Cyclotron-Induced Damping of Gould –Trivelpiece Modes in Plasma Column with Dual Ion Species

Cyclotron-Induced Damping of Gould –Trivelpiece Modes in Plasma Column with Dual Ion Species

Daljeet Kaur¹, Manjeet Kaur^2*, Ruby Gupta³, Ambika Tundwal⁴, Simmi Singh⁵

^1,2,4,5 Department of Applied Sciences, Guru Tegh Bahadur Institute of Technology,Guru Gobind Singh Indraprastha University, New Delhi

³Department of Physics, Swami Shraddhanand College, University of Delhi, Alipur, Delhi

^*Corresponding author

DOI: https://doi.org/10.51584/IJRIAS.2025.1005000109

Received: 03 June 2025; Accepted: 07 June 2025; Published: 20 June 2025

ABSTRACT

Cyclotron damping of Gould –Trivelpiece modes is analysed in this study for a cylindrical plasma column containing either SF₆⁻ or K⁺ ions. The study highlights the contrasting roles of ion polarity in mediating wave–particle interactions that govern damping behaviour. It is observed that negative ions, due to their higher mass and weaker cyclotron resonance, result in reduced damping rates of TG modes. Conversely, positive ions enable stronger cyclotron coupling, leading to enhanced damping through more effective resonance with the wave field. Furthermore, the sensitivity of cyclotron damping rates to plasma electronegativity exhibits opposite trends for negative and positive ion plasmas. These findings underscore the critical influence of ion species and plasma composition on the propagation and attenuation of electrostatic waves in electronegative plasma columns, with implications for wave control and energy dissipation in beam-plasma systems.

Keywords: Cyclotron damping, Beam–plasma interaction, Wave–particle resonance, Gould –Trivelpiece mode, Negative ions (SF₆⁻), Positive ions (K⁺), Electronegativity dependence Plasma wave attenuation.

INTRODUCTION

TG waves, often referred to as lower hybrid waves (LHWs), represent electrostatic phenomena that exist in the frequency domain lying between the ion plasma and electron cyclotron frequencies. These waves have drawn a lot of theoretical and experimental attention across many decades [1–5] due to their effective capability for heating and energy absorption from electrons, notably observed near the outer region of the plasma.

In confined plasmas, Trivelpiece–Gould (TG) waves are typically characterized by short radial wavelengths, whereas in unbounded plasmas, they manifest as modes with short azimuthal wavelengths [6]. Praburam and Sharma [7] demonstrated that TG waves at higher harmonics can be excited using low-energy electron beams. Using the linear Princeton Q-1 device, Seiler et al. [8] investigated the instability of lower hybrid waves (LHWs) induced by a spiraling ion beam. Similarly, Chang [9] analysed the dynamics of LHW instabilities resulting from the perpendicular injection of ion beams. Sharma et al. [10] explored how a modulated electron beam can excite LHWs in a cylindrical plasma configuration. In a related study, Prakash et al. [11] examined LHW generation by ion beams and reported that the fastest growth rate of instability occurs when the wave’s phase velocity, aligned with the magnetic field, closely matches the electron thermal velocity.

Recent studies have expanded to include plasmas containing dust grains [12–17]. Both theoretical and experimental investigations have been carried out in non-magnetized [12] and weakly magnetized [13] dusty plasma environments. Sharma et al. [14] proposed a model in which ion-acoustic waves (IAWs) are excited by ion beams within a magnetized dusty plasma cylinder. Kaur et al. explored the excitation of Gould–Trivelpiece modes in dusty plasmas due to streaming particles, focusing on instability growth rates and wave–particle interactions using a fluid-based approach to describe mode dynamics in the presence of dust. Barkan et al. [16] analysed ion-acoustic waves in magnetized dusty plasmas and observed that the wave phase velocity increases with a higher number density of dust grains. Additionally, Rosenberg [17], employing Vlasov theory, examined the instabilities associated with dust-ion acoustic and dust-acoustic modes in unmagnetized dusty plasmas. Annaratone et al. [18] studied the rotational dynamics of a magnetized plasma in the linear device Mistral. They noted that the injected electrons from a central source travel radially outward because of the Lorentz force exerted by an axial magnetic field. Bettega et al. [19] carried out experimental studies on ion-driven diocotron instabilities in electron plasmas confined within a Malmberg–Penning trap. David [20] offered a comprehensive tomographic analysis of linear magnetized plasmas, focusing on the spatial and temporal characterization of plasma structures. Dem’yanov et al. [21] conducted a foundational investigation into equilibrium conditions and nonlocal ion cyclotron instabilities in plasmas influenced by crossed longitudinal magnetic fields and strong radial electric fields. Dubin [22] offers an in-depth review of the physics governing non-neutral plasmas confined in Penning traps, emphasizing the theory and behaviour of collective oscillation modes.

Fajans [23] investigates a specific class of plasma instabilities that occur in non-neutral plasmas due to the presence of a small population of ions. Jaeger [24] provides an in-depth investigation into low-frequency instabilities that arise in magnetized plasma systems with crossed electric and magnetic fields. Kabantsev and Driscoll [25] presents an important experimental study on instabilities in pure electron plasmas, particularly within Penning–Malmberg traps. Levy, Daugherty, and Buneman[26] analyzes ion-induced instabilities in non-neutral plasmas, particularly focusing on conditions relevant to diocotron mode. Peurrung, Notte, and Fajans The first direct observation of ion resonance instability—a key phenomenon involving resonant interactions between ions and collective plasma oscillations—was reported in [27]. Sakawa and Joshi [28] explored both the growth and nonlinear development of the modified Simon–Hoh instability in plasmas produced by electron beams. Yeliseyev [29] analysed the spectral features of modified ion cyclotron oscillations in non-neutral plasmas formed via gas ionization. Yeliseyev [30] analysed the oscillation spectrum of an electron gas containing a small fraction of ions, revealing how even a minor ion presence can significantly influence the collective oscillation modes. Yeliseyev [31] examined Trivelpiece–Gould modes in conjunction with low-frequency electron-ion instabilities within non-neutral plasmas. Kaur, Sharma, and Pandey [32] studied the excitation of Gould–Trivelpiece (TG) modes by a relativistic electron beam in a magnetized dusty plasma. Their findings highlighted how the interplay between the electron beam and dusty plasma can initiate TG modes, uncovering significant nonlinear phenomena and wave-particle interactions in magnetically influenced complex plasma systems.

In this study, a theoretical model is developed to examine the cyclotron damping of the Gould–Trivelpiece (TG) mode by an electron beam interaction with negative ion and positive ion plasma in a finite cylindrical magnetized plasma. Section 2 presents the instability analysis for interaction with   negative ions and positive ions in finite geometry. The fluid approach is employed to determine the responses of beam electrons, plasma electrons, and plasma ions ( negative ions and positive ions) The expressions for the instability growth rates are derived using first-order perturbation theory. In Section 3, numerical analysis and discussions are given. Comparison with experimental and theoretical works in section 4 and concluding remarks are summarized in Section 5.

Instability Characterization

We examine a cylindrical plasma column of radius containing negative ions , positive ions , and electrons, with equilibrium densities of electrons, positive ions, and negative ions denoted by  ,  and respectively, where represents the overall plasma density. The electrons have temperature and mass  , while the positive ions are characterized by temperature and mass  . Similarly, the negative ions have temperature and mass  . An electrostatic wave, specifically a Gould–Trivelpiece (TG) wave, is considered to propagate at an angle relative to the external magnetic field, with the wave vector k lying in the x–z plane.

This equilibrium is perturbed by an electrostatic disturbance, and the corresponding potential associated with this perturbation is expressed as

                                                                                                                         (1)                                                                               

An electron beam travels along the z-axis, aligned with the external magnetic field, with a uniform density , radius f_b, and equilibrium velocity . Before the introduction of any perturbation, the combined beam–plasma system is assumed to be quasineutral

such that    here we have taken  

Each of the three species is modeled as a fluid and obeys the continuity and momentum (equation of motion) equations, expressed as:

Upon linearizing the equations of motion and continuity [cf. Eqs. (2) and (3)], the resulting expressions for the perturbations in electron density, positive ion density, and negative ion density are obtained as:

where is the electron gyro frequency,   is the positive ion-cyclotron frequency,  is the negative ion gyro-frequency. Substituting the expression from Eqs. (4), (5), (6) & (7) in Poisson’s equation 

and solving for finite geometry

where    are the electron, positive ion, negative ion and electron beam frequency. Solving Eq. (8)

where,

                                                                                                      (10)

In the absence of the electron beam, Eq. (10) simplifies and can be reformulated as:

Equation (11) is recognized as a standard Bessel equation, and its general solution can be expressed as , where L and M are arbitrary constants. Here J₀ denotes the zeroth-order Bessel function of the first kind, while Y₀ denotes the zeroth-order Bessel function of the second kind.

At , Y₀∞ and hence m=0,  At  must vanish, hence,,, where  represents the zeros of the Bessel function J₀(X). In the presence of the electron beam, the wave function  can be expressed as an orthogonal set of eigenfunctions.

Further, using the value of  in Equation (10) from Equation (12) and multiplying  both  the sides of  Eq.(9)  by    and  integrating  over  t from  0  to  ,  here  radius of plasma is,  Keeping solely the principal mode n = d we obtain 

                                                                                                               (13)   

where      if   

               =1 if  

Upon substitution of the value of  from Eq. (9) in Eq. (13) we obtain, 

where ,  

Now, evaluate Eq. (14) under two conditions: (i) interaction between an electron beam and a plasma that includes negative ions () and (ii) with positive () ion.

Case I: Interaction between an electron beam and a plasma that includes negative ions () 

We will reduce Eq. (14) in the absence of positive () ion to

where,  

Eq. (15) can be rewrite as

, 

where,                                                                                                                     (16)

When the beam is present, the frequency  can be expanded as as, Here,  the small frequency deviation arises from the finite term on the right-hand side of Eq. (16). Cyclotron damping is represented by the imaginary part of the frequency, when it is negative due to cyclotron resonance between the wave (e.g., Gould –Trivelpiece mode) and ion gyro-motion. 

According to Mikhailovski [32], the growth rate of the unstable mode can be expressed as

The real part of the frequency for the unstable mode is expressed as

                                                                                                        (19)

Case II: Interaction between an electron beam and a plasma that includes positive ion () 

We will reduce Eq. (14) in the absence of negative () ion to

                                                                             (20)           

Further solving Eq. (20) we will get

                                                                                                (21)

where,  

and

By applying a similar procedure as in Case I, the growth rate of the unstable mode can be expressed as

The real part of the frequency for the unstable mode is expressed as

In both cases, the real part of the unstable mode’s frequency increases with the beam voltage, consistent with the experimental findings reported by Chang [33].

Numerical Analysis 

For the numerical calculations, we have employed the experimental plasma parameters reported by Song et al. [37]. The plasma density value used is , T_e=0.2eV, T_K⁺= T_e , T = T_e ,  . Beam radius =1.2 cm, beam density ranges from    electron beam energy= 30KeV, strength of applied magnetic field B=10⁴  Gauss and mode number of Bessel function n=1 (1^st zero) . Using Equations (17), we have plotted  figure1 the growth rate (sec^-1) as a function of the longitudinal wave number (cm⁻¹) for various values of , considering the presence of both a negative ion and an electron beam and using Equations (22), we have plotted  figure 2 the growth rate (sec^-1) as a function of the longitudinal wave number (cm⁻¹) for various values of , considering the presence of both a positive ion and an electron beam.

The electron beam interaction-induced plasma growth behaviour is quite different based on whether the plasma consists almost entirely of negative ions (like ) or positive ions (like K⁺). This can be seen from the comparison between the two plots given. In the initial graph (Fig.1), representing the coupling of an electron beam with a plasma with negative ions, the rate of growth (Γ1) is seen to reduce with longitudinal wave number  (cm⁻¹). In addition, in this cylindrical geometry, the cyclotron damping amplitude of the Trivelpiece–Gould (TG) mode is quite low. Increasing electronegativity ϵ — as the density ratio between electrons and negative ions — causes a moderate increase in damping. This is largely because the free electron density decreases, which alters the plasma dielectric response and consequently changes the TG mode dispersion characteristics. Nevertheless, the massive weight and sluggish cyclotron behaviour of negative ions like restrict their capacity to resonate effectively with the wave field. Consequently, the energy exchange between the wave and the population of ions amorphously through cyclotron resonance is still inefficient. This inefficient wave–particle coupling quashes cyclotron damping, enabling the TG modes to sustain low attenuation and feeble interaction with the beam. As a result, beam-induced instabilities are damped in highly electronegative plasmas by virtue of the inertial dominance and weak resonant characteristics of the negative ion component.

By contrast, Figure 2 is related to the interaction of the same electron beam with a plasma with positive ions in the majority. In this case, the trend of the growth rate is very different from the above. The growth rate rises with rising  (cm⁻¹), and its value is considerably larger. The cause is rooted in the relatively lower mass of  ions and their greater dynamic responsiveness to perturbations created by the beam. In this arrangement, the electron beam efficiently excites electrostatic Trivelpiece–Gould (TG) modes, allowing energy transfer into the plasma and the triggering of wave instabilities. A surge in electronegativity ϵ — the electron to negative ion density ratio — is shown to enhance cyclotron damping and thus inhibit instability growth. This happens due to increased electronegativity diminishing the free electron population, which weakens collective electron-ion dynamics and degrades the requirements for robust cyclotron resonance. Consequently, resonance-mediated energy coupling between the plasma and beam diminishes, resulting in smaller instability amplitudes.

In general, the comparison highlights the prominence of ion species and electronegativity in determining beam–plasma interactions through cyclotron damping effects. Large mass and weak cyclotron sensitivity of heavy negative ions like  result in poor coupling with TG modes, resulting in increased damping and reduction of beam-driven instabilities. Conversely, plasmas with light positive ions such as allow for stronger cyclotron resonance, enabling more effective absorption of energy from the beam and consequently diminishing damping—enabling instabilities to develop more easily. In addition, the reverse tendencies exhibited by rising electronegativity in negative versus positive ion plasmas accentuate the intricate interconnection between plasma content and wave–particle interacting dynamics. These findings are important to the stability control of electronegative plasmas and also for beam-driven plasma system design for laboratory and space applications. 

Using the same equations, the growth rates for Case I (Eq. 17) and Case II (Eq. 17) have been analysed as functions of beam density  (cm⁻³) for various values of longitudinal wave number  (cm⁻¹), as shown in Figure 3 and Figure 4, respectively.

Figure 3 and Figure 4   show the electron beam density  (cm⁻³) dependence of the growth rate for various longitudinal wavenumbers  (cm⁻¹), in electronegative (and electropositive (K⁺) plasmas, respectively. The first plot (Fig. 3), that of the  plasma, shows a comparatively lower value of growth rate across the entire beam density range. The growth of Γ1 with  (cm⁻³) is sub-linear and gradual, and the rate of growth decreases with higher  (cm⁻¹). This is typical of inhibited wave growth caused by the presence of negative ions, which enhance plasma inertia and decrease overall coupling between the electron beam and plasma oscillations. On the contrary, the second plot (Fig.4) for the plasma indicates much larger growth rate for the same or even lower beam densities. The plots are more steeply increasing, indicating increased beam-plasma instability, particularly at higher  (cm⁻¹) values. This higher Γ2 indicates that the availability of light positive ions () enables more efficient energy transfer between the beam and the plasma, since the plasma is more responsive to perturbations without the added damping effects of heavy negative ions.

The striking difference in cyclotron damping behaviour of Trivelpiece–Gould modes is a result of the intrinsic ion charge polarity and mass differences. In plasmas with heavy negative ions like , poor wave–ion energy transfer due to the heavy ion mass and poor cyclotron resonance means cyclotron damping is very low. This weak coupling permits TG modes to propagate with minimal attenuation, in essence screening the plasma from beam-driven energy transfer. By contrast, positive ion plasmas composed of lighter species such as show stronger cyclotron resonance, permitting more efficient coupling between wave and ion motion. This leads to greater cyclotron damping, wherein wave energy is actively absorbed by the ion population. 

In addition, the impact of rising axial wavenumber  (cm⁻¹) differs between the two cases. In plasmas, damping continues to be weak or even diminishes with  (cm⁻¹), which means higher spatial frequency TG modes remain less damped and remain predominantly undamped. On the other hand, in plasmas, cyclotron damping gets stronger with growing  (cm⁻¹), echoing stronger wave–ion coupling and greater efficiency in energy dissipation at shorter wavelengths. For better visualization and understanding 3-D plots representing the variation of growth rates have been given in Figures 5 and 6 corresponding to 2-D plots (Figs. 3 and 4) respectively.

This comparison is essential in planning plasma systems where controlled instability or suppression are necessary, e.g., in plasma-based accelerators, thrusters, or plasma processing environment where negative ions such as are deliberately introduced.

EXPERIMENTAL AND THEORETICAL EVIDENCE SUPPORTING CYCLOTRON DAMPING BEHAVIOUR IN ELECTRONEUTRAL AND ELECTRONEGATIVE PLASMAS

Comparison of cyclotron damping characteristics of Trivelpiece–Gould (TG) modes in and  plasmas reveals significant ion mass and charge polarity roles. In heavy negative ion dominant plasmas like , the weak cyclotron resonance and enhanced plasma inertia result in significantly lower damping rates of TG modes. This weak coupling allows the TG modes to travel with little attenuation and hence screen the plasma from beam-driven energy transfer. This kind of suppression of wave excitation and damping by heavy negative ions has experimental backing in which electron beams excite ion cyclotron waves in electronegative plasmas, with observations of reduced wave-particle energy transfer caused by the heavy negative ions [34]. In the same way, experimental observation of plasma wave propagation in inductively coupled plasmas shows that adding negative ions minimizes instability growth by enhancing plasma inertia and reducing effective beam–plasma coupling [35]. In addition, the damping of beam-excited wave generation in plasmas with heavy negatively charged particles is similar to dusty plasma experimental results with negatively charged dust particles, wherein ion beam excitation of the dust acoustic wave is greatly suppressed [36].

By contrast, plasmas with lighter positive ions like have more intense cyclotron resonance, enabling improved wave–ion coupling and increased cyclotron damping. Increasing damping with increasing axial wavenumberindicates more intense wave-particle interaction and greater energy dissipation efficiency at shorter wavelengths. These differences highlight the important role of plasma composition in the stability and energy processes of beam-plasma systems.

CONCLUSION

The electron beam interaction in and plasmas compared shows different cyclotron damping properties of Trivelpiece–Gould modes, dominated mainly by ion mass and electronegativity. For  dominated plasmas, the heavier negative ions show weak cyclotron resonance, causing lower damping and very little energy transfer to the wave field. This produces relatively prolonged TG mode propagation with negligible attenuation. Conversely, plasmas of lighter positive ions show greater cyclotron coupling, which increases energy transfer and results in more efficient damping of TG modes. Increasing axial wavenumber  increases damping in plasmas, whereas damping remains weak in environments. These findings highlight that positive ion plasmas are more effective in wave energy damping through cyclotron mechanisms, while negative ion plasmas naturally inhibit such damping because of their inertial and resonant constraints.

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