Hydrodynamic Model for Particle Beam-Driven Wakefield in Carbon Nanotubes

The charged particles moving through a carbon nanotube (CNT) may be used to excite electromagnetic modes in the electron gas produced in the cylindrical graphene shell that makes up a nanotube wall. This effect has recently been proposed as a potential novel method of short-wavelength-high-gradient particle acceleration. In this contribution, the existing theory based on a linearized hydrodynamic model for a localized point-charge propagating in a single wall nanotube (SWNT) is reviewed. In this model, the electron gas is treated as a plasma with additional contributions to the fluid momentum equation from specific solid-state properties of the gas. The governing set of differential equations is formed by the continuity and momentum equations for the involved species. These equations are then coupled by Maxwell’s equations. The differential equation system is solved applying a modified Fourier-Bessel transform. An analysis has been realized to determine the plasma modes able to excite a longitudinal electrical wakefield component in the SWNT to accelerate test charges. Numerical results are obtained showing the influence of the damping factor, the velocity of the driver, the nanotube radius, and the particle position on the excited wakefields. A discussion is presented on the suitability and possible limitations of using this method for modelling CNT-based particle acceleration.


Introduction
Solid-state wakefield acceleration using crystals was proposed in the 1980s and 1990s by T. Tajima and others [1,2,3] as an alternative particle acceleration technique to sustain TV/m acceleration gradients.Solid-based acceleration media (e.g.nanostructures or crystals) could offer a possible solution to overcome the plasma wave-breaking limit, which increases with the plasma density [4], since the density of conduction band electrons in solids is four or five orders of magnitude higher than in gaseous plasma mediums.However, natural crystals have two main drawbacks: (i) the beam intensity acceptance is significantly limited by the angstrom-size channels and (ii) such small size channels are physically vulnerable to high energy interactions.
In this context, nanostructures could offer an excellent way to overcome many of the limitations of natural crystals.For instance, CNT-based structures can help to relax the constraints to more realistic regimes with respect to natural crystals.CNTs are large macromolecules that are unique for their size, shape, and physical properties, presenting the following advantages over natural crystals: (i) transverse acceptances of the order of up to 100 nm [5] (i.e. three orders of magnitude higher than a typical silicon channel); (ii) larger degree of dimensional flexibility and thermo-mechanical strength; (iii) lower dechannelling rate; and (iv) less disruptive effects such as filamentation and collisions.Consequently, CNTs are considered a robust candidate for solid-state wakefield acceleration.Wakefields in crystals or nanostructures can be induced by means of the excitation of surface plasmonic modes [6,7] (or simply plasmons), which are high-frequency collective oscillations of the conduction electrons, acting like a structured plasma through the crystalline ionic lattice.However, to properly excite wakefields, the driver dimensions need to match the spatial (∼nm) and time (sub-femtoseconds) scales of the excited plasmonic oscillations.Wakefield driving sources working on these scales are now experimentally realizable.For instance, attosecond X-ray lasers are possible thanks to the pulse compression technique [8] and, in the case of beam-driven wakefields, future facilities such as FACET-II at SLAC [9] might allow access to quasi-solid electron bunches with densities up to ∼ 10 30 m −3 and sub-micrometer bunch length scale.

Theoretical background
We are going to use a linearized hydrodynamic theory [10,11,12], in particular a model that includes charged-particle based excitations on nanotube surfaces and was described in [11].In this theory, a SWNT is modelled as an infinitesimally thin and infinitely long cylindrical shell with radius a.The delocalized electrons forming the carbon ions are considered as a twodimensional free-electron gas that is confined within the cylindrical surface of the CNT with a uniform surface density n 0 .We will consider a driving point-like charge Q travelling parallel to the cylinder axis (z-axis) inside the tube with a constant velocity v. Consequently, its position as a function of time t is r 0 (t) = (r 0 , φ 0 , vt) 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 u(r a , t) and a perturbed density per unit area n 1 (r a , t), where r a = (a, φ, 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 u is zero.
In the linearized hydrodynamic model, the electronic excitations on the tube wall can be described by three differential equations: (i) the continuity equation (ii) the Poisson's equation and (iii) the momentum-balance equation In these equations, r = (r, φ, z) is the position vector, differentiates only tangentially to the tube surface, Φ the electric scalar potential, e the elementary charge, m e the rest mass of the electron, ε 0 the vacuum electric permittivity and δ the Dirac delta function.Equation (3) shows the sum of four different contributions.The first term on the right-hand side represents the force exerted on electrons at the surface of the nanotube by the tangential component of the electric field.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 α = v 2 F /2 where v F = h(2πn 0 ) 1/2 /m e is the Fermi velocity of the two-dimensional electron gas.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 [12] and describes single-electron excitations in the electron gas, where the parameter β = 1 4 ( h me ) 2 has been defined.The last term represents a frictional force on electrons due to scattering with the ionic-lattice charges, where γ is the friction parameter.
The electric 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) taking into account that (i) the electric potential vanishes at r → ∞, (ii) is finite at the origin r = 0, and (iii) is continuous at r = a; and (iv) the discontinuity of the radial component of the electric field due to perturbed density n 1 at r = a.Hence, the total potential inside the nanotube (r < a) can be calculated as Φ in = Φ 0 + Φ ind , with the Coulomb potential Φ 0 = 1 4πε 0 Q ∥r−r 0 ∥ due to the driving charge and the induced potential Φ ind due to the perturbation of the electron fluid on the carbon nanotube surface.Thus, the corresponding longitudinal electric wakefields inside the tube is (see Eqs. ( 4)-( 6) in [13]) where ℜ and ℑ denote the real and imaginary part, respectively, ζ = z − vt is the comoving coordinate, and A m (k) is a non-dimensional function: where Ω p = e 2 n 0 ε 0 mea is the plasma frequency and Consequently, the resonant excitations occur when kv = ω m (k) if the damping factor γ vanishes.The previous integral (4) has been separated in three different terms: the first term W z0 comes from the Coulomb potential and the other terms W z1 , W z2 from the induced potential.It is very interesting to note that the third term W z2 can be analytically integrated using the residue theorem if the damping factor vanishes (γ → 0 + ):

Results
In this section, we are going to describe the obtained results.The following numerical values will be used (unless otherwise indicated): n V = n 0 /a = 10 28 m −3 , a = 0.1λ p (λ p = 2πc/Ω p ≈ λ p = 330(n V (10 28 m −3 )) −1/2 nm is the plasma wavelength), γ → 0 + , v = 0.99c and Q = −2.75pC.We will use r 0 = 0 and r = 0, since then we will only have to consider the mode m = 0 (if r 0 ̸ = 0 and r ̸ = 0 the modes with |m| > 0 are not zero, but the most important mode is still m = 0).For these parameters, the longitudinal wakefield exhibits a minimal dependence on the value of r, although for lower densities and/or velocities, the dependence can become important (the wakefield is higher near the nanotube surface, r = a).The resonant excitations are the intersections of the lines k m v with the dispersion relations ω m (k), as it is shown in Fig. 1(a).Consequently, if the driving velocity increases, the resonant conditions k m decreases.Besides, it can be easily demonstrated that, if the density n V or the CNT radius a increases, the resonant conditions k m increase as well.Figure 1(b) shows the three different contributions to the longitudinal wakefield.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 near the driving particle.The friction parameter γ produces an exponential decay of the wakefield (W z1 + W z2 ) of the type e −γζ/v .Thus, when the friction parameter γ is very small, W z2 is a cosine pattern and the longitudinal wakefield can be approximated by W z = W z0 + 2W z2 , as it can be checked in Fig. 2. Therefore, Eq. ( 7) can be used to obtain quickly optimized values for the parameters n 0 , a, v. Figure 3(a) shows the amplitude of W z2 as a function of the radius a.It is very interesting to note that this plot can be normalized (see Fig. 3(b)), which does not depend on the volumetric density n 0 /a and it is very similar to Fig. 4 in [14] performing particle-in-cell (PIC) simulations.The optimum value for the radius is directly proportional to the velocity v of the driving charge, as it can be seen in Fig. 4(a).On the other hand, the maximum W max z2 is directly proportional to the density n 0 /a, i.e. it can be expressed as W max z2 = κ(n 0 /a), where κ decreases if the velocity v increases, as it is shown in Fig. 4(b).Consequently, driving particles with low velocities can excite more efficiently the plasmonic modes.4) for small γ and the approximation using Eq. ( 7).

Conclusions and outlook
We have used a linearized hydrodynamic model to describe the electronic excitations of a 2D electron gas confined on surfaces of nanotubes.The friction parameter might be used for modelling how the electron motion constrain due to the ionic lattice affects the wakefield and it will be investigated in future works.Moreover, Eq. ( 7) can be used to obtain very quickly the optimized values {n 0 , a} for each velocity v.However, the main limitation of this method is the optimization for very high densities, since the typical saturation does not appear in our results.Further studies comparing these analytical results with PIC simulations are ongoing.

Figure 3 .
Figure 3. (a) Amplitude of W z2 as a function of the radius a for different values of n 0 /a and v.(b) Amplitude of W z2 (normalized to the maximum of Fig. 3(a)) as a function of a/λ p .