Analytical expression for an atomic lattice "muffin-tin" potential for purposes of illustration and simple scattering calculations

Question

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?

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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.

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Long term goal for background only:

• Overview of how self-consistent dynamical low-energy electron diffraction simulations are performed
• Have Finite Difference Time Domain methods made inroads into dynamical simulation of electron and/or X-ray scattering by crystals?
• Simulated low energy electron diffraction (LEED) patterns

Answer 1

The Augmented Plane Wave (APW) method, and by extension Linearly-Augmented Plane Wave method are both generalizations of the Muffin Tin Approximation.

In both the APW and LAPW methods, the potential $V(r)$ is defined as a piecewise function [1] with a single parameter: the muffin-tin radius $r_\mathrm{MT}$. $$ V(r) = % \begin{cases} \sum_{lm} V_{lm} (r) Y_{lm} (\hat{r}) & r < r_\mathrm{MT} & (\mathrm{core}) \\ V_K e^{i K r} & r > r_\mathrm{MT} & (\mathrm{interstitial}) \end{cases}$$

The values of the potential $V(r)$, the wavefunction $\phi(r)$, and the electronic density $\rho(r)$ are matched at $r = r_\mathrm{MT}$ to ensure that the derivative exists for each of them.

The following illustration is from Singh & Nordstrom (2006) [2],

On solving the non-relativistic Schrödinger equation, the same book remarks the following in ch. 5, p. 63.

These differential equations [the radial Schrödinger equation] may be solved on the radial mesh using standard, e.g. predictor-corrector methods.

On matching the two piecewise parts (ch. 4, p. 44):

Noting that from Schrödinger's equation, $$ (E_2 - E_1) ~ r ~ u_1 (r) ~ u_2 (r) = u_2 (r) ~ \frac{ \mathrm{d}^2 ~ r ~ u_1(r) }{\mathrm{d}r^2} - u_1 (r) ~ \frac{ \mathrm{d}^2 ~ r ~ u_2(r) }{\mathrm{d}r^2} $$ where $u_1 (r)$ and $u_2 (r)$ are radial solutions at different energies $E_1$ and $E_2$. The overlap is constructed using this relation and integrating by parts; the surface terms vanish if either $u_1 (r)$ or $u_2 (r)$ vanish on the sphere boundary, while the other terms cancel.

Anyway, I personally don't think solving the radial Schrödinger equation is too computationally expensive, given the current state of computers. But if you want to avoid it at all costs, there is the Kronig-Penney model, which is a lot simpler at the expense of accuracy.

References:

[1] "The Full Potential APW methods", http://susi.theochem.tuwien.ac.at/lapw/index.html

[2] Singh & Nordstrom (2006), Planewaves, Pseudopotentials, and the LAPW Method, 2nd Edition, Springer. SpringerLink