Critical size of small particles for the development of resonances

It has been often claimed that the formation of a Kondo ground state requires a minimum particle radius of r≫${\mathrm{\zeta }}_{\mathit{R}}$=ħ${\mathit{v}}_{\mathit{F}}$/Δ, where Δ=${\mathit{k}}_{\mathit{B}}$ ${\mathit{T}}_{\mathit{K}}$ is the width of the Kondo resonance. We suggest that this minimum size is an artifact of the high symmetry of the sphere for which these calculations have been performed. A similar argument applies to other resonances. This question of minimal size is investigated for a Friedel resonance in two geometries, a sphere and a parallelepiped with a noninteger ratio of its edge lengths. A numerical calculation shows that the minimum linear size of the sample can be much smaller than ${\mathrm{\zeta }}_{\mathit{R}}$.