The electron and ion Kappa-indices affect the values of the effective Debye length and DH potential at fixed values of r = 0.2, η = 10^{−3}, and Z_{d0} = 300.

## Abstract

Different plasma diagnostic methods are briefly discussed, and the framework of a test charge technique is effectively used as diagnostic tool for investigating interaction potentials in Lorentzian plasma, whose constituents are the superthermal electrons and ions with negatively charged dust grains. Applying the space-time Fourier transformations to the linearized coupled Vlasov-Poisson equations, a test charge potential is derived with a modified response function due to energetic ions and electrons. For a test charge moving much slower than the dust-thermal speed, there appears a short-range Debye-Hückel (DH) potential decaying exponentially with distance and a long-range far-field (FF) potential as the inverse cube of the distance from test charge. The FF potentials exhibit more localized shielding curves for low-Kappas, and smaller effective shielding length is observed in dusty plasma compared to electron-ion plasma. However, a wakefield (WF) potential is formed behind the test charge when it resonates with dust-acoustic oscillations, whereas a fast moving test charge leads to the Coulomb potential having no shielding around. It is revealed that superthermality and plasma parameters significantly alter the DH, FF, and WF potentials in space plasmas of Saturn’s E-ring, where power-law distributions can be used for energetic electrons and ions in contrast to Maxwellian dust grains.

### Keywords

- kinetic model
- test charge technique
- DA waves
- superthermal tails
- dynamical shielding

## 1. Plasma diagnostic methods

To understand a plasma state and its characteristics, numerous experimental techniques, mechanisms, devices, theoretical models, and computational packages have been developed as diagnostic tools for measuring the plasma parameters such as the plasma electron density and temperature [1, 2] as well as their spatial profiles and dynamics. These diagnostic techniques are used to adequately describe both low-temperature and high energy density plasmas. In some situations, the measurements by these techniques cause perturbations in plasmas and are termed as active diagnostic techniques, while passive ones do not perturb plasmas. Based on the degree of ionization, the plasmas can broadly be classified into cold and hot plasma states, which accordingly demand for various types of diagnostics to precisely estimate the plasma parameters for optimal understanding of the physics of plasmas. This includes both theoretical and experimental findings. The most common techniques for cold and hot plasmas (

On the other hand, in certain plasma sources like tokamak plasmas, strong currents are generated, which give rise to various kinds of magnetohydrodynamic (MHD) instabilities. For this, a magnetic probe is used, which is beneficial especially for measuring either local magnetic fields or its fluctuations not only in tokamaks but also in laser-produced plasmas (LPP). Furthermore, the amplitude of current flowing into the plasma can be estimated by integrating the induced magnetic field around the plasma column by utilizing the so-called Rogowski coil. However, in some plasmas (especially high temperature), it is not feasible to utilize material probes for determining the plasma parameter like plasma electron density. Therefore, a nonperturbing approach is needed to diagnose the plasma. In such a scenario, the electromagnetic spectrum is utilized. But the electromagnetic wave intensity must be low enough to the level that it will not result in plasma perturbation. For probing the high-density plasmas, a lower wavelength is required as a probe. This justifies the utilization of infrared radiation in tokamak and ultraviolet radiation for measuring the plasma electron density in LPP. The variation of the polarizing angle involving the beam probe in the presence of magnetic field can also be used for diagnostics of tokamak plasmas.

Interestingly, the evaluation of appropriate plasma parameters may be carried out by spectroscopy of emitted radiations as used generally from the beginning of plasma physics. This technique for emission measurements has been particularly making significant contributions over the past five decades for the fact that plasmas produced for nuclear fusion research exhibit intense emission in the X-ray region. Astrophysical applications further justify the wide interest in X-ray emission from plasmas. The phase soft X-ray (so defined due to their low penetrating power) indicates electromagnetic radiations with a wavelength in range

The particle measurement method is another scheme for investigating the characteristics of plasmas by using the beam of fast particles. It has received much attention in the studies of inertial confinement fusion and energy deposition in a medium driven by cluster-ion and fast heavy-ion beams, as well as in plasma accelerators and low-temperature laboratory plasmas.

## 2. Dusty plasma and test charge technique

The most common ingredient of astrospace plasmas is the dust component in addition to electrons and ions, found everywhere in earth atmospheres, in comets, in planetary rings, in interstellar clouds, in interplanetary space, in interstellar medium, etc. Dust grains may exist in the form of ice particles, metallic and dielectric materials, and are highly charged species due to different charging processes. For instance, the absorption of ambient electrons and ions on dust grain surface may lead to the negatively charged dust grains, while thermionic and secondary electron emissions as well as ultraviolet photoionization give rise to positively charged dust grains. Thus, a multispecies dusty plasma can be assumed as more complex plasma than conventional electron-ion plasma, for dust size, mass, and charge variations. Being an abundant component of the space and industrial plasmas [4, 5], dusty plasma has always attracted lots of interests for studying new distinct features of plasma modes [6, 7] with a static and dynamic background of dust grains both analytically and experimentally [8, 9, 10]. Numerous linear and nonlinear [viz., solitons, shocks, vortices, etc.] dusty modes and associated instabilities are investigated using the frameworks of perturbative and nonperturbative schemes.

The behavior of charged particles in plasmas can be described by the well-known fluid and kinetic theories [4, 11], essentially helpful for studying the basic properties of plasma waves and instabilities, depending strongly on the observed phenomena. Laboratory plasmas have effectively been modeled by fluid description, where charged fluids of plasma species are assumed in temporal and spatial configurations. But, it has been observed that fluid theory does not account for velocity space coordinate and is insufficient to study the wave phenomena in non-equilibrium plasmas, where particle distributions show significant deviations from the equilibrium states. Hence, fluid theory is unable to explain the wave-particle interactions that could lead to collisionless Landau damping phenomenon and many other interesting features of collective modes and instabilities. Conversely, kinetic theory adequately describes the physical phenomena in real time and phase space configurations, providing all information about plasma waves, instabilities, plasma equilibrium, Landau damping rate, etc.

Test charge techniques [12, 13] can be utilized to study the shielding of test charges in collisional [14] and turbulent [15] plasmas, the electric field [16], and far-field potential of a test charge in a nonuniform magnetoplasma [17], the wake-field excitations in charge fluctuating dusty plasmas [18], the two-body correlations [19], the energy loss of test charges [20], etc. If a test particle is projected into the plasma with a constant speed, its charge density is coupled with the plasma charge density by the space charge effects. Consequently, the test charge is screened by a cloud of opposite sign charges leading to the short-range Debye-Hückel (DH) potential. Of course, the speed of test charge significantly matters in plasmas when it is considered with respect to thermal speeds of plasma species. The interaction potentials and energy loss of charged particles have been recognized in many research areas, for example, in ion-cluster interaction with condensed matter [21, 22], in inertial confinement fusion [23, 24, 25], in particle acceleration [26], in low-temperature laboratory plasmas [27, 28], and in dense plasmas for heavy-ion energy deposition [29].

Montgomery et al. [13] employed the test charge technique to obtain far-field potential distribution around a test charge, which decays as the inverse cube of the distance from test charge in electron-ion Maxwellian plasmas. Subsequent investigations of shielded potentials have been phenomenally influenced by the ionic motion [30], electron-electron collisions [14, 31], and plasma turbulence [15]. The electrostatic potential [32] due to small and large test charge velocities has been investigated to display the excitation of long-range wakefields in Maxwellian plasmas. Shivamoggi and Mulser [33] examined the effects of magnetic field, collisions, and plasma inhomogeneity on the potential due to slowly and rapidly moving test charges [17, 34], and Lakshmi et al. [35] discussed the Debye shielding phenomenon in a dusty plasma by considering the Boltzmannian electrons and ions with cold negative dust grains. It was revealed that plasma parameters significantly alter the characteristics of small and large amplitude potentials. Later, Shukla [36] reported the FF potential for a slowly moving test charge in a Maxwellian dusty plasma and showed the impact of dust-charge variation on the dipole-like FF potential. Moreover, oscillatory wake-field can be excited behind the test charge [37] in a collisionless unmagnetized plasma with Maxwellian electrons and ions. Nambu et al. [38] extended this work to dusty plasmas and explained the resonant phenomenon of DA waves with a test charge, resulting in the long-range WF potential. Later, Shukla and Rao [39] analyzed the WF, DH, and FF potentials of test charge in a colloidal Maxwellian plasma accounting for the streaming ions and dust grains. It was found that external magnetic field and ion-streaming effects [40, 41] strongly affect the positive/negative potential regions in plasmas. To explore the effects of two-body correlations, dust-charge perturbations and dust-neutral collisions, various geometries have been designed for propagating test charges [18, 19, 20] in an unmagnetized Maxwellian dusty plasma. In all above investigations, the plasma particles are described by the Maxwellian distribution function.

The shielding phenomenon is one of the main objectives of this chapter to unfold many intrinsic properties of the Lorentzian space dusty plasma, which discerns it from the standard Maxwellian plasmas. It plays a key role in setting up the basic criteria for Lorentzian dusty plasmas. Any plasma medium can physically be polarized by the test charge to give rise to perfect screening if thermal agitations are absent in the plasma system. Conversely, an imperfect shielding occurs if the plasma particles get enough thermal energy to escape from the edge of screening cloud. The interaction potentials caused by the test charge are not only strongly influenced by different test charge speeds in comparison with the thermal speeds but also lead to the possibility of dust crystallization and dust coagulation in space Lorentzian dusty plasmas.

## 3. Power-law Lorentzian distribution function (df)

In some circumstances, the behavior of plasma particles cannot be described by the usual Maxwellian distribution function (df) but often modeled by the power-law df. When all or some of the plasma particles move faster than their thermal speeds, the plasma particles are known as superthermal/suprathermal species, showing high energy and velocity tails in the distribution. They are mostly accelerated by wave-particle interactions, modulational instabilities and Langmuir turbulence [42], beam-plasma interactions [43], solar wind where type III solar radio emissions occurs [44], intense microwave-plasma interactions [45], ionospheric heating experiments [46], etc. Recognizing the role of superthermal energetic particles in plasmas, the wave dynamics and instabilities need to be re-investigated with a power-law df that gives a better fit to empirical data from space plasmas. A 3D isotropic Kappa-df [47] for superthermal particles can be expressed as

The normalization constant and effective thermal speed are denoted by *ms *) stands for the temperature (mass) of the *i* for positive ions). The effective thermal speed

## 4. Kinetic model for Lorentzian dusty plasmas

To compute the potential distributions around a test charge, we consider a collisionless Lorentzian dusty plasma, containing the suprathermal electrons and ions with negatively charged dust grains following the Maxwell-df. The plasma is also assumed to be field-free in the sense that there is no external electric or magnetic field (viz., *j*th species [*j* equals *md *.

The Lorentzian dusty plasma in the presence of a test charge can be described by the following linearized coupled set of Vlasov-Poisson equations:

and

where

Taking space–time Fourier analysis of Eqs. (2), (3), and (4), we obtain the Fourier transformed potential in this form

The modified longitudinal dielectric constant can be defined by

where

The dielectric constant in terms of dielectric susceptibilities

where

and

with their corresponding arguments

To proceed further, we shall consider two limiting cases of Eq. (8) by imposing certain limitations on the test charge speed in comparison with the thermal and acoustic speeds and simplify the interaction potentials [as given by Eq. (7)] in Lorentzian dusty plasmas.

### 4.1 Slow moving test charge response

For a slow test charge propagation in a Lorentzian dusty plasma, we assume that test charge speed

The modified effective Debye length

with superthermality parameters attributed to electrons and ions as

The usual Debye length in electron-ion plasma is denoted by

For numerical analyses, we can choose the data from the dusty plasma near Saturn’s E ring, cited in Refs. [55, 56, 57, 58] and many references therein. The data essentially corresponds to the Radio and Plasma Wave Science (RPWS) instruments onboard the Cassini spacecraft, containing the plasma parameters, such as

The effect of dust concentration [denoted through the parameter

### 4.2 Short-range DH and long-range FF potentials

To study short and long-range shielded potentials of a slowly moving test charge along the

The first part of Eq. (11) corresponds to the short-range Debye-Hückel (DH) potential, which accounts for the short distances between the test charge and observer, whereas the second part represents the long-range far-field potential in the limit

The variation of the normalized DH potential

Figure 5 exhibits how suprathermal electrons and ions modify the profiles of long-range FF potential (

### 4.3 Resonating test charge response

To examine the resonant interaction of a test charge with DA waves, we first simply derive the dielectric constant of the DA waves using the limit

The effective Debye length now gets a new form

with

This is the DA resonance frequency with DA speed

### 4.4 Short-range DH and long-range WF potentials

For static or slowly moving test charge in a Lorentzian dusty plasma, its potential distributions are found spherically symmetric both in the axial and radial directions. Consequently, the DH and FF shielded potentials are appropriately solved with spherical polar coordinates. However, if the test charge moves with finite speed in a specific direction along the z-axis, the resonant interaction of test charge with the DA wave leads to the asymmetric distribution of potential in the form of WF behind the test charge. The plasma model is then preferably solved in cylindrical coordinates. Thus, following the standard techniques [37, 38, 40] for DH and WF potentials, we make use of Eq. (13) into Eq. (7) to finally arrive at

where

### 4.5 Fast moving test charge response

In this case, the test charge is assumed to be moving much faster than all the plasma species (viz., the electrons, ions and negatively charged dust grains). Consequently, Eq. (6) can be expressed in 1D form to finally arrive at

It is now clear that if the test charge is moving very fast, then there is no shielding around it in the Lorentzian dusty plasma.

## 5. Conclusion

To conclude, we have briefly discussed different plasma diagnostic techniques and specifically investigated the novel features of interaction potentials caused by a test charge moving with constant velocity

Vladimirov and Nambu [40] have already utilized the idea of WF potential for making new materials by attracting the same polarity dust grains in dusty plasmas. The physics of attractive forces between the negatively charged dust grains is completely analogous to that of Cooper pairing of electrons in superconductors [59]. The dust particle physically polarizes the plasma medium and creates attractive potential regions, where positive ions from collective interaction of DA waves can be focused. This may in turn lead to the possibility for dust crystallization and dust coagulation in both laboratory and space dusty plasmas.

## Acknowledgments

Dr. S. Ali dedicates this document to Late Prof. P.K. Shukla and Dr. B. Eliasson (University of Strathclyde, UK) who were very kind to him at many occasions during discussions on dusty plasmas, and acknowledges the partial financial assistance from USTC, Hefei, China, and ICTP, Trieste, Italy, for making his visits feasible in 2018 and 2019, respectively. Professor Y. Al-Hadeethi also acknowledges the technical support of the Deanship of Scientific Research (DSR), King Abdulaziz University, Saudi Arabia.