III. INTRA-LANDAU-BAND PLASMON SPECTRUM IN THE PRESENCE OF RSOI AT FINITE TEMPERATURE The plasma modes are readily furnished by the roots of the longitudinal plasmon disper- sion relation obtained from equation (9) as along with the condition Im Π ± 0 (¯ q, ω ) = 0 to ensure long-lived excitations. Equation (11) can be expressed as with where P is the principal value. The magnetoplasmon modes originate from two kinds of electronic transitions, those involving different Landau bands (inter-Landau band plasmons)[10–13] and those within a single Landau-band (intra-Landau band plasmons). Inter-Landau band plasmons involve the local 1D magnetoplasma mode and the Bernstein-like plasma resonances[25], all of which involve excitation frequencies greater than the Landau-band separation ( ∼ ℏ ω$_{c}$ ). On the other hand, intra-Landau band magnetoplasmons resonate at frequencies comparable to the bandwidths, and the existence of this new class of modes is due to finite width of the Landau levels caused by the modulation. In order to investigate the effects of RSOI and the modulation on the magnetoplasmon spectrum requires the study of the intra-Landau band magnetoplasmons. In the absence of modulation, the plasmon spectrum of a 2DEG with RSOI has been analyzed in detail elsewhere[10–13] with and without an external magnetic field. In the present work, we investigate the magnetoplason spectrum with RSOI in the presence of modulation. We show below that as a result of both modulation and RSOI, Weiss oscillations in the magnetoplasmon spectrum are found to exhibit a beating pattern. This occurs in the low magnetic field regime. At higher fields, Weiss oscillations are suppressed and Shubnikov de Haas (SdH) oscillations are the dominant magnetic oscillations. SdH type of oscillations result from the emptying out of electrons from successive Landau bands when 7 1 − v$_{c}$ (¯ q ) Re Π ± 0 (¯ q, ω ) = 0 (11) 1 = 2 πe 2 k ¯ q ℏ 2 m ∗ ω$_{c}$ πa ∑ n,n ′ C$_{nn}$ ′ ( l $^{2}$¯ q 2 2 )( I ± ( ω ) + I ± ( − ω )) , (12) I ± ( ω ) = P a ∫ 0 dx$_{0}$ f ± ( ε ( n, x$_{0}$ )) ℏ ω − ε ± ( n, x$_{0}$ ) + ε ± ( n ′ , x$_{0}$ + x ′ $_{0}$) , (13)