eq. (10) to predict the core density. The
results are shown by the red symbols in Figure 2. The predictions exhibit an
excellent consistency with the measurements in both the slope and the
magnitude.

The outer core radius is determined by the DM mass and the central degeneracy
with,

$r_{\rm oc}\simeq 4.5\exp(0.37\Delta_{0})\,r_{*}\,.$ (12)

Interestingly, for $r_{\rm oc}\sim 1$ kpc, the DM mass that satisfies
cosmological constraints $\gtrsim 5$ keV [37] requires $\Delta_{0}\gtrsim 19$
so that a clear outer core is induced. Thus, the central degeneracy-induced
outer core resolving the cusp-core problem for massive fermionic DM is fully
consistent in cosmology.

##  IV Discussion

The agreement between the predictions and measurements shown in Figure 2
suggests that the observed core radii for dwarf galaxies and LSB galaxies are
a sort of King radius. But unlike the classical King radius, the outer core
radius here is induced by the central inner core that is stabilized by the
fermi degenerate pressure. Thus, the central degeneracy-induced outer core
constitutes a viable and promising resolution to the cusp-core problem.

There appeared to be a dilemma in the nature of DM. On one hand, cosmological
probes are consistent with the cold dark matter paradigm [38]. On the other
hand, the cusp-core problem seemed to suggest otherwise. Our result suggests
that the observed cored DM profiles are not contradictory to the standard DM
scenario, but are pointing to the fermionic nature of DM whose mass is
consistent with cosmological constraints.

Our result also suggests that the observed $\rho_{\rm c}$-$r_{\rm c}$ relation
should fundamentally be a $\rho_{\rm c}$-$r_{\rm c}$-$\sigma_{v}$ relation.
Or, in terms of observables, it is a $\rho_{\rm c}$-$r_{\rm c}$-$M_{\rm b}$
relation assuming an empirically motivated scaling relation between the DM
velocity dispersion and the galactic baryon mass. Only viewed in the projected
$\rho_{\rm c}$-$r_{\rm c}$ plane, the product $\rho_{\rm c}r_{\rm c}$ is
nearly constant as reported in the literature [32, 33, 8].

For a given DM particle mass, a larger outer core radius corresponds to a
higher central degeneracy and a higher central density. This hints at the
connection between the central degeneracy and the halo collapse history. A
caveat is that we are considering DM-only systems, which are suitable for DM-
dominated systems such as dwarf galaxies and LSB galaxies. The interplay
between DM and baryon for larger systems may exhibit a wider diversity.
Numerical simulations are motivated to consider or mimic the quantum
statistical effects. To do so, the resolved scales are required to be smaller
than not only the observed core sizes but also the characteristic scale
$r_{*}$ that depends on DM particle mass.

###### Acknowledgements.

W. L. thanks Y. Cui and T. T. Yanagida for the useful feedback. W. L.
acknowledges that this work is supported by the “Science & Technology Champion
Project” (202005AB160002) and the “Top Team Project” (202305AT350002), both
funded by the “Yunnan Revitalization Talent Support Program.” This work is
also supported by the “Yunnan General Grant” (202401AT070489).

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