Solid-state Tube Wakefield Accelerator using Surface Waves in Crystals
Aakash A. Sahai, Toshiki Tajima, Peter Taborek, Vladimir D. Shiltsev
SSolid-state Tube Wakefield Acceleratorusing Surface Waves in Crystals
Aakash A. Sahai
College of Engineering and Applied ScienceUniversity of Colorado, Denver, CO [email protected]
Toshiki Tajima, Peter Taborek
Department of Physics & Astronomy and Applied PhysicsUniversity of California, Irvine, CA 92697
Vladimir D. Shiltsev
Accelerator Research DepartmentFermi National Accelerator Laboratory, Batavia, IL 60510
Solid-state or crystal acceleration has for long been regarded as an attractive frontier in advancedparticle acceleration. However, experimental investigations of solid-state acceleration mechanismswhich offer TVm − acceleration gradients have been hampered by several technological constraints.The primary constraint has been the unavailability of attosecond particle or photon sources suitablefor excitation of collective modes in bulk crystals. Secondly, there are significant difficulties withdirect high-intensity irradiation of bulk solids, such as beam instabilities due to crystal imperfectionsand collisions etc.Recent advances in ultrafast technology with the advent of submicron long electron bunchesand thin-film compressed attosecond x-ray pulses have now made accessible ultrafast sources thatare nearly the same order of magnitude in dimensions and energy density as the scales of collectiveelectron oscillations in crystals. Moreover, nanotechnology enabled growth of crystal tube structuresnot only mitigates the direct high-intensity irradiation of materials, with the most intense part ofthe ultrafast source propagating within the tube but also enables a high degree of control over thecrystal properties.In this work, we model an experimentally practicable solid-state acceleration mechanism usingcollective electron oscillations in crystals that sustain propagating surface waves. These surfacewaves are driven in the wake of a submicron long particle beam, ideally also of submicron transversedimensions, in tube shaped nanostructured crystals with tube wall densities, n tube ∼ − cm − .Particle-In-Cell (PIC) simulations carried out under experimental constraints demonstrate the pos-sibility of accessing average acceleration gradients of several TVm − using the solid-state tubewakefield acceleration regime. Furthermore, our modeling demonstrates the possibility that as thesurface oscillations and resultantly the surface wave transitions into a nonlinear or “crunch-in”regime under n beam /n tube > ∼ .
05, not only does the average gradient increase but strong trans-verse focusing fields extend down to the tube axis. This work thus demonstrates the near-termexperimental realizability of Solid-State Tube Wakefield Accelerator (SOTWA).The ongoing progress in nanoengineering and attosecond source technology thereby now offersthe potential to experimentally realize the promise of solid-state or crystal acceleration, opening upunprecedented pathways in miniaturization of accelerators.
INTRODUCTION
Particle acceleration techniques using collective charge density oscillations in crystals have been known to be anattractive possibility for the past many decades [1].
Solid-state Acceleration using Wakefields in Crystal Plasmas:Attosecond sources and Crystal tubes
In solid-state or crystal acceleration mechanisms a charged particle beam gains energy by extracting the electromag-netic field energy of collective electron oscillation modes excited in crystals. These solid-state collective oscillationsare known to sustain propagating charge density waves of high energy densities. These collective oscillations and theassociated waves can be efficiently excited as wakes of pulsed sources of particles or photons with pulse dimensionsthat are resonant with the scales of collective oscillations in solid-state. However, the theoretically modeled solid-stateacceleration gradients [2, 3] which are known to be orders of magnitude higher than the time-tested radio-frequency a r X i v : . [ phy s i c s . acc - ph ] J a n technology as well as the emerging gaseous plasma acceleration [4, 5] techniques, are yet to be experimentally verifiedand further studied.Experimental verification of solid-state acceleration mechanisms has been so far hampered by several technolog-ical challenges such as unavailability of pulsed particle and photon sources that are resonant with the collectiveoscillations in crystals. However, technological advances in intense particle and photon pulsed ultrafast source com-pression technologies have continued to drive the pulse dimensions towards ever shorter time and spatial scales. Thesetechnological advancements in ultrafast source compression techniques have made scales required to resonantly ex-cite collective electron modes for solid-state acceleration mechanisms experimentally accessible. Especially, recentbreakthroughs in attosecond scale photon [6] and particle [7] bunch ultrafast source technologies have opened up thepotential for experimental realization of long-sought solid state acceleration [1–3].Although attosecond source technologies provide an effective means for resonant excitation of collective modes insolid-state crystal media, there still exist other technological barriers. In addition to the barriers due to the scarcityof attosecond sources, accessing solid-state gradients has also been impeded by difficulties with direct irradiation ofsolids at high intensities using particle or photon beam. Advances in nano-structured materials and nanoengineeringof tube-like structures in crystals, such as nanotubes, however now offer the possibility of overcoming these difficultieswith direct interaction of a crystal with high-intensity sources. Direct interaction of a high-intensity particle beam isreported to undergo severe filamentation due to the deformities in the crystal structure [8]. Not only is a filamentedbeam detrimental to driving a coherent wake but it also leads to a severely uncontrolled interaction and energydissipation.Solid-state plasmas with electron densities n ∼ − cm − , sustain electron oscillations at superoptical time,177( n [10 cm − ]) − / attosec ( ∼ ω ∗− pe where ω pe = ( n e (cid:15) − m − ) / and m e the electron mass [9]) and spatial scales,330( n [10 cm − ]) − / nm ( ∼ λ pe ). Electron modes at such scales offer Tajima-Dawson (wavebreaking) accelerationgradients [4] of the order of, E wb (cid:39) . n [10 cm − ]) − / TVm − . By coupling with these superoptical scales,submicron particle bunches (e.g., planned σ z < µm [10]) or intense keV photon lasers [11] make excitation ofunprecedented TVm − average gradients experimentally feasible. Progress of Wakefields Acceleration in Gaseous Plasmas:Femtosecond sources
Over the past few decades, access to tens of femtosecond chirped pulse amplified [12] 0 . µm wavelength lasers (withfew femtosec single cycle) and particle bunches has enabled experimental verification of advanced particle accelerationtechniques that use collective electron oscillations in gaseous plasmas [4, 5]. Whereas lasers have been compressedto few cycle long pulses using the innovative chirped pulse amplification technique [12], ultrashort particle buncheshave been obtained via phase-space gymnastics [13] or self-modulation in plasma [14]. Both these ultrafast sourcetechnologies have enabled successful gaseous plasma acceleration experiments with many GVm − gradients [15, 16].These experiments have used micron-scale charge-density waves in gaseous bulk plasma.Control over bulk plasma waves in homogeneous gases by femtosecond-scale sources [4, 5] has lead to the successfuldemonstration of gaseous plasma wakefield acceleration techniques. Numerous advantages of these techniques overconventional radio-frequency acceleration techniques has now lead to them being enhanced and fine-tuned for real-world applications using commercially available femtosecond sources. Some of these enhancements include controlover: (a) wakefield profile distortions from ion motion [17], (b) dark current injection and acceleration due to secondaryionization [18], (c) accelerated beam emittance growth due to scattering off of plasma ions [19], (d) positron defocusingby the bared ions, (e) repetition rate constraints due energy coupling to long-lived ion modes [20] etc.Non-homogeneous plasmas of specific shapes have been proposed to address many of the above enhancements ofgaseous bulk plasma acceleration. The earliest shaped plasma proposal [21] for a fiber accelerator sought to keephigh-intensity laser pulses continuously focussed [22]. Utilizing this shaped plasma proposal, gaseous plasma fibersthat are excited using mechanisms such as laser-heated capillary [23] etc. are now regularly used for plasma fiberguided laser-plasma acceleration. Mechanism of beam-driven shaped gaseous hollow plasma (later labelled hollow-channel) acceleration has also been studied [25]. Experiments on beam-driven gaseous hollow plasma using intensepositron beams have observed ∼ − peak gradients ( ∼ . E wb ) [26]. Access to higher gradients in shapedgaseous hollow plasmas is currently under active research. Active areas of research include technological difficultiesin shaping a desired channel in gaseous plasmas apart from challenges due to the absence of any focusing force [25]such as control of higher-order transverse wakes excited by the drive beam due to its misalignment from channel axis[27] and beam-breakup resulting from these transverse wakes. Recent results have demonstrated that beam breakupmay be controllable via further shaping of the gaseous hollow plasmas [28].In this paper, we introduce and model a regime of experimentally realizable solid-state acceleration that usescharge density waves of submicron scale lengths in nanotube shaped solid-state plasmas. The acceleration modes inthis regime of solid-state tube wakefield acceleration take advantage of the developments in nano-fabrication as wellas submicron particle or attosecond photon pulsed source technology. We show using analytical and computationallymodeling that the crystal tube surface electron oscillations sustain an electrostatic “crunch-in” mode [11, 20, 29]. Thiselectrostatic mode supports electromagnetic surface wave modes with phase velocity close to the driver velocity andon-axis longitudinal electric fields that approach the Tajima-Dawson gradient of the tube wall electron density. Thishigh phase velocity surface wave mode supported by excitation of tube wall electron oscillations makes solid-statetube wakefield accelerator regime quite effective.Although significantly different from traveling wave modes supported by electron oscillations in solid-state, a similarelectron oscillation mode of gaseous plasma hollow channels has been computationally observed in a few previousworks. However, neither its structural and electromagnetic properties nor its acceleration characteristics have beenextensively modeled. In gaseous plasmas the “crunch-in” like mode has been observed in simulation works that haveused experimentally feasible parameter regime such as a laser-driven shaping of a hollow plasma proposal [30], aproton beam driven shaped hollow plasma acceleration proposal in externally magnetized plasma [31] and a electronor positron beam driven shaped hollow plasma [20].An important recent work [11, 40] has recently studied and modeled the excitation of modes in crystal tubes usingattosecond keV photon x-ray pulses. This work on modeling of x-ray wakefield tube accelerator has demonstratedthe potential of using x-ray wakefield acceleration mechanism in tubes for sustaining many TV-cm − gradients. Withthe advent of a few cycle high-intensity x-ray laser using thin-film compression technique, the crystal x-ray wakefieldacceleration mechanism has the potential to further advance the progress made by the Ti:Sapphire 800nm opticallaser based gaseous plasma wakefield acceleration technique.However, the mechanism of beam-driven surface modes in bulk crystals and crystal tubes, as opposed to thosedriven by an x-ray laser, has not yet been modeled and characterized. This is especially important due to the recentopening up of the availability of submicron particle bunches. The beam-driven crystal tube phenomena investigatedand the results reported here indicate that the x-ray driven crystal tube wakefields characterized in [11, 40, 41] arequite similar to that in the beam-driven crystal tube case. Therefore, our work shows that crystal tube wakefields haveboth longitudinal and focusing fields similar to the x-ray driven wakefields [11, 40, 41]. It may be noted that our workon beam-driven wakefields in a crystal tube is distinctive from previously modeled gaseous hollow-plasma wakefieldsbecause in gaseous hollow-plasma the wakefields of a relativistic particle beam are proven to have zero focusing fields[42]. The preliminary analysis and computational results presented below demonstrate the experimental realizabilityof Solid-State Tube Wakefield Accelerator (SOTWA).In the following sections on modeling of beam-driven wakefields in crystal tubes, we introduce and characterizethe beam-driven solid-state tube accelerator using surface wave wakefields in crystals. The model and significanceof solid-state collective electron or plasmon oscillation modes is presented in sec.. An analytical model of the tubewall electron oscillations extending into the tube is presented in sec.. Preliminary proof-of-principle particle-in-cellmethod based computational modeling of beam-driven solid-state tube wakefield accelerator is detailed in sec.. Wealso study the novel “crunch-in” behavior shown by the wakefields in a tube which includes wakefield amplitudes closeto the Tajima-Dawson acceleration gradient for relatively small beam to tube density ratios as well as the existenceof transverse fields that extend down to the tube axis. COLLECTIVE OSCILLATION IN QUANTUM MECHANICAL SYSTEMS:OSCILLATION MODES OF ELECTRON GAS IN CRYSTAL IONIC LATTICE
Collective electron oscillations in crystals have for long been established a critical yet physically valid simplificationof the many body interaction in solid-state materials. The many body problem of solid-state electrons can be eitherdescribed using an assembly of Fermions or using collective oscillation theory. The collective oscillation approachwas exhaustively modeled in theory [32–35] (phonon, plasmon and polaritons) and experimentally proven to result inobservable effects [36] in 1950s.Solid-state collective oscillations were first investigated with great details in the context of the modeling of thestopping power of an incident electron beam in metals with an inherent crystal structure [37]. The predictions ofenergy loss of a particle beam incident on a metal were found to be in excellent agreement with the theory of excitationof collective electron oscillations in the crystal, driven as a wake of the incident particles.The terminology of excitation of collective oscillations in the “wake” of an incident particle was introduced in 1950s.Moreover, to explain quantization in beam energy loss when interacting with a thin foil with thickness of the orderof mean free path of the bulk plasma oscillations in crystals, these oscillations where referred to as plasmons. Thetheoretical plasmon model of the collective oscillations of electrons in crystals showed good agreement with experimentson energy loss of injected beam electrons. In addition to the explanation of the quantized energy loss of the beam,the conditions for the excitation of collective oscillations in the wake of an incident particle were also detailed. Thecollective oscillations of valence electrons were demonstrated to be quite similar to the plasma oscillations observedin gaseous plasmas.Bloch [32] was the first to model the excitations of a Fermi gas as collective gas oscillations as opposed to excitedstates of single particles. Bloch treated the Fermi gas collective oscillations both with and without quantum mechanics.However, when density fluctuations were important to be considered for understanding the phenomena, then quantumaspects of the problem were critical. In Tomonaga’s work [33] on collective oscillations it was demonstrated thatmodeling and understanding the many Fermion interaction in solid-state electron gas in a crystal was greatly simplifiedby the use of collective modes of many Fermion oscillations. These collective electron oscillations were first investigatedin 1D in 1950 by Tomonaga through the use of density fluctuation method (where density if the field variable)as opposed to the conventional quantum mechanical method of computation of expectation values from the wavefunctions of the system. This was because the equations of motion of collective oscillations are linear in field variable(density fluctuation) as opposed to bilinear field variable terms in the conventional method. Moreover, linearity of thefield variables in the equation of motion holds irrespective of the presence or absence of inter-particle forces (directinteraction between single particles).Subsequently, in 1953 Pines and Bohm [34] used a collective canonical transformation method to analyze thecollective many Fermion (electron) oscillations in crystal lattices in metals. They recognized the dominance of thelong-range nature of the Coulomb forces which controls the phenomena and produces collective oscillations of cloudsof electrons over spatial scale much greater than Debye length. Here the characteristic dimension of an electron cloudis of the order of a Debye length (in quantum mechanical treatment this characteristic dimension is modified).The collective behavior is therefore critical to explain physical phenomena over micron or nanometer scales. Intheir work the term “plasmon” was introduced to describe the quantum of elementary electron excitation associatedwith this high-frequency collective motion in bulk crystals with the dimensions of the order of one plasmon oscillationwavelength. This is a quantum of energy of collective oscillations of valence electrons. The energy of a plasmon wasshown to be [34, 35], ¯ hω pe = ¯ h (cid:18) πn e m e (cid:19) (1)When the dimension of the solid-state material is below mean free path of the collective electron oscillations, quantiza-tion of electron plasma frequency is observed. Plasmon energy is greater than the energy of any individual conductionband electron. Although, these plasmonic oscillations are the quantum analog of the collective oscillations of plasmaelectrons in gaseous plasmas their extremely high energies (¯ hω pe (cid:29) k B T e ) and small spatial scales necessitate theconsideration of the quantum nature of these oscillations. Typical, valence electron density in crystals which liesin the range of n (cid:39) − cm − result in plasma energies in crystals of ¯ hω pe (cid:39) ω sp = ω pe / √ (cid:15) , where (cid:15) is the dielectric constant of the metal.The dispersion characteristics of surface plasmon and phonon modes have been well characterized in a linearizedperturbative regime [39]. SOLID-STATE SURFACE WAVES IN CRYSTAL TUBE:SURFACE ELECTRON OSCILLATION MODEL IN TUBE NANOSTRUCTURE
Using the collective electron oscillation models described above and under the condition that the dominant behaviorof solid-state media is that of an ideal electron gas, we analytically model surface electron oscillations in a tubestructure driven in the wake of an electron beam. These analytically modeled surface electron oscillations also sustaina propagating surface wave which propagates at nearly the same velocity as the drive beam. Moreover, as collision-lessbehavior dominates it is possible to treat the density fluctuations using a single particle oscillation model.Because the crystal tube under consideration here conforms with a cylindrical geometry, in our analysis we model thesurface electron oscillations in a cylindrical coordinate system. Moreover, as these surface oscillations and the surfacewave sustained by these collective oscillations co-propagate with the electron beam, the longitudinal dimension of thecylindrical coordinates will be transformed to a co-moving frame behind the drive beam. An preliminary analysis ofa similar nature has been previously attempted [43].Classifying the onset of non-linearity and wavelength of the density oscillations both require understanding of theelectron dynamics within the plasma. The plasma considered is one of density 0 for r < r tube and n for r > r tube .An electron or positron driving beam of density n b and volume V b – moving with velocity v b ˆz = cβ b ˆz – perturbselectrons within the plasma electrostatically. Subsequently, the plasma electrons oscillate freely in the radial directionas a result of the electric field that has been set up due to the no longer quasineutral plasma.This electric field is found using Gauss’ law (cid:90) S E · d S = 4 π Q enc (2)where S is a closed Gaussian surface, E is the electric field, d S is the surface area element of S , Q enc is the total chargeenclosed within S , and (cid:15) = 1 / (4 π ) is the permittivity of free space (in cgs units). As the plasma is cylindricallysymmetric, a cylinder of radius r and length l is used as the Gaussian surface S . d S can therefore be simplified to d S = r dθ dz ˆr . Assuming the electric field to be purely radial, E = E r ˆr , the left hand side of Gauss’ law is simplified(after integrating) to 2 πrlE r .The enclosed charge is given by the integral of the ion charge density over the volume enclosed by S (cylinder ofradius, r). The ions within the plasma are of density n . As the plasma density n is constant, the enclosed charge isgiven by the net volume of plasma within S multiplied by en , Q enc = en π ( r − r ) l . Gauss’ law thus gives thefollowing form for the radial electric field set up by the non-quasineutral plasma E r ( r ) = 4 πen r ( r − r ) (3)As the electric field vanishes for r = r tube , equation (3) describes the electric field for an electron situated initially onthe channel wall.The force experienced by a given plasma electron is found by multiplying the electric field by − e , the electroniccharge. Finally, a transformation to the frame of the driving beam ξ = β b ct − z is made. This is to allow for directcomparisons to be made between the model and Particle-In-Cell simulations (section ). The equation of motion is m e d rdξ + 4 πn e c β b r ( r − r ) = 0 (4)where m e is the electron mass. Defining the plasma frequency ω p = (cid:113) πn e m e and ρ = r/r tube , equation (5) isrewritten as d ρdξ + 12 β b (cid:16) ω p c (cid:17) ρ ( ρ −
1) = 0 (5)The above equation is a non-linear second order differential equation and describes the natural oscillations of aplasma electron about the channel wall. The lack of charge within the channel wall gives rise to an asymmetry inthese oscillations. Setting r tube = 0 returns the standard simple harmonic oscillations seen in homogeneous plasmasabout a cylindrical axis. Weakly driven surface charge dynamics:linear surface electron oscillations
Equation (4) is readily solvable when considering small displacements of the electron from the tube wall. Thesesmall displacements are valid for very low driving beam charges or large tube radii, when the electrostatic force ofthe beam acting on the plasma electrons is small. In this limit, r ≈ r tube , and eq.5 is linearized using r − r = ( r − r tube )( r + r tube ) ≈ r ( r − r tube ) (6)where the first term on the right hand side is a second order term and has been removed. The linearised equation ofmotion is thus d rdξ = − ( ω p /c ) β b ( r − r tube ) (7)which has the solution r ( ξ ) = r tube + A sin (cid:18) ω p cβ b ξ (cid:19) (8)where A is a constant. For an ultrarelativistic driving beam, β b = 1, and so an immediate form for the oscillationwavelength in the linear/weakly excited case is λ linear = 2 π cω p (9)which is the well-known result for plasma oscillations in homogeneous plasma. Strongly driven surface charge dynamics:non-linear surface oscillations
Solving equation (4) in general requires calculation of the plasma electron’s initial effective velocity ρ (cid:48) for a givenradial position ρ . Three basic assumptions are made to simplify the calculation to a good approximation.The first is that the driving beam is assumed to be a quasi-static point charge of total charge Q b . This assumptionis valid provided the drive beam density changes over multiple electron oscillations and its charge is conserved. Gauss’law states that the electric field intersecting a Gaussian surface S is the same regardless of the shape of the chargedistribution within S . Quasistaticity ensures the shape or size of the beam do not change significantly over time suchthat beam-plasma intersections do not arise.The second assumption requires that electrons excited by the driving beam are no longer influenced by the drivingbeam beyond the first collapse to the axis. In other words, the primary electron collapse occurs at ξ (cid:29)
0, correspondingan electric potential of approximately zero. This assumption simplifies calculation of the kinetic energy gained by theelectron due to the driving beam, as the radial position of the electron at collapse need no longer be determined.The final assumption is that the kinetic energy gained by the electron is primarily radial kinetic energy. Thissimplifies determination of the electron velocity at the crystal tube wall (section ).
Transforming Surface oscillation equation to first order
As equation (4) is an autonomous ODE (i.e. an ODE with no dependence on ξ ), the following manipulation canbe made: ddξ (cid:34) (cid:18) dρdξ (cid:19) (cid:35) = dρdξ d ρdξ (10)Using the chain rule on the left hand side of (10) ddξ = dρdξ ddρ ρ (cid:48)(cid:48) = ddρ (cid:18) ρ (cid:48) (cid:19) where ρ (cid:48) = dρ/dξ . Equation (11) can then be substituted into (4) and integrated, resulting in12 ρ (cid:48) + 12 β b (cid:16) ω p c (cid:17) (cid:18) ρ − ln ρ (cid:19) = C (11)where C is a constant of integration to be determined. Surface oscillation: Initial Condition for Velocity
To find C , one must know a value of ρ (cid:48) for a given ρ . At ξ = 0, an electron at the surface ( ρ = 1) sees the repulsivepotential (attractive potential) of the electron (positron) beam. As the electron is pulled to the axis, its energy willbe converted from potential energy between it and the beam to potential energy from the no longer neutral plasma.As it recoils back towards the crystal tube wall, the electron gains kinetic energy which will become maximized at ρ = 1 as, beyond ρ = 1, the force will be directed anti-parallel to the electron velocity. Therefore, the kinetic energyof the electron at the crystal tube wall (after the excitation from the beam) is effectively equal to the potential energyit has at ( ρ, ξ ) = (1 ,
0) due to the electron or positron beam under the assumptions given at the start of this section.By letting ρ (cid:48) ( ρ = 1) = ρ (cid:48) , i.e. some initial effective velocity to be determined later, one arrives at an equation for C after substitution into (11) C = ρ (cid:48) + 12 β b (cid:16) ω p c (cid:17) (12)Determining ρ (cid:48) requires consideration of the energy gained by the electron using the simplifying assumptions madein the introduction of this section. The potential energy of an electron due to an electron or positron beam is U ( ρ, ξ ) = − e N b π(cid:15) (cid:112) ρ r + ξ (13)where N b = Q/e is the total number of electrons or positrons in the beam. Due to the conservative nature of thepotential, the kinetic energy E gained by a surface electron due to the electron or positron beam, initially at ξ = 0, is E = U ( ρ , ξ ) − U (1 ,
0) (14)where ( ρ , ξ ) defines the position of the particle when its radial velocity is zero. If it is assumed that ξ is large, then U ( ρ , ξ ) ≈ E = e N b π(cid:15) r tube ≈ m e ˙ r (15)where ˙ r = r tube v b ρ (cid:48) is the initial condition velocity, v b is the beam velocity, and the final term on the right handside assumes that the energy gain occurs primarily in the radial direction. Rearranging equation (15) yields˙ r ≈ e n b m e (cid:15) V b πr tube = ω pb V b πr tube (16)where N b = n b V b , V b is the effective volume of the beam, and n b is the beam density. For a Gaussian beam distributionof width σ r and length σ z , n b is defined as the electron or positron density at the beam’s centre (or the peak density)with V b = σ r σ z √ π . Upon substituting ˙ r = r tube v b ρ (cid:48) into the above equation and rearranging, an approximateform for ρ (cid:48) is determined as ρ (cid:48) ≈ ω pb c β b r / (cid:114) V b π (17)and, for a Gaussian beam profile ρ (cid:48) ≈ ω pb c β b r / (cid:115) σ r σ z √ π (18) Non-linearity parameter and radial boundary conditions
Due to the oscillatory nature of the problem, it is clear that there will exist two solutions for ρ where ρ (cid:48) = 0. Aftersubstituting expressions for C and ρ (cid:48) , equation (11) reduces to ρ − ρ = 1 + 2 n b n V b πr (19)Defining α = 1 + 2 n b n V b πr , equation (19) describes a transcendental equation with two solutions: ρ + = (cid:112) − W − ( − e − α ) (20) ρ − = (cid:112) − W ( − e − α ) (21) α = 1 + 2 n b n V b πr (22) (cid:16) = 1 + 2 n b n σ r σ z √ π r , Gaussian Profile (cid:17) (23)where W − , ( x ) are the decreasing and increasing branches of the lambert W function respectively. Each solutionrespectively describes the amplitude of the crests and troughs of the plasma density oscillations. Looking at theextreme cases for α ρ + → (cid:40) α → ∞ α → ∞ (24a) ρ − → (cid:40) α → α → ∞ (24b)which suggests that, for large plasma densities and tube wall radii or physically small, low density beams, | ρ + − | ≈| ρ − − | yielding a linear wave. In the opposite case, the wave amplitudes are different and thus the wave is non-linear.It is therefore deduced that α must describe the strength of non-linearity of the wave, and that increasing n b , σ r , σ z ,or decreasing n or r tube results in increased non-linearity.Increasing the beam charge will correspond to a stronger driving potential experienced by the plasma electrons. Asa consequence, electrons have more energy to collapse closer to the axis. This is similar for the crystal tube radius;electrons will initially be closer to the driving beam and thus experience a stronger potential. Conversely, decreasingthe plasma density for a fixed beam density will reduce the number of plasma ions which weakens the restoring forceallowing the tube wall electrons to collapse closer to the axis. Wavelength of surface density oscillation: analytical model
Equation (11) can be rearranged in terms of ρ (cid:48) , leading to an integral solution ξ ( ρ ) with no closed form expression: ξ ( ρ ) − ξ = cβ b ω p (cid:90) ρ dx (cid:113) α − (cid:0) x − ln x (cid:1) (25)where ξ is a constant of integration and x is a dummy variable.Equation (25) is restricted to a range spanning half the wavelength of the density oscillation, and a domain of( ρ − , ρ + ). The wavelength is thus λ = 2 (cid:2) ξ (cid:0) ρ + ( α ) (cid:1) − ξ (cid:0) ρ − ( α ) (cid:1)(cid:3) (26a)= 2 cβ b ω p (cid:90) ρ + ( α ) ρ − ( α ) dx (cid:113) α − (cid:0) x − ln x (cid:1) = 2 cω p I ( α ) (26b)where in the last expression 26b, β b = 1. The integral I ( α ) converges to π as α →
1, which is consistent with thelinear solution. However, the wavelength of nonlinear surface oscillations is greater than that in the linear regime bythe factor 2 × I ( α ), λ crunch − in = 2 × I ( α ) 2 πcω p − (27)The ω p dependence in (26b) is as per expectations that the density oscillation wavelength is strongly affected by theplasma density in the tube walls.The parameter α also affects the wavelength in the model, suggesting a dependence of the wavelength on changing n b , σ r , σ z and r tube . Thus α the nonlinearity factor provides a correction to the oscillation wavelength under linearapproximation. The factor I ( α ) does not include the effect of relativistic enlargement of the wavelength of radialoscillations which is a well-known additional factor.This model therefore suggests the wavelength of oscillation may be tuneable by adjusting the plasma density n ,keeping α constant. Conversely, the model suggests the possibility of directly controlling the strength of non-linearitywhile maintaining a constant wavelength, simply by adjusting multiple parameters at once.The dependence on the plasma density, n , outside of the integral agrees with simulation data that adjusting theplasma frequency results in strong changes in wavelength. In addition, the solution suggests that the wavelength istunable by adjusting n while α remains constant. Conversely, the model suggests the possibility of directly controllingthe strength of non-linearity while maintaining a constant wavelength, simply by adjusting multiple parameters atonce. PROOF-OF-PRINCIPLE SIMULATIONS RESULTS:SOLID-STATE ELECTRON OSCILLATIONS WITH PARTICLE-IN-CELL SIMULATIONS
Multi-dimensional Particle-In-Cell simulations using the EPOCH code [44] have been carried out to model collectiveelectron oscillation phenomena in solid-state or crystal plasma. The use of a PIC code for analyzing collective electronoscillations at densities, n > cm − is justified under the conditions where the phenomena is collision-less as wasshown above to be the case under strong excitation of valence electrons in crystals. Moreover, as the length of thedriver particle beam is chosen to be of the order of the wavelength of collective electron oscillations, it is possibleto sustain plasmonic oscillations of electrons without triggering phonons and other mixed modes etc. Furthermore,crystal tube structures are known to naturally have mean free path lengths of several hundreds of nanometers [45].In this work we model the interaction of a crystal tube with intense sub-micron scale electron beam, especially itsbunch length being sub-micron scale. We restrict the choice of maximum crystal tube internal diameter to 1 micron.Currently, crystal tubes are grown by folding multiple layers of mono-atomic sheets into a cylinder and in the processclosing a sheet upon itself. The precise nano-engineering process of growing hundreds of nanometer tube radius isyet to be fully characterized to determine the electron density profile spanning the cross-section from the edge of thetube wall to the axis of the tube.Graphene based carbon nanotubes (CNT) are in recent years shown to be relatively straightforward to manufacturein the sense that sophisticated machinery is generally not required [45]. These crystal tubes have valence electrondensities in the range of 10 − cm − and a mean free path of about ∼ ) or molten glass based tubes withinternal diameters ranging from 200 - 1000nm are sold variously as nano-capillary [46] etc. The gradient of densityat the interface of the tube wall and hole region has been characterized using scanning electron microscope (SEM)and found to be less than its nanometer scale resolution limit. Similarly, the uniformity of the diameter is found tobe quite consistent over several meters of tube length. However, the extent of surface deformities and imperfectionsof the exact tubes grown from atomic monolayer deposition is not precisely modeled or characterized here and willform a part of the future work.Our computational modeling effort characterizes an experimentally realizable interaction scenario due to the recentlyreported (planned) availability of sub-micron scale bunch lengths at Stanford Linear Accelerator Center [10] andpossibly at other accelerator facilities in the near future.This computational modeling effort currently serves as a proof-of-principle and is an attempt to demonstrate thepossibilities that can be opened up by the solid-state tube wakefield acceleration technique. In our modeling weutilize the fact that beams of several hundred nanometer bunch lengths are accessible. For experimental relevancethe modeling effort is carried out under the following constraints:(i) In sec. - an experimentally available beam density of n b = 1 . × cm − is considered. The beam waist-sizeis σ r = 500 nm which is also chosen to be the tube radius r tube = 500 nm . The PIC simulations indicate that a0tube wall electron density of n tube = 2 . × cm − is suitable. This tube wall density may need customizednanoengineering.(ii) In sec. - under an experimental constraint that the beam waist-size exceeds the tube diameter we model abeam-tube interaction scenario such that the beam particles radially in the Gaussian wings of the tube interactwith the tube wall, σ r > r tube while the most intense part of the beam propagates within the tube.(iii) In sec. - we assume that a beam density of n b = 1 . − . × cm − is experimentally accessible. In thiscase the suitable tube wall densities of n tube = 1 . − . × cm − are known to be available commerciallyoff-the-shelf. Beam waist comparable with tube radius, r tube > ∼ σ r Using PIC simulations we model and make preliminary investigations of beam-driven solid-state tube accelerationin a parameter regime where the waist-size of the beam injected into a crystal tube is comparable to the crystal tuberadius.The electron density in the tube walls is chosen to be n = 2 . × cm − with a fixed ion background. In thesimulation results presented below a 2D cartesian grid is chosen such that it resolves the reduced plasmonic wavelengthof λ pe / (2 π ) = 38nm with 15 cells in the longitudinal and 15 cells in the transverse direction. Thus each grid cellin these simulations is about 2.5nm x 2.5nm (the Debye length, conservatively assuming a few eV thermal energy is λ D (cid:39) µ m in longitudinaldirection and at least 7 µ m in the transverse (it is wider in transverse to incorporate wider beams). The tube electronsare modeled with 10 particle per cell of the cartesian grid. Absorbing boundary conditions are used for both fieldsand particles.The electron beam has a γ b = 10 ,
000 (roughly 5.1 GeV) with a Gaussian bunch profile of a fixed bunch length with σ z = 400 nm . Typical beam density of 1 . × cm − is considered in this experimentally relevant modeling effort.The beam is initialized with 16 particle per cell. In order to analyze the interaction, the waist-size of the beam, σ r and the tube radius r tube are varied. The moving simulation box tracks the particle beam. The particle beam isinitialized in vacuum and propagates into the crystal tube before the simulations box begins to move.PIC simulation snapshots in Fig.1 correspond with solid-state tube accelerator interaction parameters of crystaltube radius, r tube of 500nm and beam waist-size, σ r = 500 nm and bunch length, σ z = 400 nm . From this snapshot weobserve that a surface wave is sustained by the oscillations of the electrons across the interface of the tube wall withdensity, n tube = 2 . × cm − . In Fig.1(a) the tube wall density snapshot in real-space shows three distinct spatialoscillations of a surface plasmon sustained by radial electron oscillations across the surface. These snapshots are at asimulation time of 250 fs which corresponds to a beam-tube interaction length of around 72 µm (the interaction hasa delayed start as the beam is initialized in vacuum and pushed to propagate into the plasma).Beam density profiles from PIC simulation for the drive beam with σ z = 400 nm at a density of n b = 1 . × cm − show that the beam electrons experience the transverse or focusing fields of the “crunch-in” wakefields of the surfacewave and exhibit betatron oscillations. This effect of beam density modulation can be prominently observed in thebeam density snapshots presented in later sections in Fig.5(b) and Fig.6(d). These coherent density modulationsof the drive beam were first modeled in an innovative plasma beam dump proposal [47]. This coherent drive beamdensity modulation has been observed and labelled as scalloping in some recent works. In the near-term, beam-tubeinteraction can be experimentally studied by observing the small spatial-scale drive beam density modulations.In Fig.1, the longitudinal (in b) and focusing forces (in c) are shown along with the density wave in real-space.The fields are normalized to the Tajima-Dawson acceleration gradient ( E wb = E = m e cω pe e − or the cold-plasmawavebreaking limit). The Tajima-Dawson limit for the tube density of n tube = 2 . × cm − is E = 13 . − .The simulation results lead to several interesting possibilities. It is observed that although the beam to tube densityratio n b /n tube is only 0.05, the longitudinal wakefields approach (cid:104) E acc (cid:105) (cid:39) . E which is an acceleration gradientof around (cid:104) E acc (cid:105) (cid:39) . − . Moreover, the “crunch-in” behavior results in the excitation of focusing fields of theorder of 0.1 E which is a focusing gradient of several 100GVm − .Fig.2 describes the longitudinal phase-space of beam-tube interaction at around 72 µm in the “crunch-in” surfacewakefields regime. From the longitudinal momentum against transverse momentum is shown in Fig.2(a) and transversereal space dimension in Fig.2(b) it can be observed that only those beam particles that are within the crystal tubeand that experience both the longitudinal as well as the transverse “crunch-in” wakefields undergo acceleration.Longitudinal phase-space plotted against longitudinal real space dimension in Fig.2(c) and the corresponding on-axisfield, electron density lineout in Fig.2(d) demonstrate that the longitudinal dimension of the wakefield is such that thebeam particles in the tail of the beam undergo acceleration. This opens up the possibility of using a single hundreds1 FIG. 1. 2.5D PIC simulation snapshot at around 72 µm of beam-tube interaction showing solid-state tube accelerator withcrystal tube radius of 500nm and beam waist-size, σ r = 500 nm and bunch length, σ z = 400 nm . The beam density is n b = 1 . × cm − whereas the channel wall density of the tube is n tube = 2 . × cm − . Solid-state tube wakefieldaccelerator dynamics extracted from a 2.5D PIC simulation at around 72 µm of beam-tube interaction showing the tube wallelectron density (in a, normalized to n tube = 2 . × cm − ), longitudinal electric field of the surface wave (in b, normalizedto E = 13 . − ) and the focusing field (in c, E − cB ). of nanometer scale bunch to observe a few TVm − acceleration gradients.The energy spectra in Fig.2(e) shows some of the particles in the tail of the beam being accelerated from theinitial beam energy centered around 5110MeV to 5360MeV, a gain of about 250MeV in 72 µm . This gives an averageacceleration gradient over 72 µm of around (cid:104) E acc (cid:105) (cid:39) . − . It is also quite evident from the snapshots in (a) and(b) that only those beam electrons that are within the tube get accelerated in the surface wave, whereas the electronsin the Gaussian wings of the beam undergo minimal energy change.The possibilities of accessing high average acceleration gradients of the order of several TVm − are unprecedentedand being based upon a modeling effort where realistic parameters are utilized call out for an experimental verificationcampaign.2 FIG. 2. Longitudinal momentum phase-space against transverse momentum (in a) and transverse real space dimension (in b)from 2.5D PIC simulation snapshot after around 72 µm of beam-tube interaction with parameters same as Fig.1. Longitudinalphase-space against longitudinal real space dimension (in c) and corresponding on-axis field, electron density lineout (in d)from 2.5D PIC simulation snapshot are also shown around 72 µm of beam-tube interaction. In Fig.3, a comparison of bulk plasma ( n = n tube = 2 × cm − ) and crystal tube wakefields is presentedby plotting side-by-side the electron density (in a,d), longitudinal electric field (in b,e) and the focusing field (in c,f)profiles. It is quite evident from the comparison of the snapshots in (a) and (d) that whereas the surface wave wakefieldamplitude for n b /n = 0 .
05 is significantly high and in the nonlinear regime for a tube radius, r tube = 500 nm , thewakefields driven in homogeneous plasma are almost non-existent due to the low beam to plasma density ratio.3 FIG. 3. Comparison of homogeneous plasma wakefield (in a,b,c) with the crystal tube wakefield (in d,e,f; repeated fromFig.1) from 2.5D PIC simulation snapshot after around 72 µm of beam-tube interaction with parameters same as Fig.1. Thecomparison of (a) and (d) shows that whereas the surface wave wakefield for n b /n = 0 .
05 is in the nonlinear regime, thewakefields driven in homogeneous plasma are almost non-existent due to the low beam to plasma density ratio.
In order to characterize the effect of the ratio of drive beam density to tube wall density, n b /n tube , in Fig.4 wecompare the PIC simulation snapshots of tube electron density for different n b /n tube ratio at a beam-tube interactionlength of 72 µm . These beam densities are: (a) n b = 0 . × cm − = 0 . n tube (b) n b = 1 . × cm − = 0 . n tube (c) n b = 2 . × cm − = 0 . n tube (d) n b = 4 . × cm − = 0 . n tube .The corresponding peak longitudinal on-axis fields or acceleration gradient over varying beam density as extractedfrom PIC simulations are summarized as follows:(a) (cid:104) E acc (cid:105) (cid:39) . E for n b = 0 . × cm − = 0 . n tube (b) (cid:104) E acc (cid:105) (cid:39) . E for n b = 1 . × cm − = 0 . n tube (c) (cid:104) E acc (cid:105) (cid:39) . E for n b = 2 . × cm − = 0 . n tube (d) (cid:104) E acc (cid:105) (cid:39) . E for n b = 4 . × cm − = 0 . n tube It is quite evident that as the drive beam density is increased in “crunch-in” regime of solid-state tube, the surfaceelectron trajectories become increasing nonlinear. The nonlinear surface wave results in wakefields that are not onlyhigher (of the order of the Tajima-Dawson acceleration gradient limit) but also result in the excitation of strongerfocusing fields within the tube.
Beam waist size larger than the tube diameter, σ r > r tube An experimentally accessible parameter regime in the short-term where the beam waist-size is a few times largerthan the crystal tube radius is investigated below using preliminary PIC simulations. In these simulations it is assumedthat the peak of the beam density coincides with the axis of the crystal tube such that the most intense part of theexternally focussed beam travels in the low density region of the tube. The PIC simulation setup and beam densityare as described in sec..From the PIC simulation results that are summarized below in Fig.5, it is quite clear that surface wave wakefieldsin the “crunch-in” regime are sustained within the tube even when σ r > r tube . From the results in this section weobserve that when the beam density is retained same, spatial profiles of the wakefields and the acceleration gradientof the order of 2 . − sustained in the case of σ r > r tube being studied here are nearly equal to the case where σ r ≤ r tube .This excitation of near Tajima-Dawson acceleration gradient (0.1 E wb ) limit in a crystal tube is quite interestingbecause the peak drive beam density of 1 . × cm − is much smaller than the tube wall density of 2 . × cm − .The simulations show an energy gain of about 50MeV in around 23 . µm which is an average acceleration gradient of (cid:104) E acc (cid:105) > . − .Moreover, the “crunch-in” regime wakefields observed in this work show that strong coherent focusing fields arealso excited within the tube of the order of several 100GVm − . It is evident from the beam density snapshots in Fig.5that only those beam particles that are within the tube and which as a result experience the (cid:104) E acc (cid:105) ∼ TVm − -scale4 FIG. 4. Comparison of the “crunch-in” surface wave modes in crystal nanotube for different drive beam densities from 2.5D PICsimulation snapshot after around 72 µm of beam-tube interaction in a crystal tube with wall density, n tube = 2 . × cm − .The drive beam densities are n b =: (a) 0 . n tube (b) 0 . n tube (c) 0 . n tube (d) 0 . n tube . fields of the surface plasmon wave undergo significant density perturbation.From comparison of the density and wakefield characteristics of crystal tube wakefields in Fig.1 for the case of σ r ∼ r tube and the same (not shown) for the case of σ r (cid:29) r tube , it is observed that the wakefield characteristics,amplitude and spatial profile, are quite similar. As the drive electron beam density in the case of Fig.1 and σ r (cid:29) r tube case are equal, n b = 1 . × cm − , it is possible to postulate that drive beams of a given density are equally effectiveat the excitation of crystal tube wakefields irrespective of their transverse properties (given that the peak of the beamspatial distribution is aligned with the axis of the tube).5 FIG. 5. 2.5D PIC simulation snapshots comparing the electron beam density at initialization and after around 23 . µm ofbeam-tube interaction. Scaling to off-the-shelf tube wall densities: n tube ∼ cm − Crystal nanotubes [45] that are currently available off-the-shelf have mass densities in the range of 1 . − . − .For purely Carbon atom based nanomaterial this mass density translates to ionic and electron densities in highlyionized states of between n tube ∼ − cm − . In this section we present our examination of the scaling of the“crunch-in” modes in crystal tubes when the crystal wall densities are around n tube ∼ − cm − . The PICsimulation setup and beam density are the same as described in sec.. These simulations show that if certain beamdensities may be experimentally within reach of existing electron beam facilities, then it may be possible to excitebeam-driven solid-state tube surface wave wakefields using off-the-shelf nanotubes.In the previous sections, sec. and sec., we have presented proof-of-principle PIC simulation results under theconstraint that the beam densities are limited to around n b ≤ × cm − whereas the beam bunch length ischaracterized by σ z (cid:39) n tube ∼ × n b .In this section, we assume that beam densities as high as n b ∼ . × cm − are experimentally accessible usingcurrently available accelerator facilities. With these range of beam densities, using PIC simulations snapshots pre-sented below, we observe that coherent “crunch-in” wakefields supported by collective oscillations of crystal electronsare accessible using nanotube structures that are available off-the-shelf. Therefore, if beam densities of the orderof n b ∼ . × cm − are experimentally accessible, then proof of concept experimental verifications of SOTWAmechanism can be carried out in the near term.In the simulations presented in this section the electron density in the tube walls is chosen to be n = 3 . × cm − with a fixed ion background. A 2D cartesian grid is chosen such that it resolves the reduced plasmonic wavelengthof λ pe / (2 π ) = 10nm with 20 cells in the longitudinal and 20 cells in the transverse direction. Thus each grid cell inthese simulations is about 500˚A x 500˚A (the Debye length is λ D ≤ r tube = 100 nm is here is modeled to have a finite thickness of 250nm, with the outerradial extent of the tube thus terminating at 350nm from the axis of the tube. The cartesian box co-propagates withthe electron beam. The box dimensions span 5 µ m in longitudinal direction and at least 3 µ m in the transverse. Thetube electrons are modeled with 4 particle per cell.The electron beam has γ b = 10 ,
000 (roughly 5.1 GeV) with a Gaussian bunch profile of a fixed bunch length with σ z = 400 nm . In the simulation snapshots presented below the beam density is 5 . × cm − which is experimentallyrelevant. The beam is initialized with 9 particle per cell. A comparison of the PIC simulation results in sec. and sec.provides enough confidence that the critical parameter in beam tube interaction is the beam density, n b with the σ r FIG. 6. Density snapshots for off-the-shelf solid-state tube parameters from 2.5D PIC simulations showing tube electron density(in a,b) with fixed background ions and drive beam electron density (in b,d) after around 36 µm of beam-tube interaction. From(a,c) it follows that “crunch-in” mode is excited and that the beam evolves as it experiences the transverse fields of this mode. to r tube ratio being relatively insignificant. In consideration of this we use a beam with σ r = 250 nm , with a good andpreviously justified approximation that a beam of higher waist-size (for example, σ r ∼ . µm ) but the same densitywill have the same characteristics of beam tube interactions and excite considerably similar wakefields.From these simulation snapshots summarized in Fig.6,7 it is possible to conclude that if beam densities as highas n b ∼ . × cm − are experimentally accessible at current accelerator facilities, then it may be possible toexcite strong “crunch-in” wakefields in off-the-shelf crystal tubes of nominal tube dimensions. In our simulations,the tube has a radius ( r tube ) of 100nm and a wall thickness (∆ r tube ) of 250nm. The beam density is initialized to n b = 5 . × cm − and tube wall density is initialized to n tube = 3 . × cm − , with the n b /n tube = 0 .
17. Thebeam properties are: γ b = 10 , σ z = 400 nm and σ r = 250 nm .From 8 we can infer that a few 10 TVm − acceleration gradient may be experimentally realizable using currentaccelerator facilities. The accelerated energy spectra shown in 8(d) shows the acceleration of a small fraction of thedrive beam from the initial beam energy centered around 5110 MeV to 7570 MeV, a gain of about 2.46GeV in 36 µm gives an average gradient of around (cid:104) E acc (cid:105) (cid:39) . − . It is quite evident from the transverse real-space vslongitudinal momentum snapshot in (a) that only those beam electrons that are within the tube ( r tube = 100 nm ) getaccelerated in the wakefield. This increase in the acceleration gradient simply follows the electron density scaling ofthe Tajima-Dawson acceleration gradient.It is quite attractive to have the possibility to accelerate a part of the 5GeV particle beam by around 2.5GeV insub millimeter-scale crystal tubes while sustaining unprecedented many tens of TVm − acceleration gradients.7 FIG. 7. 2.5D PIC simulation snapshot of the tube wall electron density (in a), longitudinal field (in b) and focusing field (in c)at around 36 µm of beam-tube interaction with exactly the same beam and tube parameters as in Fig.6. From (b,c) both thelongitudinal as well as transverse wakefields are of the order of the Tajima-Dawson acceleration gradient limit, 52TVm − . FIG. 8. Longitudinal momentum phase-space against transverse real space dimension (in a), the same against longitudinal realspace dimension (in c) and the corresponding on-axis field, electron density lineout (in b) from 2.5D PIC simulation snapshotafter 36 µm of beam-tube interaction. DISCUSSION AND FUTURE WORK
In this work we have presented a preliminary analytical and computational model of beam-driven solid-state accel-eration mechanism in crystal tubes. The solid-state tube wakefield acceleration or SOTWA mechanism presented hereutilizes collective electron oscillation modes on and across the surface of a crystal tube. These plasmonic oscillationssustain propagating surface waves driven as the wakefield of a charged particle beam of submicron bunch length(and, ideally submicron waist-size). A tube shaped nanostructured crystal is not only found to offer the possibility tominimize the direct high-intensity interaction of the beam with bulk crystal but also the possibility of excitation ofsignificantly higher wakefield amplitude compared to the direct interaction of same density particle beam with bulkcrystal.The experimentally available submicron scale bunch length (for instance, planned σ z = 400 nm [10]) is shown tohave the potential for resonant excitation of collective electron oscillations in crystal tube. The resonant excitation ofa surface mode in a crystal tube driven by the beam at a given density is shown to be experimentally realizable withina range of tube wall densities. A preliminary analytical model of the crystal tube wall surface electron oscillationshas been presented based upon the seminal works on modeling many body crystal phenomena as collective modesof collisionless Fermi electron gas. Our model currently assumes minimal ion motion over the relevant attosecondtimescales (from our preliminary mobile ion simulations) but crystal lattice ion motion effects will be a major part ofthe future work.In the computational models presented in sec. using beam densities of the order of n b ∼ cm − , as some ofthe electrons in the tail of the drive bunch experience strong wakefields of the tube surface wave, they are shown torapidly gain energy. The average acceleration gradients experienced by the tail particles are shown to be of the orderof several TVm − as per the expectations of the Tajima-Dawson acceleration gradient limit for crystals. The particlesin the tail of the beam gain several hundred MeVs in a few hundred microns under the influence of surface wakefields(sec. and sec.). The possibility of accessing average acceleration gradients that are at least two orders of magnitudehigher than the gaseous plasma wakefield acceleration techniques will pave the way forward in accelerator research.It is further demonstrated that if experimentally accessible beam densities may be of the order of n b ∼ . × cm − with other beam properties being the same, then off-the-shelf crystal tubes of a few hundred nanometer diameter canbe utilized (sec.). With the densities of the these off-the-shelf tubes being of the order of 10 cm − , the accessibleTajima-Dawson acceleration gradients are of the order of 10 TVm − . Our simulations suggest the possibility ofSOTWA fields being at least three orders of magnitude higher than gaseous plasma acceleration technology.The possibility that an increase in the drive beam density allows access to a nonlinear surface wave “crunch-in”regime has been demonstrated. In the “crunch-in” regime both strong transverse fields of the order of many 100GVm − as well as longitudinal wakefields of the order of many TVm − are excited. Thus, this regime using a crystal tube isuseful to control the accelerated bunch transverse properties while the accelerated particles do not directly experiencehigh ion density in their propagation path resulting in the minimization of associated instabilities. Controlled crystal tube photon source:
Moreover, the strong transverse fields of the crystal tube wake makecontrolled and tunable generation of gamma-ray photons as an electron or positron beam particle trajectories undergooscillations during their interaction with “crunch-in” transverse fields of the order of many 100GVm − . The use ofspecifically structured crystal tube such as with a superlattice, allows significantly higher control of the gamma-rayflux as opposed to the uncontrolled filamentation driven interaction in a metal [8]. Nano-modulation of drive beam:
In the very near-term, beam-tube interaction can be experimentally investi-gated by the observation of coherent density modulations of the drive beam [47] which may be related to the effect ofbeam scalloping observed in some gaseous plasma studies elec-Beam-Wakefield-Expt. Beam-tube interaction can beexperimentally diagnosed by observing the small spatial-scale beam density modulations after the interaction.In future work, we will extend the analytical and computational modeling of the SOTWA mechanism presentedhere. Moreover, we will determine the optimal conditions for the excitation of strong many TVm − acceleration(and in-tube focusing) gradients under various tube and beam parameters by modeling the plasmonic surface wavein crystal tube. It is also critical to understand non-ideal conditions of the interaction, such as misalignment of thebeam and tube axis, effect of limited thickness of tube wall and electron density profile of tube, secondary high-fieldionization of the channel walls, ion motion processes, modification of tube density profile due to ablation of the crystaltube to beam irradiation etc. The extent and time-scales of damage caused to the crystal tube structure by the drivebeam, the possibility of reuse in consideration of effects such as atomic stabilization as well as the effect of thesenon-ideal structural properties on collective electron oscillations will also be carefully modeled. Laser Wakefield Accelerator injector for SOTWA:
Our future work will also study the external injectionof the inherently micron-scale electron and positron beams [48] that are accelerated using laser-driven wakefields in0gaseous plasmas and are thus likely to be more accessible. Furthermore, it is well known that the radiation frommuons interacting with the high transverse or focusing field of the tube and undergoing oscillations is significantlysmaller than electrons or positrons ( ∝ ( m e /m µ ) ), we will also model the injection and acceleration of muons. Laseracceleration of muons [49], also put forth and investigated as part of this XTALS 2019 workshop, is modeled to beable to produce ultra-short micron-scale muon beams that are suitable for injection into crystal tube wakefields.Through the proposed extensive modeling effort, our work will seek the parameter regime and feasible diagnosticsfor demonstration of an experimental prototype of possibly many TVm − average acceleration gradient of the SOTWAmechanism. ACKNOWLEDGMENT
A. A. S. was supported by the College of Engineering and Applied Science, University of Colorado, Denver. V. D.S. was supported by Fermi National Accelerator Laboratory, which is operated by the Fermi Research Alliance, LLCunder Contract No. DE-AC02-07CH11359 with the United States Department of Energy. We appreciate valuablediscussions with S. Chattopadhyay, U. Winenands, G. Stupakov and V. Lebedev. This work used the Extreme Scienceand Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant numberACI-1548562 [50]. This work utilized the RMACC Summit supercomputer through the XSEDE program, which issupported by the National Science Foundation (awards ACI-1532235 and ACI-1532236), the University of ColoradoBoulder, and Colorado State University [51]. [1] Hofstadter, R.,
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