Trivelpiece-Gould Mode Excitation in Magnetized Plasmas: A Review of Electron, Ion, Relativistic Beam, and Streaming Ion Effects

Abstract

This review explores the excitation of Trivelpiece–Gould (TG) modes in bounded magnetized plasmas under different streaming particle beams, such as electron beams and ion beams, as well as relativistic electron streams. The TG mode, an electrostatic wave trapped in cylindrical geometries with an axial magnetic field, is strongly affected by beam–plasma interactions and boundary conditions. Evidence indicates that electron beams, especially those aligned along the magnetic field direction, are capable of driving TG mode instabilities effectively with growth rates highly dependent on beam velocity, density, and anisotropy of temperature. Contrarily, ion beams—more so in low-temperature dusty plasmas—may excite TG modes by resonance with the modified plasma eigen frequencies, as demonstrated in recent analytical and simulation works. The addition of relativistic electron beams results in increased coupling and more extended instability regimes from relativistic mass effects and longitudinal electric field interactions. The presence of dust grains also changes the dispersion and increases low-frequency modes, making the instability richer. These results have been validated by experimental and theoretical research which obtains dispersion relations, calculates growth rates, and investigates mode structures by employing Bessel function eigenmodes in radially confined plasmas. Overall, particle streaming excitation of TG modes presents new opportunities for manipulating plasma wave phenomena in laboratory and space plasmas.

Introduction

The Trivelpiece–Gould (TG) mode is an electrostatic wave that propagates in magnetized, bounded plasmas, typically within cylindrical geometries. Unlike free-space Langmuir waves, TG modes arise due to the influence of conducting boundaries and axial magnetic fields, resulting in discrete radial mode structures.

In Cold Plasma:
TG modes are hybrid electrostatic and magnetostatic waves, confined below the electron plasma frequency, and strongly dispersive. Cold plasma TG modes can be modeled using fluid theory, showing radial eigenmodes governed by Bessel functions. These modes are important for plasma diagnostics and have been experimentally observed, especially in Q-machines and helicon sources. They exhibit phenomena such as mode conversion and sheath effects.

In Hot Plasma:
Thermal effects introduce kinetic phenomena like Landau damping and finite Larmor radius corrections. Hot plasma TG modes require kinetic modeling (Vlasov equations) and show temperature-dependent dispersion and damping. Experimental results confirm kinetic theory predictions. Hot plasmas demonstrate complex wave-particle interactions, affecting beam-driven instabilities and wave propagation in fusion and space plasmas.

In Dusty Plasmas:
Dusty plasmas include charged dust grains that significantly alter plasma dynamics and TG mode properties by changing inertia, plasma frequency, and dielectric response. Electron beams can excite TG modes here, with dust density and charge strongly influencing mode growth and stability. These effects are relevant both in laboratories and astrophysical environments like comet tails and planetary rings.

Electron Beam Excitation:
Electron beams traveling through dusty, magnetized plasmas can resonate with TG modes, triggering beam–plasma instabilities. Dust modifies excitation thresholds and growth rates, sometimes enhancing and other times damping instabilities. Research shows complex dependencies on beam energy, dust characteristics, and magnetic field strength.

Relativistic Electron Beams:
High-energy (relativistic) electron beams interact with TG modes, with beam velocity and magnetic confinement strongly affecting wave dispersion and stability. Nonlinear phenomena such as wave breaking and mode coupling emerge. Studies reveal that high dust density increases mode frequency and growth, while very high beam energies can reduce coupling effectiveness.

Ion Beam Excitation:
Ion beams excite TG modes at lower frequencies, with dynamics linked to ion cyclotron and lower hybrid resonances. Dust and beam parameters affect mode frequencies and instability thresholds. Ion beam-driven TG modes are important for plasma device diagnostics and control, showing rich dispersion and potential nonlinear behaviors.

Recent Advances:
Recent theoretical and experimental work has focused on the combined effects of dust, relativistic streaming particles, and magnetic fields on TG mode stability and dispersion. Key findings highlight that dust density boosts TG mode frequency and growth rates, while relativistic beam effects can saturate and reduce instability due to phase mismatches. These insights are applicable to advanced plasma devices and space environments containing dust and high-energy particles.

Conclusion

This review emphasizes the rich and multiscale character of Trivelpiece–Gould (TG) mode excitation in magnetized plasmas exposed to different kinds of streaming particle beams, such as electron beams, ion beams, and relativistic electron streams. The TG mode, as an electrostatic surface-confined wave, is highly responsive to plasma conditions, beam parameters, and magnetic field geometries. The addition of charged dust grains adds further complexity by altering dispersion properties, introducing novel instability thresholds, and modifying wave–particle resonance mechanisms. Electron beams are still very efficient in exciting TG mode instabilities, particularly under aligned fields, with growth rates depending significantly on the beam energy, density, and anisotropy in temperature. Ion beams, although generally characterized by lower growth instability because of higher mass, have peculiar coupling characteristics close to ion cyclotron and hybrid resonance domains. Relativistic electron beams add more relativistic effects that transform the coupling conditions and impact the nonlinear development of the modes. Dusty plasmas introduce an additional richness of tunability by frequency shifting modes and facilitating or inhibiting instabilities based on the properties of the dust. Dust density, charge sign, and grain size are critical parameters in altering the dielectric medium and plasma inertia. Together, the excitation of TG modes in beam-driven magnetized plasmas offers an important diagnostic and control tool for exploring and manipulating wave properties in laboratory and natural plasma systems. Further investigation in this research area is critical to the continued development of applications in plasma confinement, particle acceleration, wave diagnostics, and the study of space and astrophysical plasma processes.

References

[1] W. Trivelpiece and R. W. Gould, \"Space charge waves in cylindrical plasma columns,\" J. Appl. Phys., vol. 30, no. 11, pp. 1784–1793, 1959. [2] T. H. Stix, Waves in Plasmas. New York, NY, USA: American Institute of Physics, 1992. [3] F. F. Chen, Introduction to Plasma Physics and Controlled Fusion, 3rd ed. New York, NY, USA: Springer, 2016. Fig [4] M. A. Lieberman and A. J. Lichtenberg, Principles of Plasma Discharges and Materials Processing, 2nd ed. Hoboken, NJ, USA: Wiley, 2005. [5] D. R. Nicholson, Introduction to Plasma Theory. New York, NY, USA: Wiley, 1983. [6] J. D. Callen, \"Fundamentals of plasma physics,\" Univ. Wisconsin, 2006. [Online]. Available: https://homepages.cae.wisc.edu/~callen/chap3.pdf [7] R. J. La Haye and D. W. Ross, \"Mode structures in cylindrical bounded plasmas,\" Phys. Fluids, vol. 18, pp. 1959–1965, 1975. [8] A. H. Boozer, \"Electrostatic modes in bounded plasmas,\" Phys. Fluids, vol. 17, no. 6, pp. 1218–1224, 1974. [9] T. Kaneko and R. Hatakeyama, \"Excitation of electrostatic modes in bounded plasmas,\" Phys. Rev. Lett., vol. 95, no. 8, pp. 085001, 2005. [10] M. Light and J. M. Dawson, \"Electrostatic waves in cold plasmas with boundaries,\" Phys. Rev. Lett., vol. 33, no. 12, pp. 742–745, 1974. [11] J. L. Shohet, D. N. Hill, and D. Arnush, \"Nonlinear interactions of TG modes in bounded plasmas,\" Phys. Fluids, vol. 23, pp. 399–406, 1980. [12] A. I. Smolyakov and P. H. Diamond, \"Nonlinear mode coupling of TG waves in bounded cold plasmas,\" Phys. Plasmas, vol. 7, pp. 1349–1352, 2000. [13] T. H. Stix, Waves in Plasmas. New York, NY, USA: American Institute of Physics, 1992. [14] D. R. Nicholson, Introduction to Plasma Theory. New York, NY, USA: Wiley, 1983. [15] L. D. Landau, \"On the vibrations of the electronic plasma,\" J. Phys. (USSR), vol. 10, pp. 25–34, 1946. [16] E. W. Laing and T. J. M. Boyd, \"Trivelpiece–Gould modes in hot plasmas,\" Plasma Phys., vol. 15, no. 2, pp. 141–153, 1973. [17] S. Ichimaru, Basic Principles of Plasma Physics. W.A. Benjamin, 1973. [18] B. Scott, \"FLR effects in kinetic descriptions of bounded plasmas,\" Phys. Plasmas, vol. 12, no. 6, p. 062314, 2005. [19] S. Singh and V. K. Tripathi, \"Kinetic theory of beam–excited Trivelpiece–Gould modes in a hot magnetized plasma column,\" Phys. Plasmas, vol. 14, p. 032104, 2007. [20] P. K. Shukla and A. A. Mamun, Introduction to Dusty Plasma Physics. Bristol, UK: IOP Publishing, 2002. [21] M. Y. Yu and P. K. Shukla, \"Thermal effects on electrostatic waves in bounded plasmas,\" Phys. Fluids, vol. 23, no. 8, pp. 1475–1480, 1980. [22] R. Hatakeyama, T. Kaneko, and K. Muraoka, \"Thermal modulation of electrostatic wave propagation in hot magnetized plasmas,\" Phys. Plasmas, vol. 10, no. 6, pp. 2495–2500, 2003. [23] J. L. Shohet, D. N. Hill, and D. Arnush, \"Electron beam driven instabilities in hot bounded plasmas,\" Phys. Fluids, vol. 27, pp. 343–350, 1984. [24] T. H. Stix, Waves in Plasmas. New York, NY, USA: American Institute of Physics, 1992. [25] D. G. Swanson, Plasma Waves, 2nd ed. Bristol, U.K.: Institute of Physics Publishing, 2003. [26] N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics. San Francisco, CA, USA: San Francisco Press, 1986. [27] S. C. Sharma and M. Sugawa, “The effect of dust charge fluctuations on ion cyclotron wave instability in the presence of an ion beam in a plasma cylinder,” Phys. Plasmas, vol. 6, no. 11, pp. 4264–4269, 1999 [28] V. Prakash, S. C. Sharma, V. Vijayshri, and R. Gupta, “Effect of dust grain parameters on ion beam driven ion cyclotron waves in a magnetized plasma,” Prog. Electromagn. Res. M, vol. 36, pp. 161–168, 2014 [29] P. K. Shukla and A. A. Mamun, “Introduction to dusty plasma physics,” IOP Publishing, 2002 [30] F. Verheest, Waves in Dusty Space Plasmas. Dordrecht, The Netherlands: Kluwer Academic Publishers, 2000 [31] S. V. Vladimirov and M. Nambu, “Beam–plasma instability in a dusty plasma,” Phys. Rev. E, vol. 52, no. 4, pp. R2172–R2174, 1995. [32] P. K. Shukla and A. A. Mamun, “Beam-driven electrostatic modes in magnetized dusty plasmas,” Phys. Scr., vol. T98, pp. 123–127, 2002 [33] F. Verheest, Waves in Dusty Space Plasmas, Kluwer Academic, Dordrecht, 2000. [34] P. K. Shukla and A. A. Mamun, Introduction to Dusty Plasma Physics, IOP Publishing, 2002. [35] M. Rosenberg, “The physics of dusty plasmas,” IEEE Trans. Plasma Sci., vol. 25, no. 6, pp. 1174–1181, Dec. 1997. [36] A. Barkan, N. D\'Angelo, and R. L. Merlino, “Laboratory observation of the dust-acoustic wave mode,” Phys. Plasmas, vol. 2, no. 10, pp. 3563–3565, 1995. [37] S. C. Sharma and A. Gahlot, “Excitation of upper-hybrid waves by a gyrating relativistic electron beam in a magnetized dusty plasma cylinder,” Phys. Plasmas, vol. 16, no. 12, p. 123708, 2009. [38] V. Prakash and S. C. Sharma, “Excitation of surface plasma waves by an electron beam in a magnetized dusty plasma,” Phys. Plasmas, vol. 16, p. 093703, 2009. [39] R. L. Stenzel and J. M. Urrutia, “Trivelpiece–Gould modes in a uniform unbounded plasma,” Phys. Plasmas, vol. 23, no. 9, p. 092103, 2016. [40] R. K. Bera, A. Mukherjee, S. Sengupta, and A. Das, “Excitation and breaking of relativistic electron beam driven longitudinal electron–ion modes in a cold plasma,” arXiv preprint, arXiv:2003.10490, 2020. [41] D. Kaur, S. C. Sharma, and R. S. Pandey, “Excitation of a Gould–Trivelpiece (TG) mode by relativistic electron beam in magnetized dusty plasma,” J. Atomic, Molecular, Condens. Nano Phys., vol. 5, no. 2, pp. 81–96, 2018. [42] S. Sarkar, S. Sengupta, and A. Das, “Saturation of beam–plasma instability in relativistic regimes,” Phys. Plasmas, vol. 29, no. 10, p. 102101, 2022. [43] H. Bera and S. Sengupta, “Wakefield generation by a relativistic electron beam in cylindrical geometry with dust,” J. Plasma Phys., vol. 87, p. 905870323, 2021. [44] A. Roy, A. Das, and A. Sen, “Relativistic beam-induced nonlinear structures in bounded dusty plasma,” Phys. Plasmas, vol. 30, no. 4, p. 042109, 2023. [45] N. Saini and V. K. Tripathi, “Oblique propagation of electrostatic waves excited by ion beam in a dusty plasma cylinder,” Phys. Plasmas, vol. 23, p. 013701, 2016. [46] A. Kumar and S. K. Mishra, “Low-frequency electrostatic mode excitation by ion beam in collisional dusty magnetoplasma,” Indian J. Phys., vol. 94, pp. 1323–1330, 2020. [47] D. Kaur, S. C. Sharma, R. S. Pandey, and R. Gupta, “Excitation of Gould–Trivelpiece mode by streaming particles in dusty plasma,” Laser and Particle Beams, vol. 37, pp. 122–127, 2019

Copyright

Copyright © 2025 Daljeet Kaur. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.