Before taking the deep dive (see linked questions below) into calculating the diffraction of 20 to 200 eV electrons from crystal surfaces, I'd like to generate a simple "muffin-tin potential" (see below) from some simple analytical approximation that roughly matches what might be calculated as the electrostatic potential that an incident electron would feel passing through a mid-sized atom (hydrogen << atom << uranium) arranged in a crystal.
I can start to learn how to calculate phase shifts and angular distributions with this.
Wikipedia's Muffin-tin approximation talks about this but doesn't offer any equations out-of-hand.
The zeroth order approximation would be a nuclear positive point charge and a uniform sphere of negative charge and I can certainly start with that; with vague uniformity argument based on the exclusion principle. A flat "inner potential" of 5 to 15 eV is often assumed between atoms in this context. At small distances it would have to be flattened since near the nucleus it goes to infinity.
Question: But is there a somewhat better approximation than that available?
Cross-section through a "one-muffin tin" made from a uniform $r = 1$ electron sphere and a point nucleus, arbitrarily flattened at the bottom. These would be arranged in space at each atom's location and a constant potential would fill the space between them.
Long term goal for background only: