The sharp Fermi surface gives a nonlocal oscillatory electronic response around an impurity. In a three-dimensional isotropic metal its far-field oscillations have characteristic wavevector and algebraically decaying amplitude. They are absent from the purely exponential Thomas-Fermi screening potential. The local screening formula is a useful long-wavelength model, not an exact asymptotic expression at every distance.
A single isotropic three-dimensional valley with two equal spin populations has . A density oscillation at wavevector therefore has the wavelength in the title. This depends on carrier density, independently of effective mass and dielectric screening strength. Extra valley degeneracy, spin polarization or reduced dimensionality changes the state-counting relation.
For the radial induced-charge screening kernel, integrate the sine transform twice by parts, with and vanishing remaining endpoint terms. This gives . A Cauchy principal value singularity at contributes to that integral: set and use for . Smoothly localizing around justifies the large- extension of the limits. In a three-dimensional isotropic Fermi gas, .
