Transverse Conductivity of a Degenerate System of Landau Electrons and Optical Phonons

The transverse conductivity of a system of independent electrons weakly interacting with a gas of optical phonons in an external magnetic field is obtained for the case ω=ω₀, where ω=|e|Hm* (=c=1) and where ω₀ is the optical-phonon frequency, in the limit βω1. For the densities considered (nₑ=10¹²-10¹⁵ cm^-2) the effects of the electron-electron interaction are negligible. The only mechanism used to remove the logarithmic infinity predicted, as Gurevich and Firsov have pointed out, by the usual Titeica expression is the electron-optical-phonon interaction itself (i.e., "collision broadening"). The value of σˣˣ|_ω=ω₀ is found to be ${{{σ}}ˣˣ|}_{{ω}={{ω}}₀}=3/4{{1+({2}{{}{π}}){({β}{ω})}1/2F({β}{ω}{{α}}2/3){{σ}}ˣˣ|}_{{ω}{≠}{{ω}}₀},$ where ${{{σ}}ˣˣ|}_{{ω}{≠}{{ω}}₀}=(4/3){n}ₑ{α}{β}{e}²{e}^{{-}{β}{ω}}{({m}*)}^{{-}1},$ ${α}$ being the Fr\"ohlich coupling constant and $F$ of order unity for ${β}{ω}{}1$ and ${α}{}1$.