Electron-Density Oscillations in a General Potential

The asymptotic density of a free noninteracting electron gas is discussed in the presence of a general potential $V\left(X\right)\mathrm{}$, where X is a vector in one, two, and three dimensions corresponding, respectively, to the potential of a surface barrier, an edge dislocation, and an impurity. At zero temperature, oscillations in the density have the form $A{X}^{-\frac{1}{2}(\nu +3)}cos(2\mathrm{}{k}_{f}X+\theta )$, where $\nu$ is the dimensionality of X, ${\mathrm{k}}_f$ is the Fermi momentum, and $\theta$ is a phase angle. The amplitude $A$ is determined by the backward scattering amplitude at the Fermi energy for the potential $V\left(X\right)\mathrm{}$. At a finite temperature the amplitude of the oscillations in normal metals is reduced approximately by the factor $\frac{\xi }{sinh\xi }$, where $\xi =\left(\frac{2\pi }{\beta k_f}\right)X$, and $\beta$ is the reciprocal of the thermal energy $\mathrm{KT}$. In the high-density limit, the results of the dielectric theory become the Born-approximation version of the exact scattering theory. Mild restrictions on the potential to guarantee certain analytical properties of the scattering matrix are imposed.