Kinematic Change in Conduction-Electron Density of States Due to Impurity Scattering. I. The Problem of a Single Impurity

For conduction-electron-impurity scattering the conduction-electron density of states is calculated by making use of the thermodynamical Green's-function technique with particular respect to its spatial structure. If the interaction depends on the momenta of the scattered waves in such a way that it is important only in the neighborhood of the Fermi surface (characterized by the corresponding energy width $\Delta$), a coherence length can be introduced: ${\xi }_{\Delta }=\frac{v_F}{\Delta }$, where $v_F$ is the Fermi velocity. Present experimental data obtained for different Kondo systems can be interpreted as yielding 5-50Å for ${\xi }_{\Delta }$. The spatial variation of the change in the electron density of states is found to be as follows: (a) In the short-range region $r\ll {\xi }_{\Delta }$, the change is negative definite because of the interference between the incoming and outgoing scattered waves, and (b) in the long-range region $r\gg {\xi }_{\Delta }$, it shows the Friedel oscillation. The results are expressed also by a phase shift $\delta$. Only the amplitude of change in the electron density of states depends on the scattering amplitude; however, the spatial structure is unaffected.