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• W2078703492 abstract "The influence of localized impurities in solids may be represented either by (i) the effect of randomly distributed classical scattering centers for which only averaged quantities are meaningful or, as proved herein, (ii) the effect of an equivalent quantum description which incorporates both the random character and the process of averaging. Adopting the latter viewpoint, we readily make contact with diagrammatic perturbation methods familiar in field theory and many-body studies. As an application of this formalism, various approximations, all of infinite order in the perturbation, are derived for the average density of states in an idealized solid for which the effects of the impurities prevail and the periodic lattice may be neglected. The results of these approximations are compared with the exact results obtained by Frisch and Lloyd for a special model in which the impurity potential may be represented as an attractive one-dimensional delta-function. One of the approximations discussed is a direct analog of the Brueckner model, and for the delta-function potential example this approximation can be solved exactly. In the case of a high density of impurities, our solutions demonstrate the well-known value of propagator modification, while, for a low density of impurities, several solutions exhibit an impurity band that is isolated from the conduction band. Finally, we discuss the average wave function, a quantity that is open to some arbitrariness because a wide latitude exists in the choice of a rule by which a wave function is paired with a set of impurity sites. Two distinct “pairingrules” are analyzed in detail: one rule restricts the wave and its derivative to have specified initial values but demands neither normalizability nor singlevaluedness; the other rule requires that every acceptable wave be an eigenvector whose phase, relative to the other waves, is selected in a definite manner. Differential equations are presented which are obeyed by the two differently defined average wave functions in the case of a one-dimensional delta-function potential." @default.
• W2078703492 created "2016-06-24" @default.
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• W2078703492 date "1961-01-01" @default.
• W2078703492 modified "2023-10-16" @default.
• W2078703492 title "The modification of electron energy levels by impurity atoms" @default.
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• W2078703492 doi "https://doi.org/10.1016/0003-4916(61)90051-3" @default.
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