In inertial confinement fusion plasmas, the trapped particle instability (TPI) significantly influences energy transfer dynamics in stimulated Raman scattering and two plasmon decay. This study introduces a relativistic theoretical framework that addresses the limitations of the classical Kruer–Dawson–Sudan model under weakly relativistic temperature conditions where trapped electron velocity is comparable to the speed of light. By integrating relativistic corrections to particle dynamics and wave interactions, the new model achieves improved accuracy in predicting TPI growth rates compared to the KDS model. The relativistic TPI dispersion relation, derived from the Vlasov–Poisson system, incorporates velocity-dependent bounce frequency modifications, the Maxwell–Jüttner distribution for particle dynamics, resolving the growth rate discrepancies in and decay trends observed in non-relativistic models, where δk denotes the difference between the sideband wave number and the electron plasma wave (EPW) wave number, and is the Debye length. Simulations confirm the framework’s accuracy by solving sideband growth rates triggered by externally driven EPWs, validating TPI-induced electron trapping mechanisms. The model also correctly predicts TPI growth rate dependencies on the amplitude and the wavenumber of EPW, establishing the necessity of relativistic physics in modeling TPI under high-energy-density conditions and offering a foundational tool for refining plasma instability theories and guiding inertial confinement fusion experimental designs.
Numerical solution of the linear dispersion relation in a relativistic pair plasma