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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?


uniform electron sphere with point nucleus

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:

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    $\begingroup$ A very detailed question as always by uhoh. +1 and I hope that someone here can answer it! We really need to get more diffraction people on this site. $\endgroup$ – Nike Dattani Dec 3 '20 at 4:04
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    $\begingroup$ Joly (2001) and Zwierzycki and Andersen (2008) might be useful? $\endgroup$ – TheSimpliFire Dec 3 '20 at 8:42
  • $\begingroup$ @TheSimpliFire that's interesting, thanks! It seems to be a fitting algorithm where you start with a full-blown calculation and then fit some parameters to it. While it might be possible to figure out what the functional forms of $f_i$ look like it offers no guidance on how to choose them other than to fit them to the results of a calculation. I'm hoping to find something that's plug-and-play; given $Z$ and $R$ only, provide a rough $u(r)$. $\endgroup$ – uhoh Dec 3 '20 at 9:59
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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], LAPW basis functions

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

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  • $\begingroup$ Thank you for your answer! Is it possible to point to a specific expression that I can use? For example if I have a lattice of atoms with atomic number $Z$ and lattice constant $a$, how can I make a plot of the potential $V(\mathbf{r})$? The first link shows an expansion in the form of an infinite series of radial functions and spherical harmonics, and suggests I simply solve Schroedinger's equation to find the wave functions of all of the electrons in the atom to derive the potential's coefficients. $\endgroup$ – uhoh Dec 3 '20 at 23:14
  • $\begingroup$ Completely solving an atom's electronic structure is a nontrivial task! Are there no simpler approximations than a complete solution? Do I misunderstand the 2nd half of the first link and there is some algorithm that doesn't require solving Schroedinger's equation for each electron first? I'll try to get my hands on a copy of Singh & Nordström, thanks! $\endgroup$ – uhoh Dec 3 '20 at 23:24
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    $\begingroup$ @uhoh, if you have access to APS journals, you can read Slater's original paper from 1937 describing the APW method, Phys. Rev. 51, 846 . Another paper that elaborates on this method further is Elyashar & Koelling (1976), Phys. Rev. B 13, 5362. $\endgroup$ – wyphan Dec 4 '20 at 22:11
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    $\begingroup$ If your library has access to SpringerLink, you'll want to checkout Springer's MyCopy service. Basically they will print you a Springer book for personal use, for the low price of $25 only! $\endgroup$ – wyphan Dec 4 '20 at 22:22
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    $\begingroup$ In case the system detects that a "conversation" is going on in comments, and recommends to create a new chat room, I suggest not to click that button since eventually we'll have thousands of frozen rooms that are specific to just one question each: meta.stackexchange.com/q/353643/391772. Instead I might recommend for @wyphan and uhoh to chat here: chat.stackexchange.com/rooms/112878/gpaw since that's a room for PAW calculations, which is related to APW or LAPW. $\endgroup$ – Nike Dattani Dec 5 '20 at 1:31

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