Lennard-Jones Potential inside a Spherical Cavity

The Lennard-Jones potential for an atom inside a conducting spherical cavity is derived. The interest is in possible anomalous effects near the focal point of a cavity viewed as a spherical mirror. We find a smooth dependence of the potential on the distance of the atom from the surface of the cavity. 1 Plane Mirror and the C3 coefficient Let us consider first a plane mirror, assumed to lie in the x−y plane. We locate a nucleus of charge Z at a distance L on the z axis above the plane. We suppose that there are Z electrons located at coordinates ~ ρi = (ξi, ηi, ζi) relative to the nucleus. The image of the nucleus is at the point (0, 0, −L) and the images of the electrons are located at points (ξi, ηi, −L− ζi). Assuming that the mirror is grounded, Φ(x, y, 0) = 0, the potential at any point can be written as Φ(x, y, z) = Φ0(x, y, z) + ΦI(x, y, z), where Φ0 is the potential of the charges in the absence of the mirror and ΦI is the potential of the image charges introduced to maintain the mirror potential at ground. We have ΦI(x, y, z) = − Z|e| 4π 0 1 √ x2 + y2 + (z + L)2 + ∑ i |e| 4π 0 1 √ (x− ξi) + (y − ηi) + (z + L+ ζi) . (1) The potential energy U = 1 2 ∑