First a short historical rewiev of the theory development is given. The used methods, their advantages and limitations are outlined. The main obstacle in the way of the theory development was the difficulty of proper account for scattering of vibrational excitations by many-impurity complexes (clusters). In [1, 2] a solution of this problem was proposed. The main approach of these works is the two-staged averaging procedure. First the averaging and Fourier transforming of theory equations over positions of impurity clusters as a whole are performed. Then the averaging over the distances between impurities in the cluster is carried out. In order to avoid overcounting, each sum over cluster positions is multiplied by factor 1/n, where n is the number of impurities in the cluster. The theory possesses all the needed physical properties and is capable of describing the complex structure of vibrational spectra of solid solutions. The calculated vibrational spectra of the random linear chain (using n ~ 10) are in perfect agreement with the results of the computer simulations by Dean for  a linear chain of 8000 atoms [3]. Then the properties of the theory are illustrated using frequency dependences of spectra for different cases. The spectrum changes with a change of the impurity number in the cluster are shown. Physical nature, intensity and shape of different spectral peaks are discussed. The question related to a number of sites in the cluster, which is sufficient for describing the spectra of random lattices of different dimensions, and limits of applicability of the selfconsistent and non-selfconsistent approximations are also discussed. In conclusion the examples of experimental spectra explanation by the theory are presented.       

[1] Vinogradov V S 2005 Physics of the Solid State 47 10 1937

[2] Vinogradov V S 2007 Physics of the Solid State 49 11 2163

[3] Dean P 1961 Proc. Roy. Soc. A260 1301 263

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