Large Scale Simulations for Dust Acoustic Waves in Weakly Coupled Dusty Plasmas

*Aamir Shahzad, Zamar Ahmed, Muhammad Kashif, Amjad Sohail, Alina Manzoor, Fazeelat Hanif, Rabia Waris and Sirag Ahmed*

## **Abstract**

Dust acoustic wave of three-dimensional (3D) dusty plasmas (DPs) has been computed using equilibrium molecular dynamics (EMD) simulations for plasma parameters of Coulomb coupling strength (Γ) and Debye screening (*κ*). New simulations of wave properties such as longitudinal current correlation (LCC) *CL*(*k, t*) function have been investigated for 3D weakly DPs (WCDPs), for the first time. EMD results, *CL* (*k*, *t*) have been simulated for four normalized wave numbers (*k* = 0, 1, 2, and 3). Our simulations illustrate that the frequency and amplitude of oscillation vary with increasing of Γ and *κ*. Moreover, present simulations of *CL* (*k*, *t*) illustrate that the varying behavior has been observed for changing (Γ, *κ*) and system sizes (*N*). Current investigation illustrates that amplitude of wave oscillation increases with a decrease in Γ and *N*. However, there are slightly change in the value of *CL* (*k*, *t*) and its fluctuation increases with an increasing *k*. The obtained outcomes have found to be more acceptable than those that of previous numerical, theoretical, and experimental data. EMD simulation has been performed with an increasing sequence for WCDPs and it serves to benchmark improved approach for future energy generation applications.

**Keywords:** plasma, complex (dusty) plasma, wave properties, weakly coupled (dusty) plasma, current correlation function, equilibrium molecular dynamics simulation

## **1. Introduction**

Nowadays, it has been seen complex liquids have attracted considerable interest for many researchers. Complex liquids have introduced themselves by emerging the technology of all processes. Different techniques such as experimental, theoretical, and simulation are used to study the behavior of the complex liquid. Unambiguous models have been used to describe physical properties for specific ranges of temperature and pressure. Explicit equations can also be used to calculate the thermophysical

properties of fluids when there is not any literature on complex fluids present. To study the behavior of complex fluids, information on thermophysical properties should be required. Transport properties of complex fluids play an essential role in laboratory and industrial applications. Moreover, thermophysical properties have played a very dominant role in optimization and system design. For the past few years, research on transport properties has occurred through several techniques such as experimental, theoretical, and simulation. Experimental research on transport properties has resulted in non-absolutely convergent expression because there is no accurate and precise experimental calculation of transport properties [1].

#### **1.1 Plasma**

Plasma makes up 99% of the matter in space and is referred to as the fourth state of matter. In essence, plasma is a sort of electrified gas in which atoms have split into positive ions and electrons. It is a type of matter in a variety of branches of physics, including astrophysics, technical plasma, and terrestrial plasma. Plasma is created artificially in a lab and used for a variety of technical applications, including fusion energy research, display, fluorescent lighting, and more. The term "plasma" was initially used by American scientist Irving Langmuir, who defined it as "a quasineutral gas of charged particles that exhibits collective behavior." When the number of ions equals the number of electrons (ni ≈ne ≈n), a gas is said to be "quasi-neutral," meaning it becomes electrically neutral. Where n is the number density, ni is the ion density, and ne is the electron density. The electric field and coulomb potential cause charged particles to collide when they exhibit collective behavior. The usage of plasma in research and technology is widespread. It is extremely important in daily life. Plasma is employed in many aspects of daily life, including lasers, sterilizing medical equipment, lightning, high-intensity beams, purifying water, and many more [2].

#### **1.2 History of plasma**

Plasma was initially defined in 1922 by American scientist Irving Langmuir, who was the only one to do so. Various academics began studying plasma physics in 1930 after being motivated by some particle puzzles. Hanes Alfven developed hydromagnetic waves in 1940; these waves are known as Alfven waves. He also mentioned that astrophysical plasma would be studied using these waves. Beginning around the same time in the Soviet Union, Britain, and the United States in 1950, research on magnetic fusion energy was initiated. The study of magnetic fusion energy was regarded as a subfield of thermonuclear power in 1958. This study was initially conducted under strict confidentiality, but after it became clear that the military did not appreciate controlled fusion research, the above-mentioned three countries made the research publicly available. Because of this, additional nations may take part in plasma-based fusion research. By the end of 1960, a Russian Tokomak configuration produces plasma with various plasma characteristics. Numerous sophisticated tokamaks that were created between 1970 and 1980 validated the effectiveness of the tokamak. Additionally, the Tokamak experiment nearly succeeded in achieving fusion break, and in 1990, work on DP physics started. When charged particles are absorbed by plasma, they split into four different components: electrons, ions, neutral particles, and dust particles and dust particles change plasma properties which are called as "Dusty plasma" [3].

*Large Scale Simulations for Dust Acoustic Waves in Weakly Coupled Dusty Plasmas DOI: http://dx.doi.org/10.5772/intechopen.108609*

#### **1.3 Dusty plasma**

When charged particles are submerged in plasma, the plasma properties become more complicated. This causes the plasma to be referred to as dusty plasma, which is also known as complex plasma. DP typically contains an extra charged particle and is electron-ion plasma. DPs contain dust particles of sizes ranging from tens of nanometers to hundreds of microns. The size of the particle is 3e-8g. This charged component can also be thought of as a micron-sized dust particle. Ice or metallic particles could make up dust particles. Different sizes and types of dust particles can be found throughout the universe and in the atmosphere. Although it typically takes the form of a solid, it can also be liquid or gaseous. The movement of ions and electrons can charge up dust particles. The electric potential of charged dust particles, which ranges from 1 to 10 V, is altered by the electric and magnetic fields. In a lab, dust particles can be grown. The existence of dust particles in plasma has exposed advanced fields for researchers and scientists. The majority of plasma in the universe is DP. It occurs frequently in the atmospheres of stars, the sun, planetary rings, galaxies, cosmic radiation, and the earth's magnetosphere and ionosphere. Based on the ordering of many radii and characteristic lengths between particles interacting (*rd*, *λD*), plasma with dust particles can be called either "dusty plasma" or "dust in plasma". If the *λD* > *rd* then it is called "dusty plasma" and if *λD* < *rd* then it is called "dust in plasma". Here, *λD* is dust particles' Debye length and *rd* is the interparticle distance. These are the conditions for DP [3, 4].

#### *1.3.1 History of dusty plasma*

Plasma is a very advanced field of science and technology. Many researchers have been motivated by their achievements in the field of science, medical field, and technology. In 1924, the term "plasma" first time defined by Irving Langmuir. Contemporary research on DP also has motivated many researchers and has become an interesting field of science and technology. In 1980, very exciting incident happened in the field of DP for the Saturn ring. In 1997, Mendis discovered a bright comet by a distant ancestor. The "distant ancestor" is an extraordinary comic laboratory for the investigation and study of interactions between dust particles and their physical and dynamic behaviors. Further manifestations of DPs were noctilucent clouds, origin nebula (which can be seen from the naked eye), zodiac light, etc. Images that are taken from the Saturn ring have shown that dust particles rotating around the Saturn ring have shape of spokes. The research from the last analysis has shown that these dust particles are fine particles. In 2000–2017, the present condition of DP is stable and is playing a main dominant role in science, industries, technology, energy sectors, and medical stores [5].

#### *1.3.2 Charge on dust particle*

Charging of dust grains can be drained through different processes involving the background plasma (electron and ion) being bombarded on surface of dust particles, secondary electron production, ion sputtering, photoelectronic emission by UV radiations, etc. Mostly, the charge on dust particles is negative in low-temperature laboratory plasma. When an electron from the background plasma strikes the surface of dust particle and is attached and lost from the plasma. Because electrons are more moveable than ions, the surface dust grain collects electrons, attracts ions, and repels

electrons until the state of immobility is attained. This charge is accountable for a long lifetime of particles and confinement in plasma [4]. Other collective phenomena and wave instabilities are created due to interactions between these particles.

## **1.4 Types of dusty plasma**

Dusty plasma is classified by the Coulomb coupling parameter.


## *1.4.1 Weakly coupled plasma*

Weakly coupled plasma (WCP) is described by Coulomb coupling parameter Г. Plasma is referred to as WCP when the coupling parameter's value becomes negligible. Hot plasma is another name for WCP. Plasma is referred to be hot plasma or ideal plasma when the temperature of the electron reaches the same level as the temperature of the ion (Te ffiTi). In a laboratory discharge tube with high gas pressure, hot plasma is produced. The flame, atmospheric arcs, and sparks are examples of hot plasma. WCDP lacks a defined shape due to the extremely weak interactions between interacting particles at high temperatures and low densities. The EMD simulation has been used to examine the occurrence of diffusion motion in WCDPs [6].

#### *1.4.2 Strongly coupled plasma*

Strongly coupled plasma (SCP) is defined by the Yukawa potential, also known as the screened coulomb coupling potential (Γ). Coulomb coupling potential (Γ) is the ratio of potential energy to kinetic energy. Whenever potential energy exceeds unity, then kinetic energy becomes as Γ > 1. Its mean SCP is also defined as cold plasma since potential energy increases from inter-particle kinetic energy and plasma particles change into crystalline form. Cold plasma is used for teeth sterilization and food processing. Charged particles in SCP are influenced by the electric field, but the influence of the magnetic field is ignored for this kind of cold plasma.

#### **1.5 Formation of dusty plasma in the laboratory**

Different methods have been expanded for the production of DP in laboratories. Modified Q machines, rf discharges, and dc discharges are different methods that have been used in the last few years. Modified Q machine is single-ended machine used for the production of DP which allows the dust grains to dispersion over a cylindrical plasma column portion. In dc neon glow discharge stratum, DP is produced by a micron-sized dust particle. The discharge is created with cold electrodes in cylindrical glass tube. The distance between electrodes is 40 cm. The neon pressure is varied from 0.2 to 1 Torr and the fluctuation of the discharge current is 0.4–2.5 mA. DP is restrained in a cylindrical symmetric rf plasma system. The system of rf discharges consists of grounded electrodes, hollow outer electrode capacitive coupled to a 14 MHz rf power amplifier, and glass window. Man-made DP or DP generated in laboratory is dust in fusion devices, rocket exhaust, dust in space stations, dust precipitators, and thermonuclear fireballs. Other applications of DP are microelectronics

*Large Scale Simulations for Dust Acoustic Waves in Weakly Coupled Dusty Plasmas DOI: http://dx.doi.org/10.5772/intechopen.108609*

fabrications; plasma enhanced chemical vapor deposition (PECVD), flat panel displays, solar cells, semiconductor chips, dusty plasma devices (DPDs) are used to produce DP in the laboratory. Ordinary flames, flame of candlelight, strong passion, fire, and blaze produced by burning gas are basically weakly ionized plasma that contains dust particles. Thermionic electron emission of 10 nm dust particles increases the degree of ionization in ordinary hydrocarbon flames. DP also contains volcanos and charged snow [7].

#### **1.6 Acoustic modes**

Acoustic modes are categorized into two different types such as collision-less, uniform, and unmagnetized dusty plasma. These are dust acoustic (DA) and ionacoustic (IA) waves.

#### *1.6.1 Dust acoustic (DA) wave*

Dust Acoustic Wave (DAW) is a low-frequency type of longitudinal wave. Basically, DAW is a sound wave when it travels through complex fluid which causes the oscillation between charged particles. The mass of dust particles is considered very important because it provides inertia which is used to sustain DAW, pressure of ions and electrons affords the restoring force. The dust acoustic wave is examined in DP and presented the dispersion relation using the Boltzmann distribution of ion and electron density [8].

#### *1.6.2 Ion acoustic (IA) wave*

The propagation of sound waves in ordinary gas is absurd due to the absence of collision between particles. Thus, in ionized gas, there is no possibility of the occurrence of sound waves due to the absence of collision. However, in the case of plasma, ions can transmit vibrations to each other due to their charges. IA wave is a constant velocity wave. The occurrence of IA wave depends upon the thermal motion of ions. It is a longitudinal wave in magnetized plasma. It demonstrates reflection, diffraction, and interference phenomena. The phase velocity of electron thermal speed is much greater than the IA wave. Particle displacement, particle velocity, sound intensity, and sound pressure are essential quantities of IA waves.

### **2. Models and numerical simulations**

Interactions of the dust particle are studied by many mathematical models such as "in the center of potential well and repulsion potential, in rigid sphere". There are a number of interaction potentials for the calculations of complex systems for instance DP. However, in the best of accurate experimental and theoretical balance, the Yukawa potential is more favorable interaction potential for WCDPs and is also used in many other physical systems (for instance, medicine and biology systems, chemicals, astrophysics environmental and physics of polymers, etc.) [9]. For charged particles, Yukawa potential has the following form [5]:

$$\phi(|r|) = \frac{Q^2}{4\pi\varepsilon\_o} \frac{e^{\frac{-|r|}{\varepsilon\_D}}}{|r|} \tag{1}$$

In the above equation, *r* represents inter-particle distance, Debye screening length is represented by *λD*, the permittivity of free space is represented by *ε*0, and charge on dust particle is represented by *Q*. System particle is expressed by plasma parameters. Coulomb coupling strength (Γ) is defined as the potential energy of interaction between particles and the average kinetic energy of the particles. The Γ defines the distribution of plasma. Γ is given in this form of:

$$
\Gamma = \left(\frac{Q^2}{4\pi\varepsilon\_o} \frac{1}{a\_{WS}k\_BT}\right) \tag{2}
$$

In Eq. (2),*T* is absolute temperature, *kB* is Boltzmann constant, and *aws* represents Wigner Seitz (*WS)* radius and its value is (3/4*πn*) 1/3 with *n* representing number density (*n* = *N*/*V*) of dust particle. There is another parameter, *k* equal to 2*π*/*L* which affects the system of WCDPs. Where *L* represents the computational box length and *k*\*= 2*awsπ*/*L* is the normalized value of *k*. The *Fi*= P *j Fij* is the force experienced on all dust particles. The force on *ith* particle is exerted by other particles. Debye screening strength is the ratio of the interparticle distance a (Wigner Seitz radius) to the Debye length λ<sup>D</sup> [3, 5, 7, 10]. The value of *κ* can be represented in this form:

$$\kappa = \frac{a}{\lambda\_D} \tag{3}$$

In this part, the EMD simulation is reported of *CL* (*k*, *t*) of WCDP for large domain of plasma parameter of (0.9 ≤Г≤0.1) and (1≤ *κ* ≤ 3) along with the varying value of wave number (*k* = 0, 1, 2 and 3). The accuracy and correctness of our proposed model for DAWs *CL* (*k*, *t*) is checked with different system sizes (*N* = 500, 2048, and 4000).

For the system of 3D Yukawa system, time scales in WCDP are categorized by the inverse of plasma frequency as (*ω<sup>p</sup>* �1 ) [1].

$$
\rho\_p^{-1} = \left(\frac{nQ\_d^2}{\varepsilon\_o m}\right)^{\frac{1}{2}}\tag{4}
$$

In Eq. (4), dust particle mass and number density are represented by *m* and *n*. In EMD simulation, the number of particles is *N=*500 to 4000 (the system size will be increased on the validity of computational power) placed in computational box.

#### **2.1 Current correlation function**

Local density provides information about the distribution of atoms. It is possible to study the local variations in motion of atoms. The momentum and particle current for single atomic particles are calculated by molecular dynamics simulation [9].

$$\pi(r,t) = \sum\_{j} v\_{j} \delta(r - r\_{j}(t)) \tag{5}$$

*Large Scale Simulations for Dust Acoustic Waves in Weakly Coupled Dusty Plasmas DOI: http://dx.doi.org/10.5772/intechopen.108609*

With Fourier transformation

$$\pi(r,t) = \sum\_{j} v\_{j} e^{-ik.r\_{j}(t)} \tag{6}$$

Fourier transform equation is used to measure the LCC function.

$$C\_{a\beta}(k,t) = \frac{k^2}{N\_m} \left(\pi\_a(k,\ t)\pi\_\beta(-k,0)\right) \tag{7}$$

In Eq. (7), *k* is the wave vector. The equation given above can also be stated in form of the longitudinal and transverse current correlation function.

$$\mathbf{C}\_{a\beta}(k,t) = \frac{k\_a k\_\beta}{k^2} \mathbf{C}\_L(k,t) + \left(\delta\_{a\beta} - \frac{k\_a k\_\beta}{k^2}\right) \mathbf{C}\_L(k,t) \tag{8}$$

Longitudinal and transverse current functions are acquired in X, Y, and Z directions by locating *k* are equal to *kẑ*.

## **2.2 Normalized longitudinal current correlation function C***<sup>L</sup>* **(***k***,** *t***)**

Fourier transformation is used to write Eq. (7) into LCC functions which are normalized

$$\mathbf{C}\_{L}(k,t) = \frac{k^2}{N^m} \langle \boldsymbol{\pi}\_Z(k,\ t)\boldsymbol{\pi}\_Z(-k,\ \mathbf{0})\rangle \tag{9}$$

By putting π<sup>z</sup> value in Eq. (9), we obtain

$$C\_L(k,t) = \frac{k^2}{N^m} \left\langle \sum v\_j e^{-ik \cdot Z\_j(t)}(k,\ t) \sum v\_j e^{-ik \cdot Z}(k,t) \right\rangle \tag{10}$$

Eq. (10) explains charge particle transports in longitudinal motion. Longitudinal current correlation function describes that wave is propagating in negative and positive directions of the Z-axis.

#### **2.3 Parameters and simulation techniques**

This segment shows that system contains *N =* 500–4000 particles. Yukawa potential causes the interaction between these numbers of particles which is shown in Eq. (1) and contain in the cubic region. Moreover, this segment illustrates an overview of results attained through EMD simulation for longitudinal current correlation function which is shown in Eq. (10). The simulation takes place in simulation box and the simulation box dimension is selected as LX, LY, and LZ. In our case, we have used the EMD simulation to calculate the longitudinal current correlation function for *N =* 500 number of particles at plasma coupling parameter Г = (0.1–0.9) with *κ* = (1.4, 2.0 and 3.0) at four varying values of wave numbers *k* = (0, 1, 2 and 3). The EMD simulation has been used to investigate shear viscosity and dynamical structure S(*k,ω*) of SCDPs [5, 7, 10–12].

## **3. EMD simulation results**

## **3.1 Time-dependent longitudinal current correlation function C***<sup>L</sup>* **(***k***,** *t***), at** *κ* **= 1.4**

**Figure 1.**

*The fluctuation of C*<sup>L</sup> *(*k*,* t*) as function of time of WCDP at system size* N *= 500,* κ *= 1.4, wave numbers (*k *= 0, 1, 2 and 3) for six different plasma parameters (a) Г = 0.1, (b) Г = 0.2, (c) Г = 0.3, (d) Г = 0.5, (e) Г = 0.7, and (f) Г = 0.9.*

## **3.2 Time-dependent longitudinal current correlation function C***<sup>L</sup>* **(***k***,** *t***), at** *κ* **= 2.0**

Computational results using EMD simulation have shown in **Figures 1** and **2** for weakly coupled Yukawa system LCC, C*L* (*k*, *t*) as a function of time at *κ* (= 1.4, 2.0 and 3.0) for six different values of Г (= 0.1, 0.2, 0.3, 0.5, 0.7 and 0.9) and size of system is *N*=500 and four values of wavenumbers *k* (= 0, 1, 2 and 3) which are normalized. **Figures 1** and **2** explain main results for every three different values of *Large Scale Simulations for Dust Acoustic Waves in Weakly Coupled Dusty Plasmas DOI: http://dx.doi.org/10.5772/intechopen.108609*

**Figure 2.**

*The fluctuation of C*<sup>L</sup> *(*k*,* t*) as function of time of WCDP at system size* N *= 500,* κ *= 2.0, wave numbers (*k *= 0, 1, 2 and 3) for six different plasma parameters (a) Г = 0.1, (b) Г = 0.2, (c) Г = 0.3, (d) Г = 0.5, (e) Г = 0.7, and (f) Г = 0.9.*

the Debye screening parameter (*κ* = 1.4, 2.0, 3.0) of 3D Yukawa complex liquids. It is obtained from six panels of **Figure 1** that the amplitude and wavelength of LCC C*L* (*k*, *t*) increase with decrease of Г (0.9–0.1). It is also obtained that by increasing wave numbers *k* (0, 1, 2, and 3), the amplitude of longitudinal current increases. It is noted that the C*L* (*k*, *t*) value for *k* = 2 and 3 shows maximum sinusoidal wave behavior at lower value of Г from **Figures 1a–d** and **2a–c***.* The sinusoidal wave behavior decreases with an increase of Г and damping of the wave increases with an increase of time and Г.

## **4. Summaries**

Time-dependent LCC function, C*<sup>L</sup>* (*k*, *t*) has been studied for weakly coupled Yukawa system through EMD simulation over a wider range of plasma parameters (0.9 ≤ Γ ≤ 0.1), and (1.4 ≤*κ* ≤ 3.0) along with varying wave numbers values (*k* = 0, 1, 2, 3). The behavior of the Weakly Coupled Yukawa system using EMD simulation has not been studied yet. It is observed that the amplitude and frequency C*<sup>L</sup>* (*k*, *t*) of wave of oscillation increase when the value of Г (0.9–0.1) decreases. C*<sup>L</sup>* (*k*, *t*) value also increases when wave number *k* increases. The amplitude of C*<sup>L</sup>* (*k*, *t*) steadily decreases with an increase in the number of particles *N*. It is implied that the presented simulation is based on Yukawa potential which has been employed to investigate the propagation of waves in WCDP. The presented simulation provides a great understanding of the propagation of the wave in WCDP. The employed simulation affords further respectable data for propagation in WCDP. It has been observed from these simulations that waves are repeatedly propagated at intermediate and higher values of Γ. The frequency and amplitude of waves are examined at different mode of waves.

## **Acknowledgements**

This work was partially sponsored by the National Research Program for Universities (NRPU) with (No. 20-15251/NRPU/R&D/HEC/2021) for utilization of manpower resources. Moreover, this work also was partially supported by the industrial collaborative work between Government College University Faisalabad (GCUF) and Metal Industries Development Complex (MIDC) Sialkot, TEVTA, Govt. of Punjab, Pakistan, as a position of Consultant in a proposed project, for providing facilities to test and run our computer Experiment.

## **Abbreviations**


*Large Scale Simulations for Dust Acoustic Waves in Weakly Coupled Dusty Plasmas DOI: http://dx.doi.org/10.5772/intechopen.108609*

## **Author details**

Aamir Shahzad\*, Zamar Ahmed, Muhammad Kashif, Amjad Sohail, Alina Manzoor, Fazeelat Hanif, Rabia Waris and Sirag Ahmed Modeling and Simulation Laboratory, Department of Physics, Government College University Faisalabad (GCUF), Faisalabad, Pakistan

\*Address all correspondence to: aamirshahzad\_8@hotmail.com; aamir.awan@gcuf.edu.pk

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

[1] Shahzad A, He MG. Thermal conductivity calculation of complex (dusty) plasmas. Physics of Plasmas. 2012;**19**(8):083707

[2] Chen FF. Introduction to Plasma Physics and Controlled Fusion. New York: Plenum Press; 1984. pp. 19-51

[3] Bellan PM. Fundamentals of Plasma Physics. New York, US: Cambridge University Press; 2008

[4] Rodriguez IJ. Some Assembly Required: Computational Simulations of Dusty Plasma. Portland, US: Portland State University; 2018. p. 622

[5] Shahzad A, Shakoori MA, Mao-Gang HE, Feng Y. Numerical Approach to Dynamical Structure Factor of Dusty Plasmas: Plasma Science and Technology-Basic Fundamentals and Modern Applications. Croatia/London, UK: INTECH Publisher; 2019

[6] Shahzad A, He M, Nikitin M. Thermal conductivity and non-newtonian behavior of complex plasma liquids. In: Thermoelectrics for Power Generation-A Look at Trends in the Technology. Rijeka: InTech; 2016

[7] Shukla PK, Mamun AA. Introduction to Dusty Plasma Physics. Florida, US: CRC Press; 2015

[8] Shukla PK. Twisted dust acoustic waves in dusty plasmas. Physics of Plasmas. 2012;**19**(8):083704

[9] Rapaport DC. The Art of Molecular Dynamics Simulation. New York, US: Cambridge University Press; 2004

[10] Shahzad A, He MG. Thermal conductivity of three-dimensional Yukawa liquids (Dusty Plasmas).

Contributions to Plasma Physics. 2012; **52**(8):667-675

[11] Merlino R. Dusty plasma physics: Basic theory and experiments. Summer College on Plasma Physics. 2007;**30**: 1-130

[12] Xu W, Song B, Merlino RL, D'Angelo N. A dusty plasma device for producing extended, steady state, magnetized, dusty plasma columns. Review of Scientific Instruments. 1992; **63**(11):5266-5269

## **Chapter 4**

## Future Particle Accelerators

*Javier Resta López*

## **Abstract**

Particle accelerators have enabled forefront research in high energy physics and other research areas for more than half a century. Accelerators have directly contributed to 26 Nobel Prizes in Physics since 1939 as well as another 20 Nobel Prizes in Chemistry, Medicine and Physics with X-rays. Although high energy physics has been the main driving force for the development of the particle accelerators, accelerator facilities have continually been expanding applications in many areas of research and technology. For instance, active areas of accelerator applications include radiotherapy to treat cancer, production of short-lived medical isotopes, synchrotron light sources, free-electron lasers, beam lithography for microcircuits, thin-film technology and radiation processing of food. Currently, the largest and most powerful accelerator is the Large Hadron Collider (LHC) at CERN, which accelerates protons to multi-TeV energies in a 27 km high-vacuum ring. To go beyond the maximum capabilities of the LHC, the next generation of circular and linear particle colliders under consideration, based on radiofrequency acceleration, will require multi-billion investment, kilometric infrastructure and massive power consumption. These factors pose serious challenges in an increasingly resource-limited world. Therefore, it is important to look for alternative and sustainable acceleration techniques. This chapter pays special attention to novel accelerator techniques to overcome present acceleration limitations towards more compact and cost-effective long-term future accelerators.

**Keywords:** high energy collider, Plasma wakefield, dielectric accelerator, solid-state plasma, plasmonic accelerator

### **1. Introduction**

Currently, there are more than 30,000 particle accelerators in operation around the world, with a strong impact on science, industry and economy. Although, historically high energy physics (HEP) has been the main driving force for the accelerator R&D, only 1% of particle accelerators worldwide are used for research purposes, the rest are mostly used to support commercial, industrial and medical work. For instance, active areas of accelerator applications include radiotherapy to treat cancer, production of short-lived medical isotopes, synchrotron light sources, free-electron lasers, beam lithography for microcircuits, thin-film technology and radiation processing of food.

Since the pioneering work by Gustave Ising, Rolf Widerøe, Ernest Lawrence and many others in the 1920s and 1930s, the accelerator technology has progressed immensely [1]. Particularly, during the past 70 years, an impressive development of high energy colliders, with ever-increasing luminosities and centre-of-mass energies, has resulted in a number of fundamental discoveries in particle physics. In recent decades, experiments at hadron colliders, such as the Super Proton Synchrotron (SPS) and Tevatron, at *e*þ*e*� colliders, such as the Large Electron-Positron Collider (LEP) and the Stanford Linear Collider (SLC), and at the electron-proton collider HERA (Hadron-Electron Ring Accelerator) have explored the energy range up to several 100 GeV and established beyond doubt the validity of the Standard Model in this range of energies [2]. In this tremendous progress, the development and optimisation of the radiofrequency (RF) technology has played a key role. Nowadays, most particle accelerators are driven by RF electromagnetic fields. Currently, the largest and most powerful RF-driven accelerator in operation is the Large Hadron Collider (LHC) at CERN [3], which accelerates protons to multi-TeV energies in a 27 km high-vacuum ring. Its kilometric size and complexity have earned it the name of the 'Cathedral' of the particle physics. With the discovery of the Higgs boson in 2012, the LHC has contributed to complete the Standard Model [4, 5].

To go beyond the maximum energy capabilities of the LHC, the next generation of RFdriven colliders under consideration would be among the largest and most complex facilities built on Earth, requiring intensive R&D and multi-billion Euro investments. For instance, the long-term goal of the Future Circular Collider (FCC) study hosted by CERN [6, 7] is the design and construction of a 100 TeV hadron collider in a 100 km long tunnel. Furthermore, future linear *e*þ*e*� colliders based on RF technology, such as the International Linear Collider (ILC) [8] or the Compact Linear Collider (CLIC) [9], are designed to produce acceleration gradients of between 30 MV/m (ILC) and 150 MV/m (CLIC, 30 GHz frequency operation mode). These machines must, therefore, be tens of kilometres long to reach the desired beam energies, 125 GeV (ILC) and 1.5 TeV (CLIC). Another big project proposed by the HEP community is a high luminosity multi-TeV muon collider [10]. However, practically all the projects mentioned above, based on conventional RF acceleration technology, present serious challenges in terms of size and cost per GeV of beam energy. They need large-scale infrastructure (10–100 km scale) and, in most cases, a billion investment. To afford such a cost, it usually requires a consortium of several countries. Another key challenge is the sustainability in terms of the AC wall plug power consumption. High energy colliders are really 'hungry energy' machines that require hundreds of MW AC wall power to operate. To reduce such a power consumption in the latest years there have been a trend towards more 'green accelerators'. This concept includes the investigation of different ways to reduce energy consumption. For instance, through the improvement of the efficiency of existing technologies, e.g., increasing the efficiency of klystrons [11]. Also the design of Energy-Recovery Linacs (ERLs) can be considered as part of the R&D effort aimed at increasing energy efficiency [12].

To tackle the limitations of conventional accelerators, nowadays there is also an intensive R&D program on novel advanced concepts, some of which will be briefly described in the next sections. These alternative acceleration techniques might define the long-term future of particle accelerators, thus transforming the current paradigm in collider development towards more sustainable, compact and low-cost machines. For a more exhaustive review of past, present and future accelerators and colliders, see for example Refs. [2, 13, 14].

## **2. Plasma wakefield**

Since the 1930s the RF acceleration technology has practically dictated the development of high energy colliders. Their power, cost and size have evolved with the

## *Future Particle Accelerators DOI: http://dx.doi.org/10.5772/intechopen.106340*

#### **Figure 1.**

*Conceptual drawing of a plasma wakefield accelerator [17]. A drive pulse, either an intense laser (laser-driven) or a short electron bunch (beam driven) crossing a plasma, pushes away electrons. The much heavier ions are left behind. This charge separation makes a very large electric field. As the electrons rush back to their original position they overshoot forming a plasma wave, which induces a strong electric field behind the driver (wakefield). If a trailing charged particle pulse is positioned at the right spot seeing the right longitudinal wakefield phase behind the driver, it can be accelerated.*

continuous improvements and optimisation experienced by RF cavities. However, over the recent decades, it has become apparent that RF technology is reaching its limits in terms of achievable accelerating gradients. Currently, the maximum electromagnetic fields that it can support are limited to approximately 100 MV/m due to surface breakdown [15]. To overcome this limit, Tajima and Dawson [16] proposed an alternative solution, a plasma accelerator based on laser-driven wakefields. Since plasma<sup>1</sup> is already a broken-down medium, there is no breakdown limit, compared to the conventional metallic RF cavities.

In general terms, the concept of Plasma Wakefield Acceleration (PWFA) is illustrated in **Figure 1**. A drive pulse enters the plasma and expels the electrons of the plasma outward. The plasma ions move a negligible amount due to their higher mass with respect to the electrons. In consequence, a positively charged ion channel is formed along the path of the drive pulse. The drive pulse can be either a short laser pulse (Laser Wakefield Acceleration [LWFA]) [16] or an electron bunch PWFA [18]. After the passage of the driver (laser or particle beam), the plasma electrons rush back in, attracted by the transverse restoring force of the ion channel. In this way a space charge-driven oscillation is excited, generating alternating regions of negative and positive charge, thus inducing a strong longitudinal electric field behind the driver, the so-called plasma wakefield. Therefore, if a witness charged particle bunch is injected behind the driver at the correct distance and phase, then it will be accelerated with high gradients, in some cases exceeding 100 GV/m.

In the last decades, acceleration of relativistic electrons in gaseous plasma has been successfully demonstrated in several experiments, see for example [19–21]. In the same way, PWFA of positrons has also been demonstrated [22–24]. Furthermore, the

<sup>1</sup> A plasma is defined as a fluid of positive and negative charges.

Advanced Wakefield Experiment (AWAKE) at CERN has recently shown that 10– 20 MeV electrons can be accelerated to GeV energies in a plasma wakefield driven by a highly relativistic self-modulated proton bunch [25].

Previously, we have mentioned injection of a witness beam to be accelerated by the induced wakefield. In addition, internal injection of electrons from the plasma itself is also possible in the so-called 'bubble regime' [26–28].

The wakefields generated in plasma can be evaluated by analytical expressions [29]. The maximum accelerating field in a plasma accelerator can be estimated as follows:

$$E\_\mathbf{z} \text{[V/m]} \approx \Re \delta \sqrt{n\_e \text{[cm}^{-3}]} \tag{1}$$

where *ne* is the electron density in the plasma. This density determines the plasma frequency, *<sup>ω</sup><sup>p</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *e*<sup>2</sup>*ne=meε*<sup>0</sup> p , with *me* and *e* the electron rest mass and charge, respectively, and *ε*<sup>0</sup> the electrical permittivity of free space. For example, a typical plasma density *ne* <sup>¼</sup> <sup>10</sup><sup>18</sup> cm�<sup>3</sup> gives *Ez* � 100 GV/m, that is, approximately three orders of magnitude higher than the maximum gradient obtained in RF structures.

Multi-TeV *e*þ*e*� concepts based on staged plasma cells have been proposed [30– 32], see **Figure 2**. However, one of the main drawbacks of plasma accelerators is the production of relatively poorer beam quality (higher energy spread and transverse size) with respect to conventional RF techniques. In order to tackle this problem and produce a reliable and competitive plasma-based accelerator technology, a big research consortium, EuPRAXIA, was recently established [33]. A comprehensive

#### **Figure 2.**

*Conceptual design of an LWFA-based linear collider. Both the electron and positron linacs consist of a sequence of plasma cells instead of RF cavities. The first stage consists of a plasma-based injection-acceleration module where controlled injection techniques are applied to produce a high-quality electron beam with approximately 10 GeV energy. Electrons are then accelerated to 1 TeV using 100 laser-plasma modules, each consisting of a 1 m long preformed plasma channel driven by a 30 J laser pulse giving a 10 GeV energy gain. A fresh laser pulse is injected into each module. Similarly, positrons are produced from a 10 GeV electron beam through pair creation and then trapped and accelerated in a laser-plasma acceleration module to approximately 10 GeV. Subsequent laserplasma modules would accelerate positrons to 1 TeV [32].*

review of near and long-term potential applications of plasma accelerators can be found in Ref. [34].

## **3. Solid-state-based acceleration**

To go beyond the state-of-the-art, solid-state materials might offer new paths for beam manipulation and acceleration. For example, in the field of accelerator physics, the channelling properties of silicon crystals have successfully been used for collimation and extraction of relativistic proton beams [35]. Could also solids provide an alternative medium for acceleration? Depending on their particular atomic configuration and electrical conduction nature, some solid-state micro- and nano-sized structures offer interesting properties to enhance electric field components or induce strong wakefields that could be useful for acceleration, as well as transverse particle guiding and radiation emission. Next, we review some promising concepts that could revolutionise the future generation of accelerators and light sources.

#### **3.1 Dielectric wakefield acceleration**

Since the achievable peak field in the conventional RF metallic cavities is limited by surface breakdown, one obvious way to overcome this limit is through the use of materials with better breakdown properties, such as some dielectric materials (e.g., diamond, quartz, silica and ceramics). **Figure 3** illustrates the concept of beam-driven Dielectric Wakefield Acceleration (DWA) [36]. In this case, the dielectric accelerator consists of a hollow channel covered by a layer of a dielectric material and a metallic cladding. The electric field from the driver polarises the atoms of the dielectric medium, which coherently generates high frequency electromagnetic radiation.

Alternatively, lasers can be used as a driving source. Dielectric laser-driven acceleration (DLA) has been demonstrated for both non-relativistic [37] and relativistic electrons [38, 39] and has made enormous progress in recent years. DLA is the method behind the famous concept of 'accelerator on a chip' [40, 41]. DLA and DWA are limited to a maximum gradient of approximately 10 GV/m in the THz frequency range. **Figure 4** shows a particular dual-grating DLA configuration with sub-micron aperture.

#### **Figure 3.**

*Conceptual drawing of DWA [36]. A drive beam crosses a dielectric tube of radius "a"and excites wakefields. If a trailing beam is injected with the correct phase-space parameters, then it would be accelerated by the longitudinal component of the electric wakefield (bands of colour).*

**Figure 4.**

*(a) Conceptual scheme of a dual-grating scheme of DLA. (b) Simulation of the longitudinal accelerating force in a dual-grating structure illuminated by an input laser field along the y-axis [42]. (c) Scanning electron microscope image of the longitudinal cross-section of a DLA dual grating structure with 400 nm gap [38].*

Usually, the dielectric structures have sub-mm dimensions, and there have been conceptual proposals to build a high-energy collider based on staged dielectric structures (**Figure 5**). This would allow 'table-top' high energy accelerators.

#### **3.2 Crystal channelling acceleration**

Semiconductor and metallic crystalline lattices have been proposed to generate a solid-state plasma medium to guide and accelerate positive particles, taking advantage of the channelling properties in crystals. High electron density in solids could be obtained from the conduction bands. Typical electron densities (*ne*) in solid-state plasmas lie within the range of 10<sup>19</sup> cm�<sup>3</sup> ≤ *ne* ≤ 1024 cm�<sup>3</sup> [43, 44], that is, between one and six orders of magnitude higher than the density in gaseous plasmas. Taking into account Eq. (1), solid-state-based plasmas might lead to accelerating gradients 0*:*1 TV*=*m≲*Ez*≲100 TV*=*m.

Solid-state wakefield acceleration using crystals was proposed in the 1980s and 1990s by T. Tajima and others [43, 45, 46] as a technique to sustain TV/m acceleration

#### **Figure 5.**

*Conceptual design of a staged DLA collider, consisting of dielectric structures with a tapered grating period to guarantee synchronicity with the accelerating electrons [37].*

gradients. In the original Tajima's concept [45], high energy (≃40 keV) X-rays are injected into a crystalline lattice at the Bragg angle to cause the Borrmann-Campbell effect [47, 48], yielding slow-wave accelerating fields. Then a witness beam of charged particles, e.g., muons, which is injected into the crystal with an optimal injection angle for channelling, can experience acceleration along the crystal axis.

Wakefields in crystals can be induced by means of the excitation of high-frequency collective motion of conduction electrons through the crystalline lattice (see next section on plasmonic acceleration). To reach accelerating gradients on the order of TV/m, crystals must be excited by ultrashort X-ray laser pulses within a power range of TW– PW, which makes the practical realisation of the concept very challenging. It has only recently become a realistic possibility since the invention of the so-called single-cycled optical laser compression technique by G. Mourou et al. [49, 50].

If natural crystals (e.g., silicon) are used for solid-state wakefield acceleration, the beam intensity acceptance is significantly limited by the angstrom-size channels. In addition, such small size channels increase the dechannelling rate and make the channels physically vulnerable to high energy interactions, thus increasing the damage probability by high power beams.

Over the past decade, there have been great advances in nanofabrication techniques [51–53] that could offer an excellent way to overcome many of the limitations of natural crystals. Metallic nanostructures and metamaterials [54, 55] could lead to suitable ultra-dense plasma media for wakefield acceleration or charged particle beam manipulation, that is, channelling, bending, wiggling, etc. This also includes the possibility of investigating new paths towards ultra-compact X-ray sources [54]. The possibility to excite high acceleration gradients in multilayer graphene structures is also being investigated [56].

In the next section, we describe the plasmonic acceleration, which is a novel acceleration technique based on the excitation of suitable surface plasmonic modes in conducting materials. More concretely, we focus on plasmonic acceleration using nanostructures or metamaterials based on allotropes of carbon.

#### **3.3 Plasmonic acceleration**

Plasmonics can be defined as the study of the interaction between electromagnetic fields and the free electron Fermi gas in conducting solids. External electromagnetic fields can excite plasmons, that is, collective oscillations of conduction electrons in metals [57]. To some extent, this collective effect could be exploited to generate ultrahigh acceleration gradients. **Figure 6** depicts a scheme of excited plasmons in metallic surfaces. The oscillation of induced longitudinal electric field reminds that in a sequence of a multi-cell RF cavity operating at *π*-mode.

The excitation of surface plasmonic modes [58, 59] can be driven either by charged particle beam [60] or by laser [61]. To be effective, the driver dimensions should match the spatial ( nm) and time (sub-femtoseconds) scales of the excited plasmonic oscillations. Wakefield driving sources working on these scales are now experimentally realisable. For instance, attosecond X-ray lasers are possible thanks to the pulse compression technique [50]. In the case of beam-driven wakefields, the experimental facility FACET-II at SLAC [62] will allow the access to 'quasi-solid' electron bunches with densities up to 10<sup>24</sup> cm<sup>3</sup> and submicron longitudinal size. A comparison of the range of spatial and time scales of the electric field oscillations and achievable acceleration gradient for standard and novel acceleration methods is shown

#### **Figure 6.**

*(a) Plasmonic acceleration concept. (b) Comparison with an RF cavity operating in π-mode. The drawings are not to scale. While the plasmonic structure has micrometric or submicrometric apertures and lengths on the order of 1 mm, for instance, a 9-cell RF cavity has usually apertures of tens of mm and lengths on the order of m.*

**Figure 7.**

*Schematic comparing the space and time scales of the longitudinal electric field components generated by different techniques for charged particle acceleration and their corresponding amplitude (acceleration gradient). The cases of nanostructure wakefields, plasmonic and crystal acceleration are based on theoretical and numerical predictions.*

in **Figure 7**. In principle, solid-state wakefields and plasmonics acceleration with nanostructures are predicted to have the potential to generate higher acceleration gradients than DWA and DLA, LWFA and PWFA with gaseous plasma. Plasmonic acceleration with nanostructures could be considered an intermediate step between PWFA and crystal channelling acceleration towards the PV/m gradient regime.

Due to their special thermo-mechanical and optoelectronic properties, materials based on carbon nanotubes arrays or graphene could offer an excellent medium to generate plasmonic wakefield acceleration. For instance, conduction electrons in CNTs could have densities *ne* <sup>10</sup><sup>23</sup> cm<sup>3</sup> . In principle, as proved in [63, 64], in the linear regime the plasmonic dynamics in CNT bundles can be described by classical plasma formulae. Therefore, from Eq. (1) we can estimate a maximum longitudinal wakefield *Ez* 10 TV/m.

Particle-In-Cell (PIC) simulations of beam-driven wakefield acceleration in cylindrical metallic hollow structures (**Figure 8**) with micrometric or nanometric apertures have shown the feasibility of obtaining gradients ≳100 GV/m, envisioning the possibility of an ultra-compact PeV linear collider [66–68]. Similar gradient values have

**Figure 8.**

*(a) Schematic model for beam-driven wakefield simulation using a hollow cylinder of solid-state plasma confined in a wall of thickness rout* � *rin and length Lp. In this model, the cylinder wall represents a solid made of CNT bundles. (b) an example of longitudinal wakefield as a function of the longitudinal coordinate ξ* ¼ *z* � *ct, with c the speed of light. This particular case has been computed using the PIC code FBPIC [65].*

also been predicted from PIC simulations of X-ray laser-driven wakefield acceleration in nanotubes [69].

One of the important properties of the plasmonic excitation is that the ionic lattice of the material remains strongly correlated. Unlike the plasmonic acceleration, other novel solid-state-based acceleration techniques require the complete ionisation of the ionic lattice or the ablation of solids, as described in the next section.

#### **3.4 Target normal sheath acceleration**

Solid-state plasmas can also be created by ablation of solids using high-intensity near-infrared femtosecond lasers. In this case, unlike in the plasmon wakefield excitation, the individual ions are uncorrelated. For example, the so-called target normal sheath acceleration (TNSA) technique is based on the ablation of a metallic thin foil [70]. The mechanism is schematically represented in **Figure 9**. A thin foil is irradiated by an intense laser pulse. The laser prepulse creates a pre-plasma on the target's front side. The main pulse interacts with the plasma and accelerates MeV-energy electrons mainly in the forward direction. The electrons propagate through the target, where collisions with the background material can increase the divergence of the electron current. Then the electrons leave the rear side, resulting in a dense sheath. An electric

**Figure 9.** *Target normal sheath acceleration mechanism.*

**Figure 10.**

*Laser wakefield acceleration concept with aligned CNT arrays. In this particular configuration, CNTs are arranged perpendicularly to the laser propagation [71].*

field due to charge separation is created. The field is of the order of the laser electric field (TV/m), which ionises atoms at the surface. Eventually, the ions are accelerated in this sheath field, pointing in the target normal direction.

#### **3.5 Solid-state plasma by electron field emission**

Electron field emission from nanostructures could offer another interesting way to generate solid-state plasma. A possible configuration based on two layers of aligned CNT arrays and a gap in between them is shown in **Figure 10**. Taking into account the low work function of CNTs and their emission properties, a first intense laser pulse, travelling through the gap between the CNT sheets, could be used to induce electron field emission from CNT arrays. Then extracted electrons in the gap are accelerated by the ponderomotive force of a second laser pulse. For more details see Ref. [71].

## **4. Outlook and prospects**

Currently, particle accelerators play an important role in many areas, including fundamental science, medicine, industry and society in general. During the last 100 years, the research in HEP has been the main driving force behind the development and progress of the accelerator technology. The LHC is currently the world's most advanced and powerful accelerator, and the only multi-TeV collider in operation. It consists of a 27-km ring of superconducting magnets with a number of accelerating structures based on RF electromagnetic fields to accelerate particles. The next generation of RF-driven high energy accelerators will pose serious challenges in terms of infrastructure size, technology and cost.

The current state-of-the-art of the RF techniques is limited to gradients on the order of 100 MV/m. Hence, larger and more expensive accelerator facilities are

necessary in order to go beyond the LHC capabilities. Therefore, it is important to explore alternative advanced acceleration techniques to overcome the limitations of conventional technologies, thus progressing towards more compact, sustainable and economical particle accelerators. For many years, the development of inexpensive and compact 'table-top' high energy particle accelerators and light sources has been the ultimate dream of the accelerator physics community.

Some of the most promising novel acceleration techniques have briefly been reviewed in this chapter. For instance, PWFA and LWFA using gaseous plasma have experimentally demonstrated to be able to achieve 100 GV/m acceleration [19–24]. Another path towards TV/m acceleration is the use of special solid-state media, such as micrometric and millimetric grating dielectric structures, reaching 1–10 GV/m acceleration [38, 39].

Crystals and, more recently, nanostructures have also attracted attention to create a solid-state plasma medium with electron densities 1–6 orders of magnitude higher than those in gaseous plasmas. For example, recent theoretical and numerical studies have shown the feasibility of obtaining tens of TV/m fields in nanomaterials by excitation of non-linear plasmonic modes driven by a high-density sub-micron particle bunch [66, 67]. To obtain multi-TV/m fields other techniques rely on the ablation of solids [70] or the induced electron field emission from nanomaterials [71].

It is necessary to remark that the excitation of solid-state plasma wakefields requires the use of driving sources (lasers or charged particle bunches) at the same time and spatial scales as the wakefield oscillations in the plasma. In principle, wakefield driving sources working on solid-state plasma scales are now experimentally realisable. Attosecond X-ray lasers are possible thanks to the pulse compression [49, 50]. In the case of beam-driven wakefields, the experimental facility FACET-II at SLAC [62] could deliver 'quasi-solid' electron beams in the near future. Therefore, an experimental proof-of-concept of solid-state PWFA could possibly become a reality within the next 10 years.

It is also worth mentioning that nanostructures and metamaterials, due to their special and flexible optoelectronic properties, could also provide an excellent medium to create compact X-ray sources, e.g., X-ray light sources based on CNT arrays [72].

In conclusion, the different advanced acceleration techniques described in this chapter have the potential to transform the current paradigm in accelerator physics. They offer novel pathways to multi-GV/m and multi-TV/m field regimes. This will open new horizons to the physics of extreme fields, particularly in collider physics, light sources, and in many other areas of applied sciences, medicine and industry.

## **Acknowledgements**

The author acknowledges support by the Generalitat Valenciana under grant agreement No. CIDEGENT/2019/058. This chapter is dedicated to the memory of Dr Esteban Fullana Torregrosa, a great physicist and best friend.

### **Abbreviations**



## **Author details**

Javier Resta López Institute of Materials Science (ICMUV), University of Valencia, Valencia, Spain

\*Address all correspondence to: javier2.resta@uv.es

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

[1] Sessler A, Wilson E. Engines of Discovery. A Century of Particle Accelerators. Revised and Expanded Edition. Singapore: World Scientific; 2014. 280 p. DOI: 10.1142/ 8552

[2] Zimmermann F, Shiltsev VD. Modern and future colliders. Reviews of Modern Physics. 2021;**93**(1):015006. DOI: 10.1103/RevModPhys.93.015006

[3] Bruning O, Collier P. Building a behemoth. Nature. 2007;**448**:285-289. DOI: 10.1038/nature06077

[4] Collaboration ATLAS. Observation of a new particle in the search of the standard model Higgs boson with the ATLAS detector at the LHC. Physics Letters B. 2012;**716**:1-29. DOI: 10.1016/j. physletb.2012.08.020

[5] Collaboration CMS. Observation of a new boson at the mass of 125 GeV with the CMS experiment at the LHC. Physics Letters B. 2012;**716**:30-61. DOI: 10.1016/ j.physletb.2012.08.021

[6] Benedikt M et al. FCC-ee: The Lepton collider. The European Physical Journal Special Topics. 2019;**228**:261-623. DOI: 10.1140/epjst/e2019-900045-4

[7] Benedikt M et al. FCC-hh: The hadron collider. The European Physical Journal Special Topics. 2019;**228**:755-1107. DOI: 10.1140/epjst/e2019-900087-0

[8] Michizono S. The international linear collider. Nature Reviews Physics. 2019;**1**: 244-245. DOI: 10.1038/s42254-019- 0044-4

[9] Stapnes S. The compact linear collider. Nature Reviews Physics. 2019;**1**: 235-237. DOI: 10.1038/s42254-019- 0051-5

[10] Long KR et al. Muon colliders to expand frontiers of particle physics. Nature Physics. 2021;**17**:289-292. DOI: 10.1038/s41567-020-01130-x

[11] Cai J, Syratchev I. KlyC: 1.5-D largesignal simulation code for klystrons. IEEE Transactions on Plasma Science. 2019;**47**(4):1734-1741. DOI: 10.1109/ TPS.2019.2904125

[12] Litvinenko VN, Roser T, Chamizo-Llatas M. High-energy high-luminosity e+ e� collider using energy-recovery linacs. Physics Letters B. 2020;**804**: 135394. DOI: 10.1016/j. physletb.2020.135394

[13] Shiltsev VD. High energy particle colliders: Past 20 years, next 20 years and beyond. Physics-Uspekhi. 2012; **55**(10):965-976. DOI: 10.3367/ UFNe.0182.201210d.1033

[14] Ferrario M. Present and future accelerator options beyond the LHC. Annalen der Physik. 2016;**528**(1–2): 151-160. DOI: 10.1002/andp.201500240

[15] Abe T et al. Breakdown study based on direct in situ observation of inner surfaces of an rf accelerating cavity during a high-gradient test. Physical Review Accelerators and Beams. 2016; **19**:102001. DOI: 10.1103/ PhysRevAccelBeams.19.102001

[16] Tajima T, Dawson JM. Laser Electron Accelerator. Physical Review Letters. 1979;**43**:267-270. DOI: 10.1103/ PhysRevLett.43.267

[17] Joshi C. Plasma accelerators. Scientific American. 2006;**294**(2):40-47

[18] Chen P, Dawson JM, Huff RW, Katsouleas T. Acceleration of electrons by the interaction of a bunched Electron beam with a plasma. Physical Review Letters. 1985;**54**(7):693-696. DOI: 10.1103/PhysRevLett.54.693

[19] Gordon D et al. Observation of Electron energies beyond the linear dephasing limit from a laser-excited relativistic plasma wave. Physical Review Letters. 1998;**80**:2133-2136. DOI: 10.1103/PhysRevLett.80.2133

[20] Leemans WP et al. Multi-GeV Electron beams from capillarydischarge-guided Subpetawatt laser pulses in the self-trapping regime. Physical Review Letters. 2014;**113**(24): 245002. DOI: 10.1103/ PhysRevLett.113.245002

[21] Litos M et al. High-efficiency acceleration of an electron beam in a plasma wakefield accelerator. Nature. 2015;**515**:92-95. DOI: 10.1038/ nature13882

[22] Blue BE et al. Plasma-Wakefield acceleration of an intense positron beam. Physical Review Letters. 2003;**90**: 214801. DOI: 10.1103/ PhysRevLett.90.214801

[23] Corde S et al. Multi-gigaelectronvolt acceleration of positrons in a self-loaded plasma wakefield. Nature. 2015;**524**: 442-445. DOI: 10.1038/nature14890

[24] Doche A et al. Acceleration of a trailing positron bunch in a plasma wakefield accelerator. Scientific Reports. 2017;**7**:14180. DOI: 10.1038/s41598-017- 14524-4

[25] Adli E, et al. (AWAKE Collaboration). Acceleration of electrons in the plasma wakefield of a proton bunch. Nature 2018;561:363–367. DOI: 10.1038/s41586-018-0485-4.

[26] Geddes CGR et al. High-quality Electron beams from a laser wakefield accelerator using Plasma-Channel guiding. Nature. 2004;**431**(7008): 538-541. DOI: 10.1038/nature02900

[27] Mangles SPD et al. Monoenergetic beams of relativistic electrons from intense-laser plasma interactions. Nature. 2004;**431**(7008):535-538. DOI: 10.1038/nature02939

[28] Faure J et al. A laser-plasma accelerator producing Monoenergetic Electron beams. Nature. 2004;**431**(7008): 541-544. DOI: 10.1038/nature02963

[29] Esarey E, Schroeder CB, Leemans WP. Physics of laser-driven plasma-based electron accelerators. Reviews of Modern Physics. 2009;**81**(3): 1229-1285. DOI: 10.1103/ RevModPhys.81.1229

[30] Leemans WP, Esarey E. Laserdriven plasma-wave electron accelerators. Physics Today. 2009;**62**: 44-49. DOI: 10.1063/1.3099645

[31] Schroeder CB, Esarey E, Geddes C, Benedetti C, Leemans W. Physics considerations for laser-plasma linear colliders. Physical Review Special Topics-Accelerators and Beams. 2010;**13**:101301. DOI: 10.1103/PhysRevSTAB. 13.101301

[32] Leemans W. White paper of the ICFA-ICUIL, joint taks force: High power laser technology for accelerators. In: ICFA Beam Dyn. Newslett. Vol. 56. International Committee for Future Accelerators (ICFA). 2011. pp. 10-88. Available from: https://icfausa.jlab.org/a rchive/newsletter/icfa-bd-nl-56.pdf

[33] Walker PA et al. Facility considerations for a European plasma accelerator infrastructure (EuPRAXIA). In: Proceedings of the 10th International Particle Accelerator Conference (IPAC'19); 19-24 May 2019; Melbourne, Australia: Paper THPGW025.

*Future Particle Accelerators DOI: http://dx.doi.org/10.5772/intechopen.106340*

[34] Joshi C, Corde S, Mori WB. Perspectives on the generation of electron beams from plasma-based accelerators and their near and long term applications. Physics of Plasmas. 2020; **27**:070602. DOI: 10.1063/5.0004039

[35] Scandale W, Taratin AM. Channeling and volume reflection of high-energy charged particles in short bent crystals. Crystal assisted collimation of the accelerator beam halo. Physics Reports. 2019;**815**:1-107. DOI: 10.1016/j. physrep.2019.04.003

[36] Thompson MC. Breakdown limits on Gigavolt-per-meter Electron-beamdriven Wakefields in dielectric structures. Physical Review Letters. 2008;**100**:214801. DOI: 10.1103/ PhysRevLett.100.214801

[37] Breuer J, Hommelhoff P. Laser-based acceleration of nonrelativistic electrons at a dielectric structure. Physical Review Letters. 2013;**111**:134803. DOI: 10.1103/ PhysRevLett.111.134803

[38] Peralta EA et al. Demonstration of electron acceleration in a laser driven dielectric microstructure. Nature. 2013; **503**:91-94. DOI: 10.1038/nature12664

[39] O'Shea BD et al. Observation of acceleration and deceleration in gigaelectron-volt-per-metre gradient dielectric wakefield accelerators. Nature Communications. 2016;**7**:12763. DOI: 10.1038/ncomms12763

[40] Blau SK. Particle acceleration on a chip. Physics Today. 2013;**66**(12):19. DOI: 10.1063/PT.3.2205

[41] England RJ, Hommelhoff P, Byer RL. Microchip accelerators. Physics Today. 2021;**74**(8):42-49. DOI: 10.1063/PT.3.4815

[42] Wei Y et al. Beam quality study for a grating-based dielectric laser-driven

accelerator. Physics of Plasmas. 2017;**24**: 023102. DOI: 10.1063/1.4975080

[43] Chen P, Noble RJ. A solid state accelerator. AIP Conf. Proc. 1987;**156**: 222. DOI: 10.1063/1.36458

[44] Östling D, Tománek RA. Electronic structure of single-wall, multiwall and filled carbon nanotubes. Physical Review B. 1997;**55**(20):13980-13988. DOI: 10.1103/PhysRevB.55.13980

[45] Tajima T, Cavenago M. Crystal Xray accelerator. Physical Review Letters. 1987;**59**(13):1440-1443. DOI: 10.1103/ PhysRevLett.59.1440

[46] Chen P, Noble RJ. Crystal channel collider: Ultra-high energy and luminosity in the next century. AIP Conf. Proc. 1997; **398**:273. DOI: 10.1063/1.53055

[47] Borrmann G. Die Absortion von Röntgenstrahlen im Fall der Interferenz. Zeitschrift für Physik. 1950;**127**:297-323. DOI: 10.1007/BF01329828

[48] Campbell HN. X-ray absorption in a crystal set at the Bragg angle. Journal of Applied Physics. 1951;**22**:1139. DOI: 10.1063/1.1700122

[49] Strickland D, Mourou G. Compression of amplified chirped optical pulses. Optics Communications. 1985;**56**(3):219-221. DOI: 10.1016/ 0030-4018(85)90120-8

[50] Mourou G, Mironov S, Khazanov E, Sergeev A. Single cycle thin film compressor opening the door to Zeptosecond-Exawatt physics. The European Physical Journal Special Topics. 2014;**223**:1181-1188. DOI: 10.1140/epjst/e2014-02171-5

[51] Li P et al. Recent advances in focused ion beam nanofabrication for nanostructures and devices:

Fundamentals and applications. Nanoscale. 2021;**13**:1529-1565. DOI: 10.1039/D0NR07539F

[52] Pint CL et al. Dry contact transfer printing of aligned carbon nanotube patterns and characterization of their optical properties for diameter distribution and alignment. ACS Nano. 2013;**4**(2):1131-1145. DOI: 10.1021/ nn9013356

[53] Chen X et al. Atomic layer lithography of wafer-scale nanogap arrays for extreme confinement of electromagnetic waves. Nature Communications. 2013;**4**:2361. DOI: 10.1038/ncomms3361

[54] Pizzi A et al. Graphene metamaterials for intense, Tunable, and compact extreme ultraviolet and X-ray sources. Advancement of Science. 2020; **7**:1901609. DOI: 10.1002/ advs.201901609

[55] Kong L-B, Chen Z. Plasmonic electron acceleration with the meta-surfaces. Physics of Plasmas. 2017;**24**:083111. DOI: 10.1063/ 1.4997481

[56] Bontoiu C et al. TV/M Laser-Driven Accelerating Gradients in Graphene. Proceedings of the 10th International Particle Accelerator Conference (IPAC'22); 12-17 June 2022; Bangkok, Thailand: Paper WEPOST043

[57] Sarid D, Challener W. Modern Introduction to Surface Plasmons: Theory, Mathematica Modeling, and Applications. Cambridge: Cambridge University Press; 2010. 371 p. DOI: 10.1017/CBO9781139194846

[58] Ukhtary MS, Saito R. Surface plasmon in graphene and carbon nanotubes. Carbon. 2020;**167**:455. DOI: 10.1016/j.carbon.2020.05.019

[59] Macchi A. Surface plasmons in superintense laser-solid interactions. Physics of Plasmas. 2018;**25**:031906. DOI: 10.1063/1.5013321

[60] Nejati M et al. The single wall carbon nanotube waveguides and excitations of their *σ* þ *π* plasmons by an electron beam. Physics of Plasmas. 2009;**16**: 022108. DOI: 10.1063/1.3077306

[61] Fedeli L et al. Electron acceleration by relativistic surface Plasmons in lasergrating interactions. Physical Review Letters. 2016;**116**:015001. DOI: 10.1103/ PhysRevLett.116.015001

[62] Yakimenko V et al. FACET-II facility for advanced accelerator experimental tests. Physical Review Accelerators and Beams. 2019;**22**:101301. DOI: 10.1103/ PhysRevAccelBeams.22.101301

[63] Rider AE, Ostrikov K, Furman SA. Plasmas meet plasmonics: Everything old is new again. European Physical Journal D: Atomic, Molecular, Optical and Plasma Physics. 2012;**66**:226. DOI: 10.1140/epjd/e2012-30273-3

[64] Que W. Theory of plasmons in carbon nanotube bundles. Journal of Physics: Condensed Matter. 2002;**14**: 5239-5253. DOI: 10.1088/0953-8984/14/ 20/319

[65] Lehe R, Kirchen M, Andriyash IA, Godfrey BB, Vay J-L. A spectral quasicylindrical and dispersion free particle-In-cell algorithm. Computer Communications. 2016;**203**:66-82. DOI: 10.1016/j.cpc.2016.02.00

[66] Sahai AA, Tajima T, Taborek P, Shiltsev VD. Solid-state tube wakefield accelerator using surface waves in crystals. International Journal of Modern Physics A. 2020;**34**:1943009-1943029. DOI: 10.1142/9789811217135-0009

*Future Particle Accelerators DOI: http://dx.doi.org/10.5772/intechopen.106340*

[67] Sahai AA. Nanomaterials based Nanoplasmonic accelerators and lightsources driven by particle-beams. IEEE Access. 2021;**9**:54831-54839. DOI: 10.1109/ACCESS.2021.3070798

[68] Resta-Lopez J et al. Study of Ultra-High Gradient Acceleration in Carbon Nanotube Arrays. Proceedings of the 9th International Particle Accelerator Conference (IPAC'18); 29 April-4 May 2018; Vancouver, Canada: Paper TUXGBE2. DOI: 10.18429/JACoW-IPAC2018-TUXGBE2

[69] Zhang X et al. Particle-in-cell simulation of X-ray wakefield acceleration and betatron radiation in nanotubes. Physical Review Accelerators and Beams. 2016;**19**:101004. DOI: 10.1103/PhysRevAccelBeams.19.101004

[70] Roth M, Schollmeier M. Ion Acceleration–Target Normal Sheath Acceleration. Proceedings of the CAS-CERN Accelerator School: Plasma Wake Acceleration; 23-29 November; Geneva, Switzerland. DOI: 10.5170/CERN-2016-001.231

[71] Nicks BS et al. High-density dynamics of laser wakefield acceleration from gas plasmas to nanotubes. Photonics. 2021;**8**:216. DOI: 10.3390/ photonics8060216

[72] Shin YM. Optically controlled coherent X-ray radiations from photoexcited nanotubes. Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms. 2017;**407**:276-281. DOI: 10.1016/ j.nimb.2017.07.019

## **Chapter 5**
