Ioffe-Regel criterion for Anderson localization in the model of resonant point scatterers
S. E. Skipetrov, I. M. Sokolov
Physical review B
We establish a phase diagram of a model in which scalar waves are scattered by resonant point scatterers pinned at random positions in the free three-dimensional (3D) space. A transition to Anderson localization takes place in a narrow frequency band near the resonance frequency provided that the number density of scatterers ρ exceeds a critical value ρ_c ≃ 0.08 k_0^3, where k_0 is the wave number in the free space. The localization condition ρ > ρ_c can be rewritten as k_0 ℓ_0 < 1, where ℓ_0
... the on-resonance mean free path in the independent-scattering approximation. At mobility edges, the decay of the average amplitude of a monochromatic plane wave is not purely exponential and the growth of its phase is nonlinear with the propagation distance. This makes it impossible to define the mean free path ℓ and the effective wave number k in a usual way. If the latter are defined as an effective decay length of the intensity and an effective growth rate of the phase of the average wave field, the Ioffe-Regel parameter (kℓ)_c at the mobility edges can be calculated and takes values from 0.3 to 1.2 depending on ρ. Thus, the Ioffe-Regel criterion of localization kℓ < (kℓ)_c = const∼ 1 is valid only qualitatively and cannot be used as a quantitative condition of Anderson localization in 3D.